Marcus Grote - Profile on Academia.edu (original) (raw)

Papers by Marcus Grote

Because of their robustness, efficiency and non-intrusiveness, Monte Carlo methods are probably t... more Because of their robustness, efficiency and non-intrusiveness, Monte Carlo methods are probably the most popular approach in uncertainty quantification to computing expected values of quantities of interest (QoIs). Multilevel Monte Carlo (MLMC) methods significantly reduce the computational cost by distributing the sampling across a hierarchy of discretizations and allocating most samples to the coarser grids. For time dependent problems, spatial coarsening typically entails an increased time-step. Geometric constraints, however, may impede uniform coarsening thereby forcing some elements to remain small across all levels. If explicit time-stepping is used, the time-step will then be dictated by the smallest element on each level for numerical stability. Hence, the increasingly stringent CFL condition on the time-step on coarser levels significantly reduces the advantages of the multilevel approach. To overcome that bottleneck we propose to combine the multilevel approach of MLMC with local time-stepping (LTS). By adapting the time-step to the locally refined elements on each level, the efficiency of MLMC methods is restored even in the presence of complex geometry without sacrificing the explicitness and inherent parallelism. In a careful cost comparison, we quantify the reduction in computational cost for local refinement either inside a small fixed region or towards a reentrant corner.

Stabilized Runge-Kutta (aka Chebyshev) methods are especially efficient for the numerical solutio... more Stabilized Runge-Kutta (aka Chebyshev) methods are especially efficient for the numerical solution of large systems of stiff differential equations because they are fully explicit; hence, they are inherently parallel and easily accommodate nonlinearity. For semi-discrete parabolic (or diffusion dominated) problems, for instance, stabilized Runge-Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when much of the stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depend on the remaining mildly stiff components. By applying stabilized Runge-Kutta methods to this modified equation, we then devise an explicit multirate Runge-Kutta-Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.

SIAM Journal on Numerical Analysis, 2018

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in het... more Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here optimal convergence rates are rigorously proved for the fully-discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.

Journal of Computational and Applied Mathematics, Feb 1, 2013

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

arXiv (Cornell University), Jul 30, 2021

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield sma... more Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L 2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

Robust parallel smoothing for multigrid via sparse approximate inverses

... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numer... more ... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numerical experiments show that SPAI-0 smoothing is usually preferable to damped Jacobi smooth-ing. ... and ح the upper triangular part of . Then damped Jacobi smoothing corresponds to ...

Am Rande des Unendlichen: Numerische Verfahren für unbegrenzte Gebiete

ABSTRACT

of the ordinary differential equation which occurs in the boundary condition. An exact nonreflect... more of the ordinary differential equation which occurs in the boundary condition. An exact nonreflecting boundary condition was derived previously for use with the time dependent wave equation in three Finally, we shall solve a sequence of scattering problems space dimensions. Here it is shown how to combine that boundary by using an explicit finite difference method and our condition with finite difference methods and finite element methboundary condition. We shall also solve the same problems ods. Uniqueness of the solution is proved, stability issues are disby using two of the standard artificial boundary conditions. cussed, and improvements are proposed for numerical computa-Comparison of these solutions with the ''exact'' solution, tion. Numerical examples are presented which demonstrate the improvement in accuracy over standard methods. ᮊ 1996 Academic obtained by computing in a very large domain so that Press, Inc. spurious reflections are postponed, shows that our boundary condition is much more accurate than the standard ones. Our boundary condition also has the advantage that

Springer eBooks, Jan 11, 2008

An exact nonre ecting boundary condition was derived previously for use with the time dependent M... more An exact nonre ecting boundary condition was derived previously for use with the time dependent Maxwell equations in three space dimensions 1. Here it is shown how to combine that boundary condition with the variational formulation for use with the nite element method. The fundamental theory underlying the derivation of the exact boundarycondition is reviewed. Numerical examples with the nite-di erence timedomain method are presented which demonstrate the high accuracy and con rm the expected rate of convergence of the numerical method.

Springer eBooks, 2003

A new far-field evaluation formula is pr esented which enables the efficient evaluation of the fa... more A new far-field evaluation formula is pr esented which enables the efficient evaluation of the far-field solution for wave propagation problems, if the exact nonreflecting boundary condition in [5, 6] is used for the numerical computation in the near-field. In particular, the evaluation formula permits to take advantage of the fast decay with distance of selected modes and thus to store only the minimal amount of information necessary from the past. The accuracy of this formula is illustrated via numerical experiments.

Effective Parallel Preconditioning with Sparse Approximate Inverses

PPSC, 1995

Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle*... more Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle* Abstract A parallel preconditioner is presented for the ... We also wish to thank DavidSilvester and Howard Elman who provided us with the matrices coming from incompressible ...

Explicit local time-stepping for transient electromagnetic waves

Multi-Level Runge-Kutta based Explicit Local Time-Stepping for Wave Propagation

FE-HMM for elastic waves in heterogeneous media

IAHS-AISH publication, 2017

Discontinuous Galerkin Methods and Local Time Stepping for Wave Propagation

ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, 2010

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a discontinuous Galerkin finite element discretization in space, which inherently leads to a diagonal mass matrix, the resulting numerical schemes are fully explicit. Starting from the classical Adams-Bashforth multi-step methods, local time stepping schemes of arbitrarily high accuracy are derived. Numerical experiments validate the theory and illustrate the usefulness of the proposed time integration schemes.

ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Sep 5, 2006

We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wa... more We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wave numbers. Our algebraic method is mainly based on using new multilevel incomplete LDL T techniques for symmetric indefinite systems.

De Gruyter eBooks, Sep 16, 2013

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discon... more Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leapfrog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting timemarching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.

Comptes Rendus Mathematique, Jun 1, 2013

Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'él... more Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'éléments finis multi-échelles pour l'équation des ondes dans des milieux hétérogènes sur des temps longs

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2006

An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in ... more An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in three space dimensions. It holds on a spherical surface B, outside of which the medium is assumed to be homogeneous, isotropic, and source-free. This boundary condition is local in space and time, and it does not involve high-order derivatives. Thus it is easy to incorporate into standard numerical methods. Numerical examples demonstrate the usefulness and high accuracy of this local nonreflecting boundary condition.

SIAM Journal on Scientific Computing, 2015

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for th... more Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.

Because of their robustness, efficiency and non-intrusiveness, Monte Carlo methods are probably t... more Because of their robustness, efficiency and non-intrusiveness, Monte Carlo methods are probably the most popular approach in uncertainty quantification to computing expected values of quantities of interest (QoIs). Multilevel Monte Carlo (MLMC) methods significantly reduce the computational cost by distributing the sampling across a hierarchy of discretizations and allocating most samples to the coarser grids. For time dependent problems, spatial coarsening typically entails an increased time-step. Geometric constraints, however, may impede uniform coarsening thereby forcing some elements to remain small across all levels. If explicit time-stepping is used, the time-step will then be dictated by the smallest element on each level for numerical stability. Hence, the increasingly stringent CFL condition on the time-step on coarser levels significantly reduces the advantages of the multilevel approach. To overcome that bottleneck we propose to combine the multilevel approach of MLMC with local time-stepping (LTS). By adapting the time-step to the locally refined elements on each level, the efficiency of MLMC methods is restored even in the presence of complex geometry without sacrificing the explicitness and inherent parallelism. In a careful cost comparison, we quantify the reduction in computational cost for local refinement either inside a small fixed region or towards a reentrant corner.

Stabilized Runge-Kutta (aka Chebyshev) methods are especially efficient for the numerical solutio... more Stabilized Runge-Kutta (aka Chebyshev) methods are especially efficient for the numerical solution of large systems of stiff differential equations because they are fully explicit; hence, they are inherently parallel and easily accommodate nonlinearity. For semi-discrete parabolic (or diffusion dominated) problems, for instance, stabilized Runge-Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when much of the stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depend on the remaining mildly stiff components. By applying stabilized Runge-Kutta methods to this modified equation, we then devise an explicit multirate Runge-Kutta-Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.

SIAM Journal on Numerical Analysis, 2018

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in het... more Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here optimal convergence rates are rigorously proved for the fully-discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.

Journal of Computational and Applied Mathematics, Feb 1, 2013

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

arXiv (Cornell University), Jul 30, 2021

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield sma... more Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L 2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

Robust parallel smoothing for multigrid via sparse approximate inverses

... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numer... more ... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numerical experiments show that SPAI-0 smoothing is usually preferable to damped Jacobi smooth-ing. ... and ح the upper triangular part of . Then damped Jacobi smoothing corresponds to ...

Am Rande des Unendlichen: Numerische Verfahren für unbegrenzte Gebiete

ABSTRACT

of the ordinary differential equation which occurs in the boundary condition. An exact nonreflect... more of the ordinary differential equation which occurs in the boundary condition. An exact nonreflecting boundary condition was derived previously for use with the time dependent wave equation in three Finally, we shall solve a sequence of scattering problems space dimensions. Here it is shown how to combine that boundary by using an explicit finite difference method and our condition with finite difference methods and finite element methboundary condition. We shall also solve the same problems ods. Uniqueness of the solution is proved, stability issues are disby using two of the standard artificial boundary conditions. cussed, and improvements are proposed for numerical computa-Comparison of these solutions with the ''exact'' solution, tion. Numerical examples are presented which demonstrate the improvement in accuracy over standard methods. ᮊ 1996 Academic obtained by computing in a very large domain so that Press, Inc. spurious reflections are postponed, shows that our boundary condition is much more accurate than the standard ones. Our boundary condition also has the advantage that

Springer eBooks, Jan 11, 2008

An exact nonre ecting boundary condition was derived previously for use with the time dependent M... more An exact nonre ecting boundary condition was derived previously for use with the time dependent Maxwell equations in three space dimensions 1. Here it is shown how to combine that boundary condition with the variational formulation for use with the nite element method. The fundamental theory underlying the derivation of the exact boundarycondition is reviewed. Numerical examples with the nite-di erence timedomain method are presented which demonstrate the high accuracy and con rm the expected rate of convergence of the numerical method.

Springer eBooks, 2003

A new far-field evaluation formula is pr esented which enables the efficient evaluation of the fa... more A new far-field evaluation formula is pr esented which enables the efficient evaluation of the far-field solution for wave propagation problems, if the exact nonreflecting boundary condition in [5, 6] is used for the numerical computation in the near-field. In particular, the evaluation formula permits to take advantage of the fast decay with distance of selected modes and thus to store only the minimal amount of information necessary from the past. The accuracy of this formula is illustrated via numerical experiments.

Effective Parallel Preconditioning with Sparse Approximate Inverses

PPSC, 1995

Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle*... more Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle* Abstract A parallel preconditioner is presented for the ... We also wish to thank DavidSilvester and Howard Elman who provided us with the matrices coming from incompressible ...

Explicit local time-stepping for transient electromagnetic waves

Multi-Level Runge-Kutta based Explicit Local Time-Stepping for Wave Propagation

FE-HMM for elastic waves in heterogeneous media

IAHS-AISH publication, 2017

Discontinuous Galerkin Methods and Local Time Stepping for Wave Propagation

ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, 2010

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a discontinuous Galerkin finite element discretization in space, which inherently leads to a diagonal mass matrix, the resulting numerical schemes are fully explicit. Starting from the classical Adams-Bashforth multi-step methods, local time stepping schemes of arbitrarily high accuracy are derived. Numerical experiments validate the theory and illustrate the usefulness of the proposed time integration schemes.

ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Sep 5, 2006

We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wa... more We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wave numbers. Our algebraic method is mainly based on using new multilevel incomplete LDL T techniques for symmetric indefinite systems.

De Gruyter eBooks, Sep 16, 2013

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discon... more Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leapfrog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting timemarching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.

Comptes Rendus Mathematique, Jun 1, 2013

Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'él... more Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'éléments finis multi-échelles pour l'équation des ondes dans des milieux hétérogènes sur des temps longs

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2006

An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in ... more An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in three space dimensions. It holds on a spherical surface B, outside of which the medium is assumed to be homogeneous, isotropic, and source-free. This boundary condition is local in space and time, and it does not involve high-order derivatives. Thus it is easy to incorporate into standard numerical methods. Numerical examples demonstrate the usefulness and high accuracy of this local nonreflecting boundary condition.

SIAM Journal on Scientific Computing, 2015

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for th... more Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.