Marco Donatelli | Università degli Studi dell'Insubria (original) (raw)

Papers by Marco Donatelli

Research paper thumbnail of Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem

Computers & mathematics with applications, Sep 1, 2017

We consider the solution of block-coupled large-scale linear systems of equations, arising from t... more We consider the solution of block-coupled large-scale linear systems of equations, arising from the finite element approximation of the linear elasticity problem. Due to the large scale of the problems we use properly preconditioned iterative methods, where the preconditioners utilize the underlying block matrix structures, involving inner block solvers and, when suited, broadly established tools such as the algebraic Multigrid method (AMG). For the considered problem, despite of its optimal rate of convergence, AMG, as implemented in some publicly available scientific libraries, imposes unacceptably high demands for computer resources. In this paper we propose and analyze an efficient multilevel preconditioner, based on the Generalized Locally Toeplitz framework, with a specialized transfer operator. We prove and numerically illustrate the optimal convergence rate of the proposed preconditioner, and experimentally report memory and CPU time savings. We also provide comparisons with respect to another aggregation-based algebraic multigrid algorithm.

Research paper thumbnail of An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Inverse Problems, Sep 1, 2020

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least s... more In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

Research paper thumbnail of Symbol based convergence analysis in multigrid methods for saddle point problems

arXiv (Cornell University), Mar 11, 2022

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations... more Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W-or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

Research paper thumbnail of Fractional graph Laplacian for image reconstruction

Applied Numerical Mathematics, May 1, 2023

Research paper thumbnail of A reduced-order model based on cubic B-spline basis function and SSP Runge–Kutta procedure to investigate option pricing under jump-diffusion models

Engineering Analysis With Boundary Elements, May 1, 2023

Research paper thumbnail of An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

arXiv (Cornell University), Dec 31, 2019

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least s... more In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

Research paper thumbnail of Iterated Tikhonov regularization with a general penalty term

Numerical Linear Algebra With Applications, Apr 11, 2017

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-pose... more Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice of regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance this method.

Research paper thumbnail of Analysis of level-dependent subdivision schemes near extraordinary vertices and faces

arXiv (Cornell University), Jul 6, 2017

Convergence and smoothness analysis of a bivariate level-dependent (non - stationary) subdivision... more Convergence and smoothness analysis of a bivariate level-dependent (non - stationary) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. In this paper we focus on the problem of analyzing convergence and tangent plane continuity - also known as G1G^1G1-continuity - of non-stationary subdivision schemes. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, we derive new sufficient conditions for establishing G1G^1G1-continuity of any non-stationary subdivision surface at the limit points of extraordinary vertices and/or extraordinary faces.

Research paper thumbnail of Surface subdivision algorithms and structured linear algebra : A computational approach to determine bounds of extraordinary rule weights

Surface subdivision algorithms and structured linear algebra : A computational approach to determ... more Surface subdivision algorithms and structured linear algebra : A computational approach to determine bounds of extraordinary rule weights

Research paper thumbnail of Iterated fractional Tikhonov regularization

Proceedings in applied mathematics & mechanics, Oct 1, 2015

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothin... more Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.

Research paper thumbnail of Existence of multiple solutions for a fourth-order problem with variable exponent

Discrete and Continuous Dynamical Systems-series B, 2022

We provide a new multiplicity result for a weighted p(x)-biharmonic problem on a bounded domain Ω... more We provide a new multiplicity result for a weighted p(x)-biharmonic problem on a bounded domain Ω of R n with Navier conditions on ∂Ω. Our approach, of variational nature, requires a suitable oscillating behavior of the nonlinearity and the associated weight to be compactly supported in Ω.

Research paper thumbnail of Spectral Analysis of Matrices in B-Spline Galerkin Methods for Riesz Fractional Equations

Springer INdAM series, Oct 29, 2022

Research paper thumbnail of Graph Laplacian and Neural Networks for Inverse Problems in Imaging: GraphLaNet

Research paper thumbnail of Symmetrization Techniques in Image Deblurring

arXiv (Cornell University), Dec 12, 2022

This paper presents a couple of preconditioning techniques that can be used to enhance the perfor... more This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditiong techniques, including when considering the deblurring of sparse images.

Research paper thumbnail of Convergence and normal continuity analysis of non-stationary subdivision schemes near extraordinary vertices and faces

arXiv (Cornell University), Jul 6, 2017

Convergence and normal continuity analysis of a bivariate non-stationary (leveldependent) subdivi... more Convergence and normal continuity analysis of a bivariate non-stationary (leveldependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.

Research paper thumbnail of Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

arXiv (Cornell University), Sep 19, 2022

It is known that the solution of a conservative steady-state two-sided fractional diffusion probl... more It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic nonuniform mesh. We focus on grids mapped by a smooth function which consist in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix-vector product and a detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability. Keywords two-sided fractional problems • finite volume elements methods • Toeplitz matrices • spectral distribution • multigrid methods

Research paper thumbnail of Image deblurring by sparsity constraint on the Fourier coefficients

Numerical Algorithms, Sep 23, 2015

Research paper thumbnail of Grid transfer operators for multigrid methods

Proceedings in applied mathematics & mechanics, Dec 1, 2011

Research paper thumbnail of Symbol based convergence analysis in multigrid methods for saddle point problems

Linear Algebra and its Applications, Aug 1, 2023

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations... more Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W-or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

Research paper thumbnail of Multigrid preconditioners for anisotropic space-fractional diffusion equations

Advances in Computational Mathematics, Jun 1, 2020

We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space.... more We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded-banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.

Research paper thumbnail of Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem

Computers & mathematics with applications, Sep 1, 2017

We consider the solution of block-coupled large-scale linear systems of equations, arising from t... more We consider the solution of block-coupled large-scale linear systems of equations, arising from the finite element approximation of the linear elasticity problem. Due to the large scale of the problems we use properly preconditioned iterative methods, where the preconditioners utilize the underlying block matrix structures, involving inner block solvers and, when suited, broadly established tools such as the algebraic Multigrid method (AMG). For the considered problem, despite of its optimal rate of convergence, AMG, as implemented in some publicly available scientific libraries, imposes unacceptably high demands for computer resources. In this paper we propose and analyze an efficient multilevel preconditioner, based on the Generalized Locally Toeplitz framework, with a specialized transfer operator. We prove and numerically illustrate the optimal convergence rate of the proposed preconditioner, and experimentally report memory and CPU time savings. We also provide comparisons with respect to another aggregation-based algebraic multigrid algorithm.

Research paper thumbnail of An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Inverse Problems, Sep 1, 2020

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least s... more In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

Research paper thumbnail of Symbol based convergence analysis in multigrid methods for saddle point problems

arXiv (Cornell University), Mar 11, 2022

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations... more Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W-or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

Research paper thumbnail of Fractional graph Laplacian for image reconstruction

Applied Numerical Mathematics, May 1, 2023

Research paper thumbnail of A reduced-order model based on cubic B-spline basis function and SSP Runge–Kutta procedure to investigate option pricing under jump-diffusion models

Engineering Analysis With Boundary Elements, May 1, 2023

Research paper thumbnail of An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

arXiv (Cornell University), Dec 31, 2019

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least s... more In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

Research paper thumbnail of Iterated Tikhonov regularization with a general penalty term

Numerical Linear Algebra With Applications, Apr 11, 2017

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-pose... more Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice of regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance this method.

Research paper thumbnail of Analysis of level-dependent subdivision schemes near extraordinary vertices and faces

arXiv (Cornell University), Jul 6, 2017

Convergence and smoothness analysis of a bivariate level-dependent (non - stationary) subdivision... more Convergence and smoothness analysis of a bivariate level-dependent (non - stationary) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. In this paper we focus on the problem of analyzing convergence and tangent plane continuity - also known as G1G^1G1-continuity - of non-stationary subdivision schemes. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, we derive new sufficient conditions for establishing G1G^1G1-continuity of any non-stationary subdivision surface at the limit points of extraordinary vertices and/or extraordinary faces.

Research paper thumbnail of Surface subdivision algorithms and structured linear algebra : A computational approach to determine bounds of extraordinary rule weights

Surface subdivision algorithms and structured linear algebra : A computational approach to determ... more Surface subdivision algorithms and structured linear algebra : A computational approach to determine bounds of extraordinary rule weights

Research paper thumbnail of Iterated fractional Tikhonov regularization

Proceedings in applied mathematics & mechanics, Oct 1, 2015

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothin... more Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.

Research paper thumbnail of Existence of multiple solutions for a fourth-order problem with variable exponent

Discrete and Continuous Dynamical Systems-series B, 2022

We provide a new multiplicity result for a weighted p(x)-biharmonic problem on a bounded domain Ω... more We provide a new multiplicity result for a weighted p(x)-biharmonic problem on a bounded domain Ω of R n with Navier conditions on ∂Ω. Our approach, of variational nature, requires a suitable oscillating behavior of the nonlinearity and the associated weight to be compactly supported in Ω.

Research paper thumbnail of Spectral Analysis of Matrices in B-Spline Galerkin Methods for Riesz Fractional Equations

Springer INdAM series, Oct 29, 2022

Research paper thumbnail of Graph Laplacian and Neural Networks for Inverse Problems in Imaging: GraphLaNet

Research paper thumbnail of Symmetrization Techniques in Image Deblurring

arXiv (Cornell University), Dec 12, 2022

This paper presents a couple of preconditioning techniques that can be used to enhance the perfor... more This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditiong techniques, including when considering the deblurring of sparse images.

Research paper thumbnail of Convergence and normal continuity analysis of non-stationary subdivision schemes near extraordinary vertices and faces

arXiv (Cornell University), Jul 6, 2017

Convergence and normal continuity analysis of a bivariate non-stationary (leveldependent) subdivi... more Convergence and normal continuity analysis of a bivariate non-stationary (leveldependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.

Research paper thumbnail of Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

arXiv (Cornell University), Sep 19, 2022

It is known that the solution of a conservative steady-state two-sided fractional diffusion probl... more It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic nonuniform mesh. We focus on grids mapped by a smooth function which consist in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix-vector product and a detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability. Keywords two-sided fractional problems • finite volume elements methods • Toeplitz matrices • spectral distribution • multigrid methods

Research paper thumbnail of Image deblurring by sparsity constraint on the Fourier coefficients

Numerical Algorithms, Sep 23, 2015

Research paper thumbnail of Grid transfer operators for multigrid methods

Proceedings in applied mathematics & mechanics, Dec 1, 2011

Research paper thumbnail of Symbol based convergence analysis in multigrid methods for saddle point problems

Linear Algebra and its Applications, Aug 1, 2023

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations... more Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W-or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

Research paper thumbnail of Multigrid preconditioners for anisotropic space-fractional diffusion equations

Advances in Computational Mathematics, Jun 1, 2020

We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space.... more We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded-banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.