This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient method, preconditioned with hierarchical interpolative factorization. Stiffness matrices arising in the discretization of parabolic equations typically have large condition numbers, and therefore preconditioning becomes essential, especially for large time steps. We propose to use the hierarchical interpolative factorization as the preconditioning for the conjugate gradient iteration. Computed only once, the hierarchical interpolative factorization offers an efficient and accurate approximate inverse of the linear system. As a result, the preconditioned conjugate gradient iteration converges in a small number of iterations. Compared to other classical exact and approximate factorizations such as Cholesky or incomplete Cholesky, the hierarchical interpolative factorization can be computed in linear time and the application of its inverse has linear complexity. Numerical experiments demonstrate the performance of the method and the reduction of conjugate gradient iterations.
This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. H-IDBF-LU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iter…
A new preconditioner based on a block $LDU$ factorization with algebraic multigrid subsolves for scalability is introduced for the large, structured systems appearing in implicit Runge-Kutta time integration of parabolic partial differential equations. This preconditioner is compared in condition number and eigenvalue distribution, and in numerical experiments with others in the literature: block Jacobi, block Gauss-Seidel, and the optimized block Gauss-Seidel method of Staff, Mardal, and Nilssen [{\em Modeling, Identification and Control}, 27 (2006), pp. 109-123]. Experiments are run on two t…
We generalize previous work by Mardal, Nilssen, and Staff (2007, SIAM J. Sci. Comp. v. 29, pp. 361-375) and Rana, Howle, Long, Meek, and Milestone (2021, SIAM J. Sci. Comp. v. 43, p. 475-495) on order-optimal preconditioners for parabolic PDEs to a larger class of differential equations and methods. The problems considered are those of the forms $u_{t}=-\mathcal{K}u+g$ and $u_{tt}=-\mathcal{K}u+g$, where the operator $\mathcal{K}$ is defined by $\mathcal{K}u:=-\nabla\cdot\left(α\nabla u\right)+βu$ and the functions $α$ and $β$ are restricted so that $α>0$, and $β\ge0$. The methods considered a…
Fast multilevel solvers for a class of discrete fourth order parabolic problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Southwest Jiaotong University（西南交通大学）；Tufts University（塔夫茨大学）；University of California, Irvine（加州大学尔湾分校）；University of Maryland（马里兰大学）；Pennsylvania State University（宾夕法尼亚州立大学）
In this work we focus on the preconditioning of a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By i…
Runge-Kutta (RK) schemes, especially Gauss-Legendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their A-stability and high-order accuracy. However, it is significantly more challenging to construct optimal preconditioners for them compared to diagonally implicit RK (or DIRK) schemes. To address this challenge, we first introduce mathematically optimal preconditioners called block complex Schur decomposition (BCSD), block real Schur decomposition (BRSD), and block Jordan form (BJF), motivated by block…
Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
Johannes Kepler University（约翰内斯·开普勒大学）；Institute of Computational Mathematics, Johannes Kepler University（约翰内斯·开普勒大学计算数学研究所）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）
The DD-CPM software library provides a set of tools for the discretization and solution of problems arising from the closest point method (CPM) for partial differential equations on surfaces. The solvers are built on top of the well-known PETSc framework, and are supplemented by custom domain decomposition (DD) preconditioners specific to the CPM. These solvers are fully compatible with distributed memory parallelism through MPI. This library is particularly well suited to the solution of elliptic and parabolic equations, including many reaction-diffusion equations. The software is detailed he…
This paper introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a hierarchical matrix representation is constructed for the hermitian matrix arising from composing the Fourier integral operator with its adjoint. This representation is inverted efficiently with a new algorithm based on the hierarchical interpolative factorization. By combining these two factorizations, an approximate inverse factorization for the Fourier integral o…
Time-parallel iterative solvers for parabolic evolution equations
Institute of Computational Mathematics, Johannes Kepler University Linz（林茨约翰·开普勒大学计算数学研究所）；Department of Mathematics, University College London（伦敦大学学院数学系）
In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time re…
The Sinc-Nyström method is a high-order numerical method based on Sinc basis functions for discretizing evolutionary differential equations in time. But in this method we have to solve all the time steps in one-shot (i.e. all-at-once), which results in a large-scale nonsymmetric dense system that is expensive to handle. In this paper, we propose and analyze preconditioner for such dense system arising from both the parabolic and hyperbolic PDEs. The proposed preconditioner is a low-rank perturbation of the original matrix and has two advantages. First, we show that the eigenvalues of the preco…
We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system of linear equations obtained by using finite element discretization of a distributed optimal control problem together with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solvin…
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states wi…
Fully implicit Runge-Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but high-order IRK methods are not commonly used in practice with numerical PDEs due to the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic preconditioning framework for solving the systems of equations that arise from IRK methods applied to linear numerical PDEs (without algebraic constraints). This framework also naturally applies to discontinuous Galerkin discretizations in time. Under quite general assumptions…
