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  • 1.
    Abdullah Al Ahad, Muhammed
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Non-linearstates in parallel Blasius boundary layer2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    There is large theoretical, experimental and numerical interest in studying boundary layers, which develop around any body moving through a fluid. The simplest of these boundary layers lead to the theoretical abstraction of a so-called Blasius boundary layer, which can be derived under the assumption of a flat plate and zero external pressure gradient. The Blasius solution is characterised by a slow growth of the boundary layer in the streamwise direction. For practical purposes, in particular related to studying transition scenarios, non-linear finite-amplitude states (exact coherent states, edge states), but also for turbulence, a major simplification of the problem could be attained by removing this slow streamwise growth, and instead consider a parallel boundary layer. Parallel boundary layers are found in reality, e.g. when applying suction (asymptotic suction boundary layer) or rotation (Ekman boundary layer), but not in the Blasius case. As this is only a model which is not an exact solution to the Navier-Stokes (or boundary-layer) equations, some modifications have to be introduced into the governing equations in order for such an approach to be feasible. Spalart and Yang introduced a modification term to the governing Navier-Stokes equations in 1987. In this thesis work, we adapted the amplitude of the modification term introduced by Spalart and Yang to identify the nonlinear states in the parallel Blasius boundary layer. A final application of this modification was in determining the so-called edge states for boundary layers, previously found in the asymptotic suction boundary layer

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  • 2. Abdulle, A.
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Multiscale Methods for Wave Problems in Heterogeneous Media2017In: Handbook of Numerical Analysis, Elsevier B.V. , 2017, p. 545-576Conference paper (Refereed)
    Abstract [en]

    In this chapter we give a survey on various multiscale methods for the numerical solution of second-order hyperbolic equations in highly heterogeneous media. We concentrate on the wave equation and distinguish between two classes of applications. First we discuss numerical methods for the wave equation in heterogeneous media without scale separation. Such a setting is for instance encountered in the geosciences, where natural structures often exhibit a continuum of different scales, that all need to be resolved numerically to get meaningful approximations. Approaches tailored for these settings typically involve the construction of generalized finite element spaces, where the basis functions incorporate information about the data variations. In the second part of the chapter, we discuss numerical methods for the case of structured media with scale separation. This setting is for instance encountered in engineering sciences, where materials are often artificially designed. If this is the case, the structure and the scale separation can be explicitly exploited to compute appropriate homogenized/upscaled wave models that only exhibit a single coarse scale and that can be hence solved at significantly reduced computational costs. 

  • 3. Abdulle, Assyr
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Localized orthogonal decomposition method for the wave equation with a continuum of scales2017In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 304, p. 549-587Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

  • 4.
    Adler, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    GPU Monte Carlo scatter calculations for Cone Beam Computed Tomography2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    A GPU Monte Carlo code for x-ray photon transport has been implemented and extensively tested. The code is intended for scatter compensation of cone beam computed tomography images.

    The code was tested to agree with other well known codes within 5% for a set of simple scenarios. The scatter compensation was also tested using an artificial head phantom. The errors in the reconstructed Hounsfield values were reduced by approximately 70%.

    Several variance reduction methods have been tested, although most were found infeasible on GPUs. The code is nonetheless fast, and can simulate approximately 3 ·109 photons per minute on a NVIDIA Quadro 4000 graphics card. With the use of appropriate filtering methods, the code can be used to calculate patient specific scatter distributions for a full CBCT scan in approximately one minute, allowing scatter reduction in clinical applications.

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  • 5.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Computational methods for microfluidics2013Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

    The first paper of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an explicit interface tracking method, wherein the interface between two fluids is decomposed into segments, which are represented locally on an Eulerian grid. The framework of this method provides a natural setting for solving the advection-diffusion equation governing the surfactant concentration on the interface. Open interfaces and moving contact lines are also incorporated into the method in a natural way, though we show that care must be taken when regularizing interface forces to the grid near the boundary of the computational domain.

    In the second paper we present a boundary integral formulation for sedimenting particles in periodic Stokes flow, using the completed double layer boundary integral formulation. The long-range nature of the particle-particle interactions lead to the formulation containing sums which are not absolutely convergent if computed directly. This is solved by applying the method of Ewald summation, which in turn is computed in a fast manner by using the FFT-based spectral Ewald method. The complexity of the resulting method is O(N log N), as the system size is scaled up with the number of discretization points N. We apply the method to systems of sedimenting spheroids, which are discretized using the Nyström method and a basic quadrature rule.

    The Ewald summation method used in the boundary integral method of the second paper requires a decomposition of the potential being summed. In the introductory chapters of the thesis we present an overview of the available methods for creating Ewald decompositions, and show how the methods and decompositions can be related to each other.

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  • 6.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Ewald summation for the rotlet singularity of Stokes flow2016Report (Other academic)
    Abstract [en]

    Ewald summation is an efficient method for computing the periodic sums that appear when considering the Green's functions of Stokes flow together with periodic boundary conditions. We show how Ewald summation, and accompanying truncation error estimates, can be easily derived for the rotlet, by considering it as a superposition of electrostatic force calculations.

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  • 7.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast and accurate integral equation methods with applications in microfluidics2016Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

    The first, and smaller, part of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an interface decomposition resulting in local, Eulerian grid representations. This provides a natural setting for solving the PDE governing the surfactant concentration on the interface.

    The second, and larger, part of the thesis is concerned with a framework for simulating large systems of rigid particles in three-dimensional, periodic viscous flow using a boundary integral formulation. This framework can solve the underlying flow equations to high accuracy, due to the accurate nature of surface quadrature. It is also fast, due to the natural coupling between boundary integral methods and fast summation methods.

    The development of the boundary integral framework spans several different fields of numerical analysis. For fast computations of large systems, a fast Ewald summation method known as Spectral Ewald is adapted to work with the Stokes double layer potential. For accurate numerical integration, a method known as Quadrature by Expansion is developed for this same potential, and also accelerated through a scheme based on geometrical symmetries. To better understand the errors accompanying this quadrature method, an error analysis based on contour integration and calculus of residues is carried out, resulting in highly accurate error estimates.

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  • 8.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindbo, Dag
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant2014In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, p. 50-63Article in journal (Refereed)
    Abstract [en]

    The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

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  • 9.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Saffar Shamshirgar, Davoud
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for free-space Stokes potentials2017In: Research in the Mathematical Sciences, ISSN 2197-9847, Vol. 4, no 1Article in journal (Refereed)
    Abstract [en]

    We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23):8994–9010, 2010. doi: 10.1016/j.jcp.2010.08.026 ), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191–203, 2016. doi: 10.1016/j.jcp.2016.07.028 ). Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of $$O(N \log N)$$ O ( N log N ) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive.

  • 10.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Sorgentone, Chiara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions2022In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 111, p. 1-19Article in journal (Refereed)
    Abstract [en]

    The quadrature error associated with a regular quadrature rule for evaluation of a layer potential increases rapidly when the evaluation point approaches the surface and the integral becomes nearly singular. Error estimates are needed to determine when the accuracy is insufficient and a more costly special quadrature method should be utilized.& nbsp;The final result of this paper are such quadrature error estimates for the composite Gauss-Legendre rule and the global trapezoidal rule, when applied to evaluate layer potentials defined over smooth curved surfaces in R-3. The estimates have no unknown coefficients and can be efficiently evaluated given the discretization of the surface, invoking a local one-dimensional root-finding procedure. They are derived starting with integrals over curves, using complex analysis involving contour integrals, residue calculus and branch cuts. By complexifying the parameter plane, the theory can be used to derive estimates also for curves in R3. These results are then used in the derivation of the estimates for integrals over surfaces. In this procedure, we also obtain error estimates for layer potentials evaluated over curves in R2. Such estimates combined with a local root-finding procedure for their evaluation were earlier derived for the composite Gauss-Legendre rule for layer potentials written in complex form [4]. This is here extended to provide quadrature error estimates for both complex and real formulations of layer potentials, both for the Gauss-Legendre and the trapezoidal rule.& nbsp;Numerical examples are given to illustrate the performance of the quadrature error estimates. The estimates for integration over curves are in many cases remarkably precise, and the estimates for curved surfaces in R-3 are also sufficiently precise, with sufficiently low computational cost, to be practically useful.

  • 11.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A fast integral equation method for solid particles in viscous flow using quadrature by expansionManuscript (preprint) (Other academic)
    Abstract [en]

    Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

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  • 12.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    A fast integral equation method for solid particles in viscous flow using quadrature by expansion2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 326, p. 420-445Article in journal (Refereed)
    Abstract [en]

    Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

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  • 13.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions2018In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 3, p. A1225-A1249Article in journal (Refereed)
    Abstract [en]

    When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX—adaptive quadrature by expansion—which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss–Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard.

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  • 14.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Error estimation for quadrature by expansion in layer potential evaluation2017In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 43, no 1, p. 195-234Article in journal (Refereed)
    Abstract [en]

    In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

  • 15.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Estimation of quadrature errors in layer potential evaluation using quadrature by expansionManuscript (preprint) (Other academic)
    Abstract [en]

    In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

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  • 16.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for Stokesian particle suspensions2014In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 76, no 10, p. 669-698Article in journal (Refereed)
    Abstract [en]

    We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to O(N log N), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. 

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  • 17.
    Alathur Srinivasan, Prem Anand
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Deep Learning models for turbulent shear flow2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Deep neural networks trained with spatio-temporal evolution of a dynamical system may be regarded as an empirical alternative to conventional models using differential equations. In this thesis, such deep learning models are constructed for the problem of turbulent shear flow. However, as a first step, this modeling is restricted to a simplified low-dimensional representation of turbulence physics. The training datasets for the neural networks are obtained from a 9-dimensional model using Fourier modes proposed by Moehlis, Faisst, and Eckhardt [29] for sinusoidal shear flow. These modes were appropriately chosen to capture the turbulent structures in the near-wall region. The time series of the amplitudes of these modes fully describe the evolution of flow. Trained deep learning models are employed to predict these time series based on a short input seed. Two fundamentally different neural network architectures, namely multilayer perceptrons (MLP) and long short-term memory (LSTM) networks are quantitatively compared in this work. The assessment of these architectures is based on (i) the goodness of fit of their predictions to that of the 9-dimensional model, (ii) the ability of the predictions to capture the near-wall turbulence structures, and (iii) the statistical consistency of the predictions with the test data. LSTMs are observed to make predictions with an error that is around 4 orders of magnitude lower than that of the MLP. Furthermore, the flow fields constructed from the LSTM predictions are remarkably accurate in their statistical behavior. In particular, deviations of 0:45 % and 2:49 % between the true data and the LSTM predictions were obtained for the mean flow and the streamwise velocity fluctuations, respectively.

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  • 18.
    Alexei, Iupinov
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Implementation of the Particle Mesh Ewald method on a GPU2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The Particle Mesh Ewald (PME) method is used for efficient long-range electrostatic calculations in molecular dynamics (MD).

    In this project, PME is implemented for a single GPU alongside the existing CPU implementation, using the code base of an open source MD software GROMACS and NVIDIA CUDA toolkit. The performance of the PME GPU implementation is then studied.

    The motivation for the project is examining the PME algorithm’s parallelism, and its potential benefit for performance scalability of MD simulations on various hardware.

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  • 19.
    Altmann, Robert
    et al.
    Univ Augsburg, Dept Math, Univ Str 14, D-86159 Augsburg, Germany..
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Ruhr Univ Bochum, Fac Math, D-44801 Bochum, Germany..
    Peterseim, Daniel
    Univ Augsburg, Dept Math, Univ Str 14, D-86159 Augsburg, Germany..
    Localization And Delocalization Of Ground States Of Bose-Einstein Condensates Under Disorder2022In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 82, no 1, p. 330-358Article in journal (Refereed)
    Abstract [en]

    This paper studies the localization behavior of Bose-Einstein condensates in disorder potentials, modeled by a Gross-Pitaevskii eigenvalue problem on a bounded interval. In the regime of weak particle interaction, we are able to quantify exponential localization of the ground state, depending on statistical parameters and the strength of the potential. Numerical studies further show delocalization if we leave the identified parameter range, which is in agreement with experimental data. These mathematical and numerical findings allow the prediction of physically relevant regimes where localization of ground states may be observed experimentally.

  • 20.
    Altmann, Robert
    et al.
    Univ Augsburg, Inst Math, D-86159 Augsburg, Germany..
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Ruhr Univ Bochum, Fak Math, D-44801 Bochum, Germany..
    Peterseim, Daniel
    Univ Augsburg, Inst Math, D-86159 Augsburg, Germany..
    Numerical homogenization beyond scale separation2021In: Acta Numerica, ISSN 0962-4929, E-ISSN 1474-0508, Vol. 30, p. 1-86, article id PII S0962492921000015Article in journal (Refereed)
    Abstract [en]

    Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

  • 21.
    Altmann, Robert
    et al.
    Univ Augsburg, Dept Math, Univ Str 14, D-86159 Augsburg, Germany..
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Peterseim, Daniel
    Univ Augsburg, Dept Math, Univ Str 14, D-86159 Augsburg, Germany..
    Quantitative Anderson localization of Schrodinger eigenstates under disorder potentials2020In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 30, no 5, p. 917-955Article in journal (Refereed)
    Abstract [en]

    This paper analyzes spectral properties of linear Schrodinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps among the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local operator preconditioning by domain decomposition.

  • 22.
    Altmann, Robert
    et al.
    Univ Augsburg, Dept Math, Univ Str 14, D-86159 Augsburg, Germany..
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Ruhr Univ Bochum, Dept Math, D-44801 Bochum, Germany..
    Peterseim, Daniel
    Univ Augsburg, Dept Math, Univ Str 14, D-86159 Augsburg, Germany..
    The J-method for the Gross-Pitaevskii eigenvalue problem2021In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 148, no 3, p. 575-610Article in journal (Refereed)
    Abstract [en]

    This paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross-Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the J-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the J-method's unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.

  • 23.
    Andersson, Magnus
    et al.
    KTH, School of Engineering Sciences (SCI), Applied Physics, Materials and Nanophysics.
    Karlander, Johan
    KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Theoretical Computer Science, TCS.
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tibert, Gunnar
    KTH, School of Engineering Sciences (SCI), Engineering Mechanics, Vehicle Engineering and Solid Mechanics.
    Admission to master programmes: What are the indicators for successful study performance?2023In: Bidrag från den 9:e utvecklingskonferensen för Sveriges ingenjörsutbildningar / [ed] Joel Midemalm, Amir Vadiee, Elisabeth Uhlemann, Fredrik Georgsson, Gunilla Carlsson-Kvarnlöf, Jonas Månsson, Kristina Edström, Lennart Pettersson och Pedher Johansson, Västerås: Mälardalens universitet, 2023, p. 9-18Conference paper (Refereed)
    Abstract [en]

    Admission of applicants to higher education in a fair, reliable, transparent, and efficient way is a real challenge, especially if there are more eligible applicants than available places and if there are applicants from many different educational systems. Previous research on best practices for admission to master’s programmes identified the key question about an applicant’s potential for success in studies, but was not able to provide an answer about how to rate the merits of the applicants. In this study, indicators for study success are analysed by comparing the study performance of 228 students in master’s programmes with their merits at the time of admission. The null hypothesis was that the applicant’s average grade at the time of admission is the only indictor for study success. After testing for potential bias using almost 20 possible other indicators, the null hypothesis had to be rejected for four indicators (in order of importance): (i) university ranking, (ii) length of bachelor’s studies within subject, (iii) English language test and (iv) subject matching between bachelor’s and master’s education. Evaluation of quality of prior education is tricky and results from this study clearly indicate that students from higher ranked universities possess better knowledge and stronger skills for our master’s programmes. Work is ongoing to improve the merit rating model by involving more master’s programmes at KTH and analysing performance data from a larger number of students.

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    Master-admission
  • 24.
    Appelo, Daniel
    et al.
    Univ Colorado, Dept Appl Math, Boulder, CO 80309 USA..
    Garcia, Fortino
    Univ Colorado, Dept Appl Math, Boulder, CO 80309 USA..
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Waveholtz: Iterative solution of the helmholtz equation via the wave equation2020In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 42, no 4, p. A1950-A1983Article in journal (Refereed)
    Abstract [en]

    A new iterative method, the WaveHoltz iteration, for solution of the Helmholtz equation is presented. WaveHoltz is a fixed point iteration that filters the solution to the solution of a wave equation with time periodic forcing and boundary data. The WaveHoltz iteration corresponds to a linear and coercive operator which, after discretization, can be recast as a positive definite linear system of equations. The solution to this system of equations approximates the Helmholtz solution and can be accelerated by Krylov subspace techniques. Analysis of the continuous and discrete cases is presented, as are numerical experiments.

  • 25. Appelö, D.
    et al.
    Garcia, F.
    Alvarez Loya, A.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    El-WaveHoltz: A time-domain iterative solver for time-harmonic elastic waves2022In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 401, article id 115603Article in journal (Refereed)
    Abstract [en]

    We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that filters the solution of the wave equation with time-harmonic forcing and boundary data. As in the original WaveHoltz method, we reformulate the fixed point iteration as a positive definite linear system of equations that is iteratively solved by a Krylov method. We present two time-stepping schemes, one explicit and one (novel) implicit, which completely remove time discretization error from the WaveHoltz solution by performing a simple modification of the initial data and time-stepping scheme. Numerical experiments indicate an iteration scaling similar to that of the original WaveHoltz method, and that the convergence rate is dictated by the shortest (shear) wave speed of the problem. We additionally show that the implicit scheme can be advantageous in practice for meshes with disparate element sizes.

  • 26.
    Appelö, Daniel
    et al.
    University of Colorado, Boulder.
    Garcia, Fortino
    University of Colorado, Boulder.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    WaveHoltz: Parallel and scalable solution of the Helmholtz equation via wave equation iteration2020In: SEG International Exposition and Annual Meeting 2019, Society of Exploration Geophysicists , 2020, p. 1541-1545Conference paper (Refereed)
    Abstract [en]

    We introduce a novel idea, the WaveHoltz iteration, for solving the Helmholtz equation inspired by recent work on exact controllability (EC) methods. As in EC methods our method make use of time domain methods for wave equations to design frequency domain Helmholtz solvers but unlike EC methods we do not require adjoint solves. We show that the WaveHoltz iteration we propose is symmetric and positive definite in the continuous setting. We also present numerical examples, using various discretization techniques, that show that our method can be used to solve problems with rather high wave numbers. 

  • 27. Arakelyan, A.
    et al.
    Shah Gholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Multi-Phase Quadrature Domains and a Related Minimization Problem2016In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, p. 1-21Article in journal (Refereed)
    Abstract [en]

    In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers to a generalized type of mean value property for harmonic functions. The well-established and developed theory of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of the multi-phase version of the problem, and prove existence as well as several properties of such solutions. In particular, we discuss possibilities of multi-junction points.

  • 28. Ariel, G.
    et al.
    Kim, S. J.
    Tsai, Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Parareal multiscale methods for highly oscillatory dynamical systems2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 6, p. A3540-A3564Article in journal (Refereed)
    Abstract [en]

    We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solutions, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.

  • 29.
    Arjmand, Doghonay
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations2015Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.

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    Thesis
  • 30.
    Arjmand, Doghonay
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations2013Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers.

    The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q  + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.     

    In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities.

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    Thesis_Intro
  • 31. Arjmand, Doghonay
    et al.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A time dependent approach for removing the cell boundary error in elliptic homogenization problems2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 314, p. 206-227Article in journal (Refereed)
    Abstract [en]

    This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(epsilon/eta) error in the computation, where epsilon is the size of the microscopic variations in the media and eta is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of epsilon/eta in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O(epsilon/eta) error to O(epsilon) in general non-periodic media.

  • 32.
    Arjmand, Doghonay
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Analysis of heterogeneous multiscale methods for long time wave propagation problems2014In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 12, no 3, p. 1135-1166Article in journal (Refereed)
    Abstract [en]

    In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time O(epsilon(-2)) wave propagation, where e represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale wave equations exhibit O(1) dispersive effects which are not observed in short time scales. A typical HMM has two main components: a macromodel and a micromodel. The macromodel is incomplete and lacks a set of local data. In the setting of multiscale PDEs, one has to solve for the full oscillatory problem over local microscopic domains of size eta = O(epsilon) to upscale the parameter values which are missing in the macroscopic model. In this paper, we prove that if the microproblems are consistent with the macroscopic solutions, the HMM approximates the unknown parameter values in the macromodel up to any desired order of accuracy in terms of epsilon/eta..

  • 33.
    Arjmand, Doghonay
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Analysis of HMM for Long Time Multiscale Wave Propagation Problems in Locally-Periodic MediaManuscript (preprint) (Other academic)
    Abstract [en]

    Multiscale wave propagation problems are difficult to solve numerically due to the interaction of different scales inherent in the problem. Extracting information about the average behaviour of the system requires resolving small scales in the problem. This leads to a tremendous computational burden if the size of microscopic variations are much smaller than the size of scales of interest. Heterogeneous multiscale methods (HMM) is a tool to avoid resolving the small scales everywhere. Nevertheless, it approximates the average part of the solution by upscaling the microscopic information on a small part of the domain. This leads to a substantial improvement in the computational cost. In this article, we analyze an HMM-based numerical method which approximates the long time behaviour of multiscale wave equations. In particular, we consider theoretically challenging case of locally-periodic media where fast and slow variations are allowed at the same time. We are interested in the long time regime (T=O(e^{-1})), where e represents the wavelength of the fast variations in themedia. We first use asymptotic expansions to derive effective equations describing the long time effects of the multiscale waves in multi-dimensional locally-periodic media. We then show that HMM captures these non-trivial long time eects. All the theoretical statements are general in terms of dimension. Two dimensional numericale xamples are considered to support our theoretical arguments

  • 34. Arjmand, Doghonay
    et al.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media2017In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 15, no 2, p. 948-976Article in journal (Refereed)
    Abstract [en]

    This paper concerns the analysis of a multiscale method for wave propagation problems in microscopically nonhomogeneous media. A direct numerical approximation of such problems is prohibitively expensive as it requires resolving the microscopic variations over a much larger physical domain of interest. The heterogeneous multiscale method (HMM) is an efficient framework to approximate the solutions of multiscale problems. In the HMM, one assumes an incomplete macroscopic model which is coupled to a known but expensive microscopic model. The micromodel is solved only locally to upscale the parameter values which are missing in the macro model. The resulting macroscopic model can then be solved at a cost independent of the small scales in the problem. In general, the accuracy of the HMM is related to how good the upscaling step approximates the right macroscopic quantities. The analysis of the method that we consider here was previously addressed only in purely periodic media, although the method itself is numerically shown to be applicable to more general settings. In the present study, we consider a more realistic setting by assuming a locally periodic medium where slow and fast variations are allowed at the same time. We then prove that the HMM captures the right macroscopic effects. The generality of the tools and ideas in the analysis allows us to establish convergence rates in a multidimensional setting. The theoretical findings here imply an improved convergence rate in one dimension, which also justifies the numerical observations from our earlier study.

  • 35.
    Arjmand, Doghonay
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Stohrer, Christian
    ENSTA ParisTech.
    A Finite Element Heterogenous Multiscale Method with Improved Control Over the Modeling ErrorManuscript (preprint) (Other academic)
    Abstract [en]

    Multiscale partial dierential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic,and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macromodel. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating by the coupling between the dierent scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well

  • 36.
    Bagge, Joar
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Accurate quadrature and fast summation in boundary integral methods for Stokes flow2023Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis concerns accurate and efficient numerical methods for the simulation of fluid flow on the microscale, known as Stokes flow or creeping flow. Such flows are important, for example, in understanding the swimming of microorganisms, spreading of dust particles, as well as in developing new nano-materials, and microfluidic devices that can be used for on-the-fly analysis of blood samples, among other things.

    Flow on the microscale is dominated by viscous forces, meaning that a fluid such as water or air will behave as a very viscous fluid, like e.g. honey. The equations governing the flow, known as the Stokes equations, are linear PDEs, which permits the use of boundary integral methods (BIMs). In these methods, the PDE is reformulated as a boundary integral equation, thus reducing the dimensionality of the computational problem from three dimensions to two dimensions. The boundary integral formulation is well-conditioned, so that high accuracy can be achieved.

    We consider two main challenges related to BIMs. The first challenge is that the integrals in the formulation contain integrands that vary rapidly for evaluation points close to the boundary, and cannot be accurately resolved using a standard method for numerical integration. Therefore, special quadrature methods are needed. We consider two such methods: quadrature by expansion (QBX) and the “line extrapolation/interpolation method” (also known as the Hedgehog method). In particular, we consider these methods applied to simulations involving rigid rodlike particles and surrounding walls.

    The second challenge is that discretizing the boundary integral formulation leads to a dense linear system, which requires O(N2) operations to solve iteratively, where N is the number of unknowns. This becomes too expensive for large systems. A fast summation method, such as the Spectral Ewald (SE) method considered in this thesis, reduces the number of operations required, for example to O(N log N). The SE method can also be used for problems with periodic boundary conditions in any number of the spatial directions (arbitrary periodicity).

    We also consider an application of these methods to a flow problem involving an inertial spheroid in a parabolic flow profile, and analyze the lateral drift of this spheroidal particle.

    The numerical methods studied in this thesis enable fast and accurate computer simulations of e.g. suspensions of rigid particles in three-dimensional Stokes flow, including surrounding walls and arbitrary periodicity.

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    fulltext
  • 37.
    Bagge, Joar
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Numerical simulation of an inertial spheroidal particle in Stokes flow2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Particle suspensions occur in many situations in nature and industry. In this master’s thesis, the motion of a single rigid spheroidal particle immersed in Stokes flow is studied numerically using a boundary integral method and a new specialized quadrature method known as quadrature by expansion (QBX). This method allows the spheroid to be massless or inertial, and placed in any kind of underlying Stokesian flow.

     

    A parameter study of the QBX method is presented, together with validation cases for spheroids in linear shear flow and quadratic flow. The QBX method is able to compute the force and torque on the spheroid as well as the resulting rigid body motion with small errors in a short time, typically less than one second per time step on a regular desktop computer. Novel results are presented for the motion of an inertial spheroid in quadratic flow, where in contrast to linear shear flow the shear rate is not constant. It is found that particle inertia induces a translational drift towards regions in the fluid with higher shear rate.

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    fulltext
  • 38.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Rosén, Tomas
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences in Chemistry, Biotechnology and Health (CBH), Centres, Wallenberg Wood Science Center. KTH, School of Engineering Sciences (SCI), Engineering Mechanics.
    Lundell, Fredrik
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences in Chemistry, Biotechnology and Health (CBH), Centres, Wallenberg Wood Science Center. KTH, School of Engineering Sciences (SCI), Engineering Mechanics.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Parabolic velocity profile causes shape-selective drift of inertial ellipsoids2021In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 926, article id A24Article in journal (Refereed)
    Abstract [en]

    Understanding particle drift in suspension flows is of the highest importance in numerous engineering applications where particles need to be separated and filtered out from the suspending fluid. Commonly known drift mechanisms such as the Magnus force, Saffman force and Segre-Silberberg effect all arise only due to inertia of the fluid, with similar effects on all non-spherical particle shapes. In this work, we present a new shape-selective lateral drift mechanism, arising from particle inertia rather than fluid inertia, for ellipsoidal particles in a parabolic velocity profile. We show that the new drift is caused by an intermittent tumbling rotational motion in the local shear flow together with translational inertia of the particle, while rotational inertia is negligible. We find that the drift is maximal when particle inertial forces are of approximately the same order of magnitude as viscous forces, and that both extremely light and extremely heavy particles have negligible drift. Furthermore, since tumbling motion is not a stable rotational state for inertial oblate spheroids (nor for spheres), this new drift only applies to prolate spheroids or tri-axial ellipsoids. Finally, the drift is compared with the effect of gravity acting in the directions parallel and normal to the flow. The new drift mechanism is stronger than gravitational effects as long as gravity is less than a critical value. The critical gravity is highest (i.e. the new drift mechanism dominates over gravitationally induced drift mechanisms) when gravity acts parallel to the flow and the particles are small.

  • 39.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Accurate quadrature methods with application to Stokes flow with particles in confined geometries2017In: Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11) / [ed] David J. Chappell, Nottingham: Nottingham Trent University, 2017, p. 15-24Conference paper (Refereed)
    Abstract [en]

    Boundary integral methods are attractive for simulating Stokes flow with particles or droplets due to the reduction in dimensionality and natural handling of the geometry. In many problems walls are present, and it becomes necessary to evaluate singular or nearly singular layer potentials over the wall. In this paper we show how this can be done using quadrature by expansion (QBX), a relatively new method based on local expansions of the layer potential. We present results for the Laplace single layer potential and the Stokes double layer potential. QBX can be used to evaluate the potentials to high accuracy arbitrarily close to the wall and on the wall. We also discuss how some quantities can be precomputed and how geometric symmetries can be used to reduce precomputation and storage.

  • 40.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Accurate quadrature via line extrapolation and rational approximation with application to boundary integral methods for Stokes flow2023Report (Other academic)
    Abstract [en]

    In boundary integral methods, special quadrature methods are needed to approximate layer potentials, integrals where the integrand is singular or sharply peaked for evaluation points on or close to the boundaries. In this paper, we study a method based on extrapolation or interpolation along a line, sometimes called the Hedgehog method. In this method, the layer potential is evaluated with a regular quadrature method for evaluation points along a line, and an approximant is constructed and evaluated in an area of interest where the original layer potential is difficult to evaluate due to it being singular or sharply peaked.

    We analyze the errors in the Hedgehog method with polynomial approximation, and use this to construct optimal distributions of sample points. Furthermore, rational approximation is introduced in the Hedgehog method, and compared with polynomial approximation. It is found that rational approximation can typically achieve a lower error than polynomial approximation, and does not increase the computational cost of the method significantly. Strategies for avoiding and dealing with spurious poles in rational approximation are discussed.

    We compare extrapolation (no sample point on the boundary) with interpolation (sample point present) in the Hedgehog method, and find that the error in our example is lower in the interpolation case by around one order of magnitude, compared to the extrapolation case.

    We consider a specific test case, consisting of two rigid rodlike particles in Stokes flow. Parameter selection and error estimation for the Hedgehog method is discussed for this test case. The accuracy and computational cost of the Hedgehog method is examined, and compared with another special quadrature method, namely quadrature by expansion (QBX). We find that the Hedgehog method should be able to compete with QBX in this context, but further investigation is needed for strict tolerances.

  • 41.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Fast Ewald summation for Stokes flow with arbitrary periodicityManuscript (preprint) (Other academic)
    Abstract [en]

    A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(N log N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

  • 42.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Fast Ewald summation for Stokes flow with arbitrary periodicity2023In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 493, p. 112473-, article id 112473Article in journal (Refereed)
    Abstract [en]

    A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(Nlog⁡N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

  • 43.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries2021In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 93, no 7, p. 2175-2224Article in journal (Refereed)
    Abstract [en]

    Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This article presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centerpiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion, accelerated using a precomputation scheme. The method is demonstrated for rodlike and spheroidal particles, with the confining geometry given by a pipe or a pair of flat walls. A parameter selection strategy for the special quadrature method is presented and tested. Periodic interactions are computed using the spectral Ewald fast summation method, which allows our method to run in O(n log n) time for n grid points in the primary cell, assuming the number of geometrical objects grows while the grid point concentration is kept fixed.

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  • 44.
    Bai, Bing
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A Study of Adaptive Random Features Models in Machine Learning based on Metropolis Sampling2021Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Artificial neural network (ANN) is a machine learning approach where parameters, i.e., frequency parameters and amplitude parameters, are learnt during the training process. Random features model is a special case of ANN that the structure of random features model is as same as ANN’s but the parameters’ learning processes are different. For random features model, the amplitude parameters are learnt during the training process but the frequency parameters are sampled from some distributions. If the frequency distribution of the random features model is well-chosen, both models can approximate data well. Adaptive random Fourier features with Metropolis sampling is an enhanced random Fourier features model which can select appropriate frequency distribution adaptively. This thesis studies Rectified Linear Unit and sigmoid features and combines them with the adaptive idea to generate another two adaptive random features models. The results show that using the particular set of hyper-parameters, adaptive random Rectified Linear Unit features model can also approximate the data relatively well, though the adaptive random Fourier features model performs slightly better.

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  • 45.
    Balmus, Maximilian
    et al.
    Kings Coll London, Kings Hlth Partners, Sch Imaging Sci & Biomed Engn, Dept Biomed Engn, London SE1 7EH, England..
    Massing, Andre
    Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.;Norwegian Univ Sci & Technol, Dept Math Sci, NO-7491 Trondheim, Norway..
    Hoffman, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Razavi, Reza
    Kings Coll London, Kings Hlth Partners, Sch Imaging Sci & Biomed Engn, Dept Biomed Engn, London SE1 7EH, England..
    Nordsletten, David A.
    Kings Coll London, Kings Hlth Partners, Sch Imaging Sci & Biomed Engn, Dept Biomed Engn, London SE1 7EH, England.;Univ Michigan, Dept Biomed Engn & Cardiac Surg, Ann Arbor, MI 48109 USA..
    A partition of unity approach to fluid mechanics and fluid-structure interaction2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112842Article in journal (Refereed)
    Abstract [en]

    For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

  • 46.
    Baloglu, Maximilian Volkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Parallelisation and Performance Analysis of a TreeSPH Code for Galaxy Simulations2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In cosmological simulations, the Lagrangian method Smoothed Particle Hydrodynamics is often applied to cover gas dynamics and combined with tree algorithms for long-range potentials like the Barnes-Hut method to include self-gravity and derive the nearest neighbour lists efficiently. In this thesis, a so-called TreeSPH code is parallelized by using MPI and subsequently the performance is analysed. For the domain decomposition to the processes, the structure of an octree is examined and space filling curves are applied to achieve well-working dynamical load balancing. For an efficient parallel SPH calculation, a novel method with a localised boundary handling is proposed to reduce communication overhead

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  • 47. Bartuschat, D.
    et al.
    Fischermeier, E.
    Gustavsson, Katarina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Rüde, U.
    Two computational models for simulating the tumbling motion of elongated particles in fluids2016In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 127, p. 17-35Article in journal (Refereed)
    Abstract [en]

    Suspensions with fiber-like particles in the low Reynolds number regime are modeled by two different approaches that both use a Lagrangian representation of individual particles. The first method is the well-established formulation based on Stokes flow that is formulated as integral equations. It uses a slender body approximation for the fibers to represent the interaction between them directly without explicitly computing the flow field. The second is a new technique using the 3D lattice Boltzmann method on parallel supercomputers. Here the flow computation is coupled to a computational model of the dynamics of rigid bodies using fluid-structure interaction techniques. Both methods can be applied to simulate fibers in fluid flow. They are carefully validated and compared against each other, exposing systematically their strengths and weaknesses regarding their accuracy, the computational cost, and possible model extensions.

  • 48.
    Bastian, Peter
    et al.
    University of Heidelberg.
    Berninger, Heiko
    Dedner, Andreas
    University of Warwick.
    Engwer, Christian
    University of Münster.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Kornhuber, Ralf
    FU Berlin.
    Kröner, Dietmar
    University of Freiburg.
    Ohlberger, Mario
    University of Münster.
    Sander, Oliver
    TU Dresden.
    Schiffler, Gerd
    Shokina, Nina
    Smetana, Kathrin
    University of Münster.
    Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management2012In: Progress in Industrial Mathematics at ECMI 2010 / [ed] Michael Günther and Andreas Bartel and Markus Brunk and Sebastian Schöps and Michael Striebel, Springer Berlin/Heidelberg, 2012, p. 561-567Conference paper (Refereed)
    Abstract [en]

    This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated–unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet–Neumann method, using the dune-grid-glue module in the numerics software DUNE.

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  • 49. Bayer, C.
    et al.
    Hoel, Håkon
    Kadir, Ashraful
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Plechác, Petr
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces2015In: Applied Mathematics Research eXpress, ISSN 1687-1200, E-ISSN 1687-1197, no 2, p. 329-417Article in journal (Refereed)
    Abstract [en]

    The difference of the values of observables for the time-independent Schrödinger equation, with matrix-valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states, and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem.

  • 50. Bayer, Christian
    et al.
    Hoel, Håkon
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    von Schwerin, Erik
    Tempone, Raul
    On nonasymptotic optimal stopping criteria in monte carlo simulations2014In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, no 2, p. A869-A885Article in journal (Refereed)
    Abstract [en]

    We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.

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