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  • 1.
    Burman, Erik
    et al.
    UCL, Dept Math, London WC1E 6BT, England..
    Hansbo, Peter
    Jonkoping Univ, Dept Mech Engn, SE-55111 Jonkoping, Sweden..
    Larson, Mats G.
    Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden..
    Massing, Andre
    Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden..
    Zahedi, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A stabilized cut streamline diffusion finite element method for convection-diffusion problems on surfaces2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 358, article id UNSP 112645Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut finite element method for the stationary convection-diffusion problem on a surface embedded in R-d. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

  • 2.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Krylov methods for nonlinear eigenvalue problems and matrix equations2020Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Nonlinear eigenvalue problems (NEPs) arise in many fields of science and engineering. Such problems are often defined by large matrices, which have specific structures, such as being sparse, low-rank, etc. Like the linear eigenvalue problem, the eigenvector appears in a linear form, whereas the eigenvalue appears in a nonlinear form. This feature allows for an extension of several methods, which are originally derived for the linear eigenvalue problem, to the nonlinear case. Among these methods, Krylov algorithms have been successfully extended in various ways. These methods are designed to take advantage of the matrix structures mentioned above. In this thesis, we present two Krylov-based methods for solving NEPs: the tensor infinite Arnoldi (TIAR), with its restarting variant, and infinite Lanczos (ILAN). We illustrate the flexibility of TIAR by adapting it for solving a NEP which comes from the study of waves propagating in periodic mediums. Despite the fact that Krylov methods are, in a sense, globally convergent, the convergence to the targeted eigenvalues, in certain cases, may be slow. When an accurate solution is required, the obtained approximations are refined with methods which have higher convergence order, e.g., Newton-like methods, which are also analyzed in this thesis. In the context of eigenvalue computation, the framework used to analyse Newton methods can be combined with the Keldysh theorem in order to better characterize the convergence factor. We also show that several well-established methods, such as residual inverse iteration and Ruhe’s method of successive linear problems, belong to the class of Newton-like methods. In this spirit, we derive a new quasi-Newton method, which is, in terms of convergence properties, equivalent to residual inverse iteration, but does not require the solution of a nonlinear system per iteration. The mentioned methods are implemented in NEP-PACK, which is a registered Julia package for NEPs that we develop. This package consists of: many state-of-the-art, but also well-established, methods for solving NEPs, a vast problem collection, and types and structures to efficiently represent and do computations with NEPs.Many problems in control theory, and many discretized partial differential equations, can be efficiently solved if formulated as matrix equations. Moreover, matrix equations arise in a very large variety of areas as intrinsic problems. In our framework, for certain applications, solving matrix equations is a part of the process of solving a NEP. In this thesis we derive a preconditioning technique which is applicable to linear systems which can be formulate as generalized Sylvester equation. More precisely, we assume that the matrix equation can be formulated as the sum of a Sylvester operator and another term which can be low-rank approximated. Such linear systems arise, e.g., when solving certain NEPs which come from wave propagation problems.We also derive an algorithm, which consists of applying a Krylov method directly to the the matrix equation rather then to the vectorized linear system, that exploits certain structures in the matrix coefficients.

  • 3.
    Frachon, Thomas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Zahedi, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A cut finite element method for incompressible two-phase Navier–Stokes flows2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 384, p. 77-98Article in journal (Refereed)
    Abstract [en]

    We present a space–time Cut Finite Element Method (CutFEM) for the time-dependent Navier–Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. The numerical method is able to accurately capture the strong discontinuity in the pressure and the weak discontinuity in the velocity field across evolving interfaces without re-meshing processes or regularization of the problem. We combine the strategy proposed in P. Hansbo et al. (2014) [14] for the Stokes equations with a stationary interface and the space–time strategy presented in P. Hansbo et al. (2016) [20]. We also propose a strategy for computing high order approximations of the surface tension force by computing a stabilized mean curvature vector. The presented space–time CutFEM uses a fixed mesh but includes stabilization terms that control the condition number of the resulting system matrix independently of the position of the interface, ensure stability and a convenient implementation of the space–time method based on quadrature in time. Numerical experiments in two and three space dimensions show that the numerical method is able to accurately capture the discontinuities in the pressure and the velocity field across evolving interfaces without requiring the mesh to be conformed to the interface and with good stability properties.

  • 4.
    Martin, Lindsay
    et al.
    Univ Texas Austin, Dept Math, Austin, TX 78712 USA..
    Tsai, Yen-Hsi R.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Univ Texas Austin, Dept Math, Austin, TX 78712 USA.;Univ Texas Austin, Oden Inst Computat Engn & Sci, Austin, TX 78712 USA.
    A multiscale domain decomposition algorithm for boundary value problems for eikonal equations2019In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 17, no 2, p. 620-649Article in journal (Refereed)
    Abstract [en]

    In this paper, we present a new multiscale domain decomposition algorithm for computing solutions of static Eikonal equations. In our new method, the decomposition of the domain does not depend on the slowness function in the Eikonal equation or the boundary conditions. The novelty of our new method is a coupling of coarse grid and fine grid solvers to propagate information along the characteristics of the equation efficiently. The method involves an iterative parareal-like update scheme in order to stabilize the method and speed up convergence. One can view the new method as a general framework where an effective coarse grid solver is computed "on the fly" from coarse and fine grid solutions that are computed in previous iterations. We study the optimal weights used to define the effective coarse grid solver and the stable update scheme via a model problem. To demonstrate the framework, we develop a specific scheme using Cartesian grids and the fast sweeping method for solving Eikonal equations. Numerical examples are given to show the method's effectiveness on Eikonal equations involving a variety of multiscale slowness functions.

  • 5.
    Paul, Seema
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Sustainable development, Environmental science and Engineering, Water and Environmental Engineering.
    Oppelstrup, Jesper
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Thunvik, Roger
    KTH, School of Architecture and the Built Environment (ABE), Sustainable development, Environmental science and Engineering, Water and Environmental Engineering.
    Mango Magero, John
    Makerere University.
    DDumba Walakira, David
    Makerere University.
    Cvetkovic, Vladimir
    KTH, School of Architecture and the Built Environment (ABE), Sustainable development, Environmental science and Engineering, Water and Environmental Engineering.
    Bathymetry Development and Flow Analyses Using Two-Dimensional Numerical Modeling Approach for Lake Victoria2019In: Fluids, ISSN 2311-5521, Vol. 4, no 4, p. 1-21Article in journal (Refereed)
    Abstract [en]

    This study explored two-dimensional (2D) numerical hydrodynamic model simulations of Lake Victoria. Several methods were developed in Matlab to build the lake topography. Old depth soundings taken in smaller parts of the lake were combined with more recent extensive data to produce a smooth topographical model. The lake free surface numerical model in the COMSOL Multiphysics (CM) software was implemented using bathymetry and vertically integrated 2D shallow water equations. Validated by measurements of mean lake water level, the model predicted very low mean flow speeds and was thus close to being linear and time invariant, allowing long-time simulations with low-pass filtered inflow data. An outflow boundary condition allowed an accurate simulation to achieve the lake’s steady state level. The numerical accuracy of the linear measurement of lake water level was excellent.

  • 6.
    Pålsson, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Boundary integral methods for fast and accurate simulation of droplets in two-dimensional Stokes flow2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Accurate simulation of viscous fluid flows with deforming droplets creates a number of challenges. This thesis identifies these principal challenges and develops a numerical methodology to overcome them. Two-dimensional viscosity-dominated fluid flows are exclusively considered in this work. Such flows find many applications, for example, within the large and growing field of microfluidics; accurate numerical simulation is of paramount importance for understanding and exploiting them.

    A boundary integral method is presented which enables the simulation of droplets and solids with a very high fidelity. The novelty of this method is in its ability to accurately handle close interactions of drops, and of drops and solid boundaries, including boundaries with sharp corners. The boundary integral method is coupled with a spectral method to solve a PDE for the time-dependent concentration of surfactants on each of the droplet interfaces. Surfactants are molecules that change the surface tension and are therefore highly influential in the types of flow problems which are considered herein.

    A method’s usefulness is not dictated by accuracy alone. It is also necessary that the proposed method is computationally efficient. To this end, the spectral Ewald method has been adapted and applied. This yields solutions with computational cost O(N log N ), instead of O(N^2), for N source and target points.

    Together, these innovations form a highly accurate, computationally efficient means of dealing with complex flow problems. A theoretical validation procedure has been developed which confirms the accuracy of the method.

  • 7.
    Havtun, Hans
    et al.
    KTH, School of Industrial Engineering and Management (ITM), Energy Technology, Applied Thermodynamics and Refrigeration.
    Wingård, Lars
    KTH, School of Industrial Engineering and Management (ITM), Production Engineering.
    Carlsund, Ninni
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Kann, Viggo
    KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.
    Continuous assessment: Time and effort well spent for students and teachers?2019In: KTH SoTL 2019, 2019Conference paper (Refereed)
  • 8. Engwer, C.
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Målqvist, A.
    Peterseim, D.
    Efficient implementation of the localized orthogonal decomposition method2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 123-153Article in journal (Refereed)
    Abstract [en]

    In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.

  • 9.
    Hanke, Michael
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    März, Roswitha
    Tischendorf, Caren
    Weinmüller, E.
    Wurm, S.
    Least-Squares Collocation for Higher-Index Linear Differential-Algebraic Equations: Estimating the Instability Threshold2019In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 88, no 318, p. 1647-1683Article in journal (Refereed)
    Abstract [en]

    Differential-algebraic equations with higher-index give rise to essentially ill-posed problems. The overdetermined least-squares collocation for differential-algebraic equations which has been proposed recently is not much more computationally expensive than standard collocation methods for ordinary differential equations. This approach has displayed impressive convergence properties in numerical experiments, however, theoretically, till now convergence could be established merely for regular linear differential-algebraic equations with constant coefficients. We present now an estimate of the instability threshold which serves as the basic key for proving convergence for general regular linear differential-algebraic equations.

  • 10.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Wärnegård, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    NUMERICAL COMPARISON OF MASS-CONSERVATIVE SCHEMES FOR THE GROSS-PITAEVSKII EQUATION2019In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 12, no 6, p. 1247-1271Article in journal (Refereed)
    Abstract [en]

    In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM,42(3):934-952,2004).

  • 11.
    Wendt, Fabian
    et al.
    NREL, 15013 Denver West Pkwy, Golden, CO 80401 USA..
    Nielsen, Kim
    Ramboll Grp AS, Hannemanns Alle 53, DK-2300 Copenhagen S, Denmark.;Aalborg Univ, Dept Civil Engn, Thomas Mann Vej 23, DK-9220 Aalborg O, Denmark..
    Hoffman, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Jansson, Johan
    KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST).
    Leoni, Massimiliano
    KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST).
    Yasutaka, Imai
    Saga Univ, Inst Ocean Energy, Honjo 1, Saga 8408502, Japan..
    Ocean Energy Systems Wave Energy Modelling Task: Modelling, Verification and Validation of Wave Energy Converters2019In: Journal of Marine Science and Engineering, E-ISSN 2077-1312, Vol. 7, no 11, article id 379Article in journal (Refereed)
    Abstract [en]

    The International Energy Agency Technology Collaboration Programme for Ocean Energy Systems (OES) initiated the OES Wave Energy Conversion Modelling Task, which focused on the verification and validation of numerical models for simulating wave energy converters (WECs). The long-term goal is to assess the accuracy of and establish confidence in the use of numerical models used in design as well as power performance assessment of WECs. To establish this confidence, the authors used different existing computational modelling tools to simulate given tasks to identify uncertainties related to simulation methodologies: (i) linear potential flow methods; (ii) weakly nonlinear Froude-Krylov methods; and (iii) fully nonlinear methods (fully nonlinear potential flow and Navier-Stokes models). This article summarizes the code-to-code task and code-to-experiment task that have been performed so far in this project, with a focus on investigating the impact of different levels of nonlinearities in the numerical models. Two different WECs were studied and simulated. The first was a heaving semi-submerged sphere, where free-decay tests and both regular and irregular wave cases were investigated in a code-to-code comparison. The second case was a heaving float corresponding to a physical model tested in a wave tank. We considered radiation, diffraction, and regular wave cases and compared quantities, such as the WEC motion, power output and hydrodynamic loading.

  • 12.
    Koskela, Antti
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    On a generalization of neumann series of bessel functions using Hessenberg matrices and matrix exponentials2019In: European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Springer, 2019, Vol. 126, p. 205-214Conference paper (Refereed)
    Abstract [en]

    The Neumann expansion of Bessel functions (of integer order) of a function g: ℂ→ ℂ corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., g(s)=∑ℓ=0 ∞wℓφℓ(s), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

  • 13.
    Pålsson, Sara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Siegel, Michael
    New Jersey Inst Technol, Dept Math Sci, Newark, NJ 07102 USA..
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 386, p. 218-247Article in journal (Refereed)
    Abstract [en]

    Performing highly accurate simulations of droplet systems is a challenging problem. This is primarily due to the interface dynamics which is complicated further by the addition of surfactants. This paper presents a boundary integral method for computing the evolution of surfactant-covered droplets in 2D Stokes flow. The method has spectral accuracy in space and the adaptive time-stepping scheme allows for control of the temporal errors. Previously available semi-analytical solutions (based on conformal-mapping techniques) are extended to include surfactants, and a set of algorithms is introduced to detail their evaluation. These semi-analytical solutions are used to validate and assess the accuracy of the boundary integral method, and it is demonstrated that the presented method maintains its high accuracy even when droplets are in close proximity. 

  • 14.
    Figueiredo, Isabel N.
    et al.
    Univ Coimbra, CMUC, Dept Math, Fac Sci & Technol, Coimbra, Portugal..
    Pinto, Luis
    Univ Coimbra, CMUC, Dept Math, Fac Sci & Technol, Coimbra, Portugal..
    Figueiredo, Pedro N.
    Univ Coimbra, Fac Med, Coimbra, Portugal.;CHUC, Dept Gastroenterol, Coimbra, Portugal.;Ctr Cirurg Coimbra, Coimbra, Portugal..
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Univ Texas Austin, Dept Math, Austin, TX 78712 USA.;Univ Texas Austin, Oden Inst Computat Engn & Sci, Austin, TX 78712 USA..
    Unsupervised segmentation of colonic polyps in narrow-band imaging data based on manifold representation of images and Wasserstein distance2019In: Biomedical Signal Processing and Control, ISSN 1746-8094, E-ISSN 1746-8108, Vol. 53, article id UNSP 101577Article in journal (Refereed)
    Abstract [en]

    Colorectal cancer (CRC) is one of the most common cancers worldwide and after a certain age (>= 50) regular colonoscopy examination for CRC screening is highly recommended. One of the most prominent precursors of CRC are abnormal growths known as polyps. If a polyp is detected during colonoscopy examination the endoscopist needs to decide whether the polyp should be discarded, removed, or biopsied for further examination. However, the last two options involve some risks for the patient, while not all the polyps are precancerous. On the other hand, discarding a polyp has the risk of failing to detect CRC. We propose an automatic and unsupervised method for the segmentation of colonic polyps for in vivo Narrow-Band-Imaging (NBI) data. Polyp segmentation is a crucial step towards an automatic real-time polyp classification system, that could help the endoscopist in the diagnosis of CRC. The proposed method is a histogram based two-phase segmentation model, involving the Wasserstein distance. These histograms incorporate fused information about suitable image descriptors, namely semi-local texture, geometry and color. To test the proposed segmentation methodology we use a dataset consisting of 86 NBI polyp frames: the 83% sensitivity, 95% specificity, and 93% accuracy suggest a better performance compared to the results obtained with other methods.

  • 15.
    Nguyen, Van Dang
    et al.
    KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).
    Jansson, Johan
    KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).
    Frachon, Thomas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Degirmenci, Cem
    Hoffman, Johan
    KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).
    A fluid-structure interaction model with weak slip velocity boundary conditions on conforming internal interfaces2018Conference paper (Other (popular science, discussion, etc.))
    Abstract [en]

    We develop a PUFEM–Partition of Unity Finite Element Method to impose slip velocity boundary conditions on conforming internal interfaces for a fluid-structure interaction model. The method facilitates a straightforward implementation on the FEniCS/FEniCS-HPC platform. We show two results for 2D model problems with the implementation on FEniCS: (1) optimal convergence rate is shown for a stationary Navier-Stokes flow problem, and (2) the slip velocity conditions give qualitatively the correct result for the Euler flow. 

  • 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.
    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.

  • 17. Zhong, Yimin
    et al.
    Ren, Kui
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. University of Texas, United States.
    An implicit boundary integral method for computing electric potential of macromolecules in solvent2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 359, p. 199-215Article in journal (Refereed)
    Abstract [en]

    A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equation that arises in mathematical models for the electrostatics of molecules in solvent. The proposed method uses an implicit boundary integral formulation to derive a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separates the molecule and the solvent. The needed implicit surface is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.

  • 18.
    Wang, Siyang
    et al.
    Dept Informat Technol, Div Sci Comp, Box 337, SE-75105 Uppsala, Sweden.;Chalmers Univ Technol, Dept Math Sci, SE-41296 Gothenburg, Sweden.;Univ Gothenburg, SE-41296 Gothenburg, Sweden..
    Nissen, Anna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Univ Bergen, Dept Math, POB 7803, N-5020 Bergen, Norway.
    Kreiss, Gunilla
    Dept Informat Technol, Div Sci Comp, Box 337, SE-75105 Uppsala, Sweden..
    Convergence Of Finite Difference Methods For The Wave Equation in Two Space Dimensions2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 314, p. 2737-2763Article in journal (Refereed)
    Abstract [en]

    When using a finite difference method to solve an initial-boundary-value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.

  • 19.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Koskela, Antti
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Disguised and new quasi-Newton methods for nonlinear eigenvalue problems2018In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 79, no 1, p. 311-335Article in journal (Refereed)
    Abstract [en]

    In this paper, we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type M(λ)v = 0, where (Formula presented.) is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh’s theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby, we provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier’s residual inverse iteration and Ruhe’s method of successive linear problems.

  • 20.
    Srinivasan, Shriram
    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 Green's functions of Stokes flow in a half-space2018In: RESEARCH IN THE MATHEMATICAL SCIENCES, ISSN 2197-9847, Vol. 5, article id 35Article in journal (Refereed)
    Abstract [en]

    Recently, Gimbutas et al. (J Fluid Mech, 2015. https://doi.org/10.1017/jfm.2015.302) derived an elegant representation for the Green's functions of Stokes flow in a half-space. We present a fast summation method for sums involving these half-space Green's functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al. (Res Math Sci 4(1): 1, 2017. https://doi.org/10.1186/s40687-016-0092-7) for the corresponding free-space Green's functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real space (short-range interactions) and one in Fourier space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison with the free-space evaluation, greater computational savings is also achieved when compared to their respective direct sums.

  • 21.
    Saffar Shamshirgar, Davood
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast methods for electrostatic calculations in molecular dynamics simulations2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis deals with fast and efficient methods for electrostatic calculations with application in molecular dynamics simulations. The electrostatic calculations are often the most expensive part of MD simulations of charged particles. Therefore, fast and efficient algorithms are required to accelerate these calculations. In this thesis, two types of methods have been considered: FFT-based methods and fast multipole methods (FMM).

    The major part of this thesis deals with fast N.log(N) and spectrally accurate methods for accelerating the computation of pairwise interactions with arbitrary periodicity. These methods are based on the Ewald decomposition and have been previously introduced for triply and doubly periodic problems under the name of Spectral Ewald (SE) method. We extend the method for problems with singly periodic boundary conditions, in which one of three dimensions is periodic. By introducing an adaptive fast Fourier transform, we reduce the cost of upsampling in the non periodic directions and show that the total cost of computation is comparable with the triply periodic counterpart. Using an FFT-based technique for solving free-space harmonic problems, we are able to unify the treatment of zero and nonzero Fourier modes for the doubly and singly periodic problems. Applying the same technique, we extend the SE method for cases with free-space boundary conditions, i.e. without any periodicity.

    This thesis is also concerned with the fast multipole method (FMM) for electrostatic calculations. The FMM is very efficient for parallel processing but it introduces irregularities in the electrostatic potential and force, which can cause an energy drift in MD simulations. In this part of the thesis we introduce a regularized version of the FMM, useful for MD simulations, which approximately conserves energy over a long time period and even for low accuracy requirements. The method introduces a smooth transition over the boundary of boxes in the FMM tree and therefore it removes the discontinuity at the error level inherent in the FMM.

  • 22. Ohtsuka, T.
    et al.
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. University of Texas at Austin, TX, United States.
    Giga, Y.
    Growth Rate of Crystal Surfaces with Several Dislocation Centers2018In: Crystal Growth & Design, ISSN 1528-7483, E-ISSN 1528-7505, Vol. 18, no 3, p. 1917-1929Article in journal (Refereed)
    Abstract [en]

    We studied analytically and numerically the growth rate of a crystal surface growing by several screw dislocations. We observed some discrepancy between the growth rates computed by our level set method (J. Sci. Comput. 2015, 62.

  • 23.
    Nissen, Anna
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. University of Bergen, Norway.
    Keilegavlen, Eirik
    Sandve, Tor Harald
    Berre, Inga
    Nordbotten, Jan Martin
    Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs2018In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, Vol. 22, no 2, p. 451-467Article in journal (Refereed)
    Abstract [en]

    In simulation of fluid injection in fractured geothermal reservoirs, the characteristics of the physical processes are severely affected by the local occurence of connected fractures. To resolve these structurally dominated processes, there is a need to develop discretization strategies that also limit computational effort. In this paper, we present an upscaling methodology for geothermal heat transport with fractures represented explicitly in the computational grid. The heat transport is modeled by an advection-conduction equation for the temperature, and solved on a highly irregular coarse grid that preserves the fracture heterogeneity. The upscaling is based on different strategies for the advective term and the conductive term. The coarse scale advective term is constructed from sums of fine scale fluxes, whereas the coarse scale conductive term is constructed based on numerically computed basis functions. The method naturally incorporates the coupling between solution variables in the matrix and in the fractures, respectively, via the discretization. In this way, explicit transfer terms that couple fracture and matrix solution variables are avoided. Numerical results show that the upscaling methodology performs well, in particular for large upscaling ratios, and that it is applicable also to highly complex fracture networks.

  • 24.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Palitta, Davide
    Univ Bologna, Dipartimento Matemat, Piazza Porta S Donato,5, I-40127 Bologna, Italy..
    Ringh, Emil
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Krylov methods for low-rank commuting generalized Sylvester equations2018In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 25, no 6, article id e2176Article in journal (Refereed)
    Abstract [en]

    We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator pi with a particular structure. More precisely, the commutators of the matrix coefficients of the operator pi and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low-rank approximability of this problem, that is, the solution to this matrix equation can be approximated with a low-rank matrix. Projection methods have successfully been used to solve other matrix equations with low-rank approximability. We propose a new projection method for this class of matrix equations. The choice of the subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low-rank commutators. We illustrate the effectiveness of our method by solving large-scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.

  • 25.
    Gallistl, Dietmar
    et al.
    Univ Twente, Fac Elect Engn Math & Comp Sci, POB 217, NL-7500 AE Enschede, Netherlands..
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Verfuerth, Barbara
    Westfalische Wilhelms Univ Munster, Appl Math, D-48149 Munster, Germany..
    NUMERICAL HOMOGENIZATION OF H(CURL)-PROBLEMS2018In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, no 3, p. 1570-1596Article in journal (Refereed)
    Abstract [en]

    If an elliptic differential operator associated with an H (curl)- problem involves rough (rapidly varying) coefficients, then solutions to the corresponding H (curl)- problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest-order Nedelec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, H (curl)- stable, and quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh size) in the H (curl) norm are obtained provided the right-hand side belongs to H (div). With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first-order corrector, including corresponding quantitative error estimates without the requirement of scale separation. The constructed generalized finite element method falls into the class of localized orthogonal decomposition methods, which have not been studied for H (curl)- problems so far.

  • 26.
    Mele, Giampaolo
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    On restarting the tensor infinite Arnoldi method2018In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 58, no 1, p. 133-162Article in journal (Refereed)
    Abstract [en]

    An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated. 

  • 27.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lehto, Erik
    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.
    Partition of unity extension of functions on complex domains2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 57-79Article in journal (Refereed)
    Abstract [en]

    We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in 2D. Function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques. Overlapping partitions are placed along the boundaries, and a local extension of the function is computed on each patch using smooth radial basis functions; a trivially parallel process. A partition of unity method blends the local extrapolations into a global one, where weight functions impose compact support. The regularity of the extended function can be controlled by the construction of the partition of unity function. We evaluate the performance of the PUX method in the context of solving the Poisson equation on multiply connected domains using a boundary integral method and a spectral solver. With a suitable choice of parameters the error converges as a tenth order method down to 10−14.

  • 28.
    Ringh, Emil
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Sylvester-based preconditioning for the waveguide eigenvalue problem2018In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 542, no 1, p. 441-463Article in journal (Refereed)
    Abstract [en]

    We consider a nonlinear eigenvalue problem (NEP) arising from absorbing boundary conditions in the study of a partial differential equation (PDE) describing a waveguide. We propose a new computational approach for this large-scale NEP based on residual inverse iteration (Resinv) with preconditioned iterative solves. Similar to many preconditioned iterative methods for discretized PDEs, this approach requires the construction of an accurate and efficient preconditioner. For the waveguide eigenvalue problem, the associated linear system can be formulated as a generalized Sylvester equation AX+XB+A1XB1+A2XB2+K(ring operator)X=C, where (ring operator) denotes the Hadamard product. The equation is approximated by a low-rank correction of a Sylvester equation, which we use as a preconditioner. The action of the preconditioner is efficiently computed by using the matrix equation version of the Sherman-Morrison-Woodbury (SMW) formula. We show how the preconditioner can be integrated into Resinv. The results are illustrated by applying the method to large-scale problems.

    The full text will be freely available from 2020-05-03 09:14
  • 29.
    Malenova, Gabriela
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Uncertainty quantification for high frequency waves2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    We consider high frequency waves satisfying the scalar wave equation with highly oscillatory initial data represented by a short wavelength ε. The speed of propagation of the medium as well as the phase and amplitude of the initial data is assumed to be uncertain, described by a finite number of independent random variables with known probability distributions. We introduce quantities of interest (QoIs) as spatial and/or temporal averages of the squared modulus of the wave solution, or its derivatives. The main focus of this work is on fast computation of the statistics of those QoIs in the form of moments like the average and variance. They are difficult to obtain numerically by standard methods, as the cost grows rapidly with ε−1 and the dimension of the stochastic space. We therefore propose a fast approximation method consisting of three techniques: the Gaussian beam method to approximate the wave solution, the numerical steepest descent method to compute the QoIs and the sparse stochastic collocation to evaluate the statistics.

    The Gaussian beam method is introduced to avoid the considerable cost of approximating the wave solution by direct methods. A Gaussian beam is an asymptotic solution to the wave equation localized around rays, bicharacteristics of the wave equation. This setup allows us to replace the PDE by a set of ODEs that can be solved independently of ε.

    The computation of QoIs includes evaluations of highly oscillatory integrals. The idea of the numerical steepest descent method is to change the integration path in the complex plane such that the integrand is non-oscillatory along it. Standard quadrature methods can be then utilized. We construct such paths for our case and show error estimates for the integral approximation by the Gauss-Laguerre and Gauss-Hermite quadrature.

    Finally, the evaluation of statistical moments of the QoI may suffer from the curse of dimensionality.  The sparse grid collocation method introduces a framework where certain large group of points can be neglected while only slightly reducing the convergence rate. The regularity of the QoIs in terms of the input random parameters and the wavelength is important for the method to be efficient.  In particular, the size of the derivatives should be bounded independently of ε. We show that the QoIs indeed have this property, despite the highly oscillatory character of the waves.

  • 30. Chu, J.
    et al.
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Volumetric variational principles for a class of partial differential equations defined on surfaces and curves: In memory of Heinz-Otto Kreiss2018In: Research in Mathematical Sciences, ISSN 2522-0144, Vol. 5, no 2, article id 19Article in journal (Refereed)
    Abstract [en]

    In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace–Beltrami equations and surface wave equations. The approach is to systematically formulate extensions of the variational integrals and derive the Euler–Lagrange equations of the extended problem, including the boundary conditions that can be easily discretized on uniform Cartesian grids or adaptive meshes. In our approach, the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problems. For elliptic and parabolic problems, our boundary closure mostly yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher-order regularization term or by periodically but infrequently “reinitializing” the computed solutions. Some numerical examples for each representative surface PDEs are presented.

  • 31.
    Lehto, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Shankar, Varun
    Wright, Grady B.
    A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 5, p. A2129-A2151Article in journal (Refereed)
    Abstract [en]

    We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in R-d. The novelty of the method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds.

  • 32.
    Zahedi, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A Space-Time Cut Finite Element Method with Quadrature in Time2017In: Geometrically Unfitted Finite Element Methods and Applications: Proceedings of the UCL Workshop 2016, Cham: Springer, 2017, p. 281-306Chapter in book (Refereed)
  • 33.
    Sorgentone, Chiara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Favini, B.
    A systematic method to construct mimetic Finite-Difference schemes for incompressible flows2017In: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 14, no 3, p. 419-436Article in journal (Refereed)
    Abstract [en]

    We present a general procedure to construct a non-linear mimetic finite-difference operator. The method is very simple and general: it can be applied for any order scheme, for any number of grid points and for any operator constraints. In order to validate the procedure, we apply it to a specific example, the Jacobian operator for the vorticity equation. In particular we consider a finite difference approximation of a second order Jacobian which uses a 9x9 uniform stencil, verifies the skew-symmetric property and satisfies physical constraints such as conservation of energy and enstrophy. This particular choice has been made in order to compare the present scheme with Arakawa’s renowned Jacobian, which turns out to be a specific case of the general solution. Other possible generalizations of Arakawa’s Jacobian are available in literature but only the present approach ensures that the class of solutions found is the widest possible. A simplified analysis of the general scheme is proposed in terms of truncation error and study of the linearised operator together with some numerical experiments. We also propose a class of analytical solutions for the vorticity equation to compare an exact solution with our numerical results.

  • 34.
    Pålsson, Sara
    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.
    Sorgentone, Chiara
    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.
    Adaptive time-stepping for surfactant-laden drops2017In: Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11) / [ed] Chappell, D.J., 2017Conference paper (Refereed)
    Abstract [en]

    An adaptive time-stepping scheme is presented aimed at computing the dynamics of surfactant-covered deforming droplets. This involves solving a coupled system, where one equation corresponds to the evolution of the drop interfaces and one to the surfactant concentration. The first is discretised in space using a boundary integral formulation which can be treated explicitly in time. The latter is a convection-diffusion equation solved with a spectral method and is advantageously solved with a semi-implicit method in time. The scheme is adaptive with respect to drop deformation as well as surfactant concentration and the adjustment of time-steps takes both errors into account. It is applied and demonstrated for simulation of the deformation of surfactant-covered droplets, but can easily be applied to any system of equations with similar structure. Tests are performed for both 2D and 3D formulations and the scheme is shown to meet set error tolerances in an efficient way.

  • 35. Chen, C.
    et al.
    Kublik, C.
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An implicit boundary integral method for interfaces evolving by Mullins-Sekerka dynamics2017In: Springer Proceedings in Mathematics and Statistics, Springer New York LLC , 2017, p. 1-21Conference paper (Refereed)
    Abstract [en]

    We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations. The computation of the dynamics involves solving Laplace’s equation with Dirichlet boundary conditions on multiply connected and unbounded domains and propagating the interface using a normal velocity obtained from the solution of the PDE at each time step. Our method is based on a simple formulation for implicit interfaces, which rewrites boundary integrals as volume integrals over the entire space. The resulting algorithm thus inherits the benefits of both level set methods and boundary integral methods to simulate the nonlocal front propagation problem with possible topological changes. We present numerical results in both two and three dimensions to demonstrate the effectiveness of the algorithm.

  • 36.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Peterseim, D.
    Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials2017In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 27, no 11, p. 2147-2184Article in journal (Refereed)
    Abstract [en]

    This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

  • 37. Figueiredo, Isabel N.
    et al.
    Leal, Carlos
    Pinto, Luis
    Figueiredo, Pedro N.
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Univ Texas Austin, Dept Math, Austin, TX 78712 USA; Univ Texas Austin, ICES, Austin, TX 78712 USA.
    Dissimilarity Measure of Consecutive Frames in Wireless Capsule Endoscopy Videos: a way of searching for abnormalities2017In: 2017 IEEE 30TH INTERNATIONAL SYMPOSIUM ON COMPUTER-BASED MEDICAL SYSTEMS (CBMS) / [ed] Bamidis, PD Konstantinidis, ST Rodrigues, PP, IEEE , 2017, p. 702-707Conference paper (Refereed)
    Abstract [en]

    In a previous work we have shown that the curve representing the dissimilarity measure between consecutive frames of a wireless capsule endoscopic video of the small bowel, obtained by means of an image registration method, can be regarded as a rough indicator of the speed of the capsule, and simultaneously, it is also a valuable auxiliary medical tool. In effect, this curve enables a global and fast interpretation of the video, in the sense that it clearly divides the video frames into two main categories: consecutive frames with similar content, which correspond to low values in the curve, and consecutive frames displaying abrupt changes in the image content, which are depicted by peaks, i.e. high values, in the curve. As the main goal of a wireless capsule video examination consists in searching for abnormal features in the images, the purpose of the present work is to analyse whether this curve can also be used to search, quickly, for abnormalities. The experiments performed focus on bleeding identification in small bowel images.

  • 38.
    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.

  • 39. 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.

  • 40.
    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.

  • 41. Chen, Chieh
    et al.
    Tsai, Yen-Hsi Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Implicit boundary integral methods for the Helmholtz equation in exterior domains2017In: RESEARCH IN THE MATHEMATICAL SCIENCES, ISSN 2197-9847, Vol. 4, article id UNSP 19Article in journal (Refereed)
  • 42.
    Hanke, Michael
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    März, R.
    Tischendorf, C.
    Weinmüller, E.
    Wurm, S.
    Least-squares collocation for linear higher-index differential–algebraic equations2017In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 317, p. 403-431Article in journal (Refereed)
    Abstract [en]

    Differential–algebraic equations with higher index give rise to essentially ill-posed problems. Therefore, their numerical approximation requires special care. In the present paper, we state the notion of ill-posedness for linear differential–algebraic equations more precisely. Based on this property, we construct a regularization procedure using a least-squares collocation approach by discretizing the pre-image space. Numerical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index-1 differential–algebraic equations. Convergence is shown for a limited class of linear higher-index differential–algebraic equations.

  • 43. 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.

  • 44.
    Brocke, Ekaterina
    et al.
    KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Djurfeldt, M.
    Bhalla, U. S.
    Hellgren Kotaleski, Jeanette
    KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST).
    Hanke, Michael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Multirate method for co-simulation of electrical-chemical systems in multiscale modeling2017In: Journal of Computational Neuroscience, ISSN 0929-5313, E-ISSN 1573-6873, Vol. 42, no 3, p. 245-256Article in journal (Refereed)
    Abstract [en]

    Multiscale modeling by means of co-simulation is a powerful tool to address many vital questions in neuroscience. It can for example be applied in the study of the process of learning and memory formation in the brain. At the same time the co-simulation technique makes it possible to take advantage of interoperability between existing tools and multi-physics models as well as distributed computing. However, the theoretical basis for multiscale modeling is not sufficiently understood. There is, for example, a need of efficient and accurate numerical methods for time integration. When time constants of model components are different by several orders of magnitude, individual dynamics and mathematical definitions of each component all together impose stability, accuracy and efficiency challenges for the time integrator. Following our numerical investigations in Brocke et al. (Frontiers in Computational Neuroscience, 10, 97, 2016), we present a new multirate algorithm that allows us to handle each component of a large system with a step size appropriate to its time scale. We take care of error estimates in a recursive manner allowing individual components to follow their discretization time course while keeping numerical error within acceptable bounds. The method is developed with an ultimate goal of minimizing the communication between the components. Thus it is especially suitable for co-simulations. Our preliminary results support our confidence that the multirate approach can be used in the class of problems we are interested in. We show that the dynamics ofa communication signal as well as an appropriate choice of the discretization order between system components may have a significant impact on the accuracy of the coupled simulation. Although, the ideas presented in the paper have only been tested on a single model, it is likely that they can be applied to other problems without loss of generality. We believe that this work may significantly contribute to the establishment of a firm theoretical basis and to the development of an efficient computational framework for multiscale modeling and simulations.

  • 45.
    Lindholm, Love
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Numerical methods for the calibration problem in finance and mean field game equations2017Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis contains five papers and an introduction. The first four of the included papers are related to financial mathematics and the fifth paper studies a case of mean field game equations. The introduction thus provides background in financial mathematics relevant to the first four papers, and an introduction to mean field game equations related to the fifth paper.

    In Paper I, we use theory from optimal control to calibrate the so called local volatility process given market data on options. Optimality conditions are in this case given by the solution to a Hamiltonian system of differential equations. Regularization is added by mollifying the Hamiltonian in this system and we solve the resulting equation using a trust region Newton method. We find that our resulting algorithm for the calibration problem is both accurate and robust.

    In Paper II, we solve the local volatility calibration problem using a technique that is related to - but also different from - the Hamiltonian framework in Paper I. We formulate the optimization problem by means of a Lagrangian multiplier and add a Tikhonov type regularization directly on the parameter we are trying to estimate. The resulting equations are solved with the same trust region Newton method as in Paper II, and again we obtain an accurate and robust algorithm for the calibration problem.

    Paper III formulates the problem of calibrating a local volatility process to option prices in a way that differs entirely from what is done in the first two papers. We exploit the linearity of the Dupire equation governing the prices to write the optimization problem as a quadratic programming problem. We illustrate by a numerical example that method can indeed be used to find a local volatility that gives good match between model prices and observed market prices on options.

    Paper IV deals with the hedging problem in finance. We investigate if so called quadratic hedging strategies formulated for a stochastic volatility model can generate smaller hedging errors than obtained when hedging with the standard Black-Scholes framework. We thus apply the quadratic hedging technique as well as the Black-Scholes hedging to observed option prices written on an equity index and calculate the empirical errors in the two cases. Our results indicate that smaller errors can be obtained with quadratic hedging in the models used than with hedging in the Black-Scholes framework.

    Paper V describes a model of an electricity market consisting of households that try to minimize their electricity cost by dynamic battery usage. We assume that the price process of electricity depends on the aggregated momentaneous electricity consumption. With this assumption, the cost minimization problem of each household is governed by a system of mean field game equations. We also provide an existence and uniqueness result for these mean field game equations. The equations are regularized and the approximate equations are solved numerically. We illustrate how the battery usage affects the electricity price.

  • 46.
    Malenova, Gabriela
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Motamed, Mohammad
    Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131, USA.
    Runborg, Olof
    KTH, Centres, SeRC - Swedish e-Science Research Centre. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Regularity of stochastic observables in Gaussian beam superposition of high-frequency waves2017In: Research in mathematical sciences, ISSN 2197-9847Article in journal (Other academic)
    Abstract [en]

    We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described by a finite number of random variables with known probability distributions. We define quantities of interest (QoIs), or observables, as local averages of the squared modulus of the wave solution. We aim to quantify the regularity of these QoIs in terms of the input random parameters, and the wave length, i.e., to estimate the size of their derivatives. The regularity is important for uncertainty quantification methods based on interpolation in the stochastic space. In particular, the size of the derivatives should be bounded independently of the wave length. In this paper, we are able to show that when these QoIs are approximated by Gaussian beam superpositions, they indeed have this property, despite the highly oscillatory character of the waves.

  • 47.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Malqvist, Axel
    The finite element method for the time-dependent gross-pitaevskii equation with angular momentum rotation2017In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, no 2, p. 923-952Article in journal (Refereed)
    Abstract [en]

    We consider the time-dependent Gross Pitaevskii equation describing the dynamics of rotating Bose Einstein condensates and its discretization with the finite element method. We analyze a mass conserving Crank-Nicolson-type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the L-2- and energy-norm in space. The estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the Gross Pitaevskii equation. We demonstrate the performance of the method in numerical experiments.

  • 48.
    Gaaf, Sarah W.
    et al.
    TU Eindhoven.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    The infinite Bi-Lanczos method for nonlinear eigenvalue problems2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, p. S898-S919Article in journal (Refereed)
  • 49.
    Saffar Shamshirgar, Davoud
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    The Spectral Ewald method for singly periodic domains2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 347, p. 341-366Article in journal (Refereed)
    Abstract [en]

    We present a fast and spectrally accurate method for efficient computation of the three dimensional Coulomb potential with periodicity in one direction. The algorithm is FFT-based and uses the so-called Ewald decomposition, which is naturally most efficient for the triply periodic case. In this paper, we show how to extend the triply periodic Spectral Ewald method to the singly periodic case, such that the cost of computing the singly periodic potential is only marginally larger than the cost of computing the potential for the corresponding triply periodic system. In the Fourier space contribution of the Ewald decomposition, a Fourier series is obtained in the periodic direction with a Fourier integral over the non-periodic directions for each discrete wave number. We show that upsampling to resolve the integral is only needed for modes with small wave numbers. For the zero wave number, this Fourier integral has a singularity. For this mode, we effectively need to solve a free-space Poisson equation in two dimensions. A very recent idea by Vico et al. makes it possible to use FFTs to solve this problem, allowing us to unify the treatment of all modes. An adaptive 3D FFT can be established to apply different upsampling rates locally. The computational cost for other parts of the algorithm is essentially unchanged as compared to the triply periodic case, in total yielding only a small increase in both computational cost and memory usage for this singly periodic case.

  • 50.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Mele, Giampaolo
    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.
    The waveguide eigenvalue problem and the tensor infinite Arnoldi method2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 3, p. A1062-A1088Article in journal (Refereed)
    Abstract [en]

    We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is two fold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the waveguide eigenvalue problem. The waveguide eigenvalue problem arises from a finite-element discretization of a partial differential equation used in the study waves propagating in a periodic medium. The algorithm is successfully applied to accurately solve benchmark problems as well as complicated waveguides. We study the complexity of the specialized algorithm with respect to the number of iterations "m" and the size of the problem "n", both from a theoretical perspective and in practice. For the waveguide eigenvalue problem, we establish that the computationally dominating part of the algorithm has complexity O(nm^2+sqrt(n)m^3). Hence, the asymptotic complexity of TIAR applied to the waveguide eigenvalue problem, for n→ ∞, is the same as for Arnoldi’s method for standard eigenvalue problems.

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