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  • 1. Babuska, I.
    et al.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zouraris, Georgios
    Galerkin finite element approximations of stochastic elliptic partial differential equations2004In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, no 2, p. 800-825Article in journal (Refereed)
    Abstract [en]

    We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

  • 2. Babuska, Ivo
    et al.
    Nobile, Fabio
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    A stochastic collocation method for elliptic partial differential equations with random input data2007In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 45, no 3, p. 1005-1034Article in journal (Refereed)
    Abstract [en]

    In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 ( 2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

  • 3.
    Bogfjellmo, Geir
    et al.
    Chalmers & Univ Gothenburg, Math Sci, SE-41296 Gothenburg, Sweden.;CSIC, Inst Ciencias Matemat, Madrid, Spain..
    Modin, Klas
    Chalmers & Univ Gothenburg, Math Sci, SE-41296 Gothenburg, Sweden..
    Verdier, Olivier
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Western Norway Univ Appl Sci, Dept Comp Math & Phys, N-5020 Bergen, Norway..
    A NUMERICAL ALGORITHM FOR C-2-SPLINES ON SYMMETRIC SPACES2018In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, no 4, p. 2623-2647Article in journal (Refereed)
    Abstract [en]

    Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.

  • 4. Chung, E.
    et al.
    Engquist, Björn
    Princeton University.
    Convergence analysis of fully discrete finite volume methods for Maxwell's equations in nonhomogeneous media2005In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 43, no 1, p. 303-317Article in journal (Refereed)
    Abstract [en]

    We will consider both explicit and implicit fully discrete finite volume schemes for solving three-dimensional Maxwell's equations with discontinuous physical coefficients on general polyhedral domains. Stability and convergence for both schemes are analyzed. We prove that the schemes are second order accurate in time. Both schemes are proved to be first order accurate in space for the Voronoi-Delaunay grids and second order accurate for nonuniform rectangular grids. We also derive explicit expressions for the dependence on the physical parameters in all estimates.

  • 5.
    Chung, Eric T.
    et al.
    Department of Mathematics, The Chinese University of Hong Kong, Hong Kong.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions2009In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 47, no 5, p. 3820-3848Article in journal (Refereed)
    Abstract [en]

    In this paper, we developed and analyzed a new class of discontinuous Galerkin (DG) methods for the acoustic

    wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these

    schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches

    are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be

    seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally

    and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume

    method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally

    convergent. Furthermore, in order to apply the new method for unbounded domains, we proposed a new way to handle the

    second order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.

  • 6. Chung, Eric T.
    et al.
    Engquist, Björn
    Optimal discontinuous Galerkin methods for wave propagation2006In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 44, no 5, p. 2131-2158Article in journal (Refereed)
    Abstract [en]

    We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.

  • 7. Dorobantu, M
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Wavelet-based numerical homogenization1998In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 35, no 2, p. 540-559Article in journal (Refereed)
    Abstract [en]

    A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in find and coarse scale components followed by the elemination of the fine scale contributions. If the operator is in divergence form, this is preserved by the homogenization procedure. For periodic problems, results similar to classical effective coefficient theory is proved. The procedure can be applied to problems that are not cell-periodic.

  • 8.
    Efraimsson, Gunilla
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    A Remark on Numerical Errors Downstream of Slightly Viscous Shocks1999In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 36, no 3, p. 853-863Article in journal (Refereed)
    Abstract [en]

    Lower-order errors downstream of a shock layer have been detected in computations with nonconstant solutions when using higher-order shock capturing schemes in one and two dimensions [B. Engquist and B. Sjögreen, {SIAM J. Numer. Anal., 35 (1998), pp. 2464--2485].By analyzing the steady-state solution of slightly viscous hyperbolic systems of conservation laws we find that the solution can have an ${\cal O}(h)$-dependence downstream of a shock layer, although the viscous term in that region is of ${\cal O}(h^2)$. Numerical examples illustrate the analysis.

  • 9.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Hou, T
    Particle method approximation of oscillatory solutions to hyperbolic differential equations1989In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, ISSN 0036-1429, Vol. 26, no 2, p. 289-319Article in journal (Refereed)
    Abstract [en]

    Particle methods approximating hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid. Highly oscillatory solutions to the Broadwell and variable coefficients Carleman models are considered. Homogenization results are given and the approximations of more general systems are discussed. Numercial exampels are presented

  • 10.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Luo, E
    Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients1997In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 34, no 6, p. 2254-2273Article in journal (Refereed)
    Abstract [en]

    Standard multigrid methods are not so effective for equations with highly oscillatory coefficients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite difference approximations are used for the discretization of the equations. Convergence analysis is based on the homogenization theory. Proofs are given for a two-level multigrid method with the homogenized coarse grid operator for two classes of two-dimensional elliptic equations with Dirichlet boundary conditions.

  • 11.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Sjögreen, B
    The convergence rate of finite difference schemes in the presence of shocks1998In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 35, no 6, p. 2464-2485Article in journal (Refereed)
    Abstract [en]

    : Finite difference approximations generically have O(1) pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.

  • 12.
    Gaidashev, Denis
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Khmelev, Dmitry
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On numerical algorithms for the solution of a Beltrami equation2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 5, p. 2238-2253Article in journal (Refereed)
    Abstract [en]

    The paper concerns numerical algorithms for solving the Beltrami equation f (z) over bar (z) = mu( z) fz( z) for a compactly supported mu. First, we study an e. cient algorithm that has been proposed in [ P. Daripa, J. Comput. Phys., 106 ( 1993), pp. 355 - 365] and [ P. Daripa and D. Mashat, Numer. Algorithms, 18 ( 1998), pp. 133 - 157] and present its rigorous justi. cation. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of easier implementation by avoiding the use of the Hilbert transform. The present paper can also be viewed as a prologue to one important application of the Beltrami equation: it provides a detailed description of the algorithm that has been used in [ D. Gaidashev, Nonlinearity, 20 ( 1998), pp. 713 - 741] and [ D. Gaidashev and M. Yampolsky, Experiment. Math., 16 ( 2007), pp. 215 - 226] to address an important issue in complex dynamics - conjectural universality for Siegel disks.

  • 13.
    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.

  • 14. Hansbo, Peter
    et al.
    Larson, Mats G.
    Zahedi, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES2015In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 53, no 4, p. 1806-1832Article in journal (Refereed)
    Abstract [en]

    The mean curvature vector of a surface is obtained by letting the Laplace-Beltrami operator act on the embedding of the surface in R-3. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in L-2. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.

  • 15.
    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.

  • 16.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Ohlberger, M.
    Verfürth, B.
    A new heterogeneous multiscale method for time-harmonic Maxwell's equations2016In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, no 6, p. 3493-3522Article in journal (Refereed)
    Abstract [en]

    In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the H(curl)-and the H-1-norm, and we derive reliable and efficient localized residual-based a posteriori error estimates. Numerical experiments are presented to verify the a priori convergence results.

  • 17.
    Kreiss, G.
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Efraimsson, Gunilla
    KTH, Superseded Departments, Aeronautical and Vehicle Engineering.
    Nordstrom, J.
    FFA, Sweden; Uppsala University, Department of Scientific Computing, Sweden.
    Elimination of first order errors in shock calculations2001In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 38, no 6, p. 1986-1998Article in journal (Refereed)
    Abstract [en]

    First order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions. Based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy.

  • 18. Kreiss, H O
    et al.
    Petersson, N A
    Yström, Jacob
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Difference approximations of the Neumann problem for the second order wave equation2004In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, no 3, p. 1292-1323Article in journal (Refereed)
    Abstract [en]

    Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate approximations of the boundary conditions are presented. Away from the boundary, the basic second order method can be corrected to achieve fourth order spatial accuracy. To integrate in time, we present both a second order and a fourth order accurate explicit method. The stability of the method is ensured by adding a small fourth order dissipation operator, locally modified near the boundary to allow its application at all grid points inside the computational domain. Numerical experiments demonstrate the accuracy and long-time stability of the proposed method.

  • 19.
    Mordecki, Ernesto
    et al.
    Universidad de la República, Iguá 4225, Montevideo, Uruguay.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tempone, Raúl
    Zouraris, Georgios
    Div of Applied Math - Statistics, Univ of Crete.
    Adaptive weak approximation of diffusions with jumps2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 4, p. 1732-1768Article in journal (Refereed)
    Abstract [en]

    This work develops Monte Carlo Euler adaptive time stepping methods for the weak approximation problem of jump diffusion driven stochastic differential equations. The main result is the derivation of a new expansion for the omputational error, with computable leading order term in a posteriori form, based on stochastic flows and discrete dual backward problems which extends the results in [STZ]. These expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or quasi-deterministic time steps are described. Numerical examples show the performance of the proposed error approximation and of the described adaptive time-stepping methods.

  • 20. Nobile, F.
    et al.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Webster, C. G.
    A sparse grid stochastic collocation method for partial differential equations with random input data2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 5, p. 2309-2345Article in journal (Refereed)
    Abstract [en]

    This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coeffcients and forcing terms ( input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using L-q norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem ( number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.

  • 21. Nobile, F.
    et al.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Webster, C. G.
    An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 5, p. 2411-2442Article in journal (Refereed)
    Abstract [en]

    This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms ( input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are Karhunen-Loeve truncations of "smooth" random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates ( sub) exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.

  • 22. Nordstrom, J.
    et al.
    Svärd, Magnus
    Well-posed boundary conditions for the Navier-Stokes equations2005In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 43, no 3, p. 1231-1255Article in journal (Refereed)
    Abstract [en]

    In this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differentia equations. With those, well-posedness can be proven for a general initial-boundary value problem. The procedure is exemplifie on the linearized Navier-Stokes equations in two and three space dimensions on a general domain.

  • 23.
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Convergence of the forward euler method for nonconvex differential inclusions2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 47, no 1, p. 308-320Article in journal (Refereed)
    Abstract [en]

    The convergence of reachable sets for nonconvex differential inclusions is considered. When the right-hand side in the differential inclusion is a compact-valued, Lipschitz continuous set-valued function it is shown that the convergence in Hausdorff distance of reachable sets for a forward Euler discretization is linear in the time step.

  • 24.
    Siklosi, Malin
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Efraimsson, Gunilla
    KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Aeroacoustics.
    Analysis of first order errors in shock calculations in two space dimensions2005In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 43, no 2, p. 672-685Article in journal (Refereed)
    Abstract [en]

    Numerical computations show that solutions of hyperbolic conservation laws obtained by second or higher order shock capturing methods in many cases are only first order accurate downstream of shocks (see, e.g., [M. H. Carpenter and J.H. Casper, AIAA J., 37 (1999), pp. 1072 1079]). We use matched asymptotic expansions to analyze the degeneration in order of accuracy for stationary solutions of hyperbolic conservation laws in two space dimensions.

  • 25.
    Zahedi, Sara
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Wadbro, Eddie
    Kreiss, Gunilla
    Berggren, Martin
    A uniformly well-conditioned, unfitted Nitsche method for interface problems: PartIIn: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170Article in journal (Other academic)
    Abstract [en]

    A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented.The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and a variant of Nitsche's method enforces the jump conditions at the interface.In this method, the solutions on each side of the interface are extended to the entire domain, which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. Numerical experiments are presented showing optimal convergence order in the energy and $L^2$ norms, and also for pointwise errors. The presented results also show that the condition number of the system matrix is independent of the position of the interface relative to the grid.

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