Mathematical models for the description, in a quantitative way, of the damages induced on the monuments by the action of specific pollutants are often systems of nonlinear, possibly degenerate, parabolic equations. Although some the asymptotic properties of the solutions are known, for a short window of time, one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest. In this paper a fully implicit numerical method is proposed, analyzed and numerically tested for parabolic equations of porous media type and on a systems of two PDEs that models the sulfation of marble in monuments. Due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures. Numerical experiments for the validation of the analysis complement this contribution.
We develop a theoretical foundation for the application of Nesterov's accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requ…
We propose a preconditioner to accelerate the convergence of the GMRES iterative method for solving the system of linear equations obtained from discretize-then-optimize approach applied to optimal control problems constrained by a partial differential equation. Eigenvalue distribution of the preconditioned matrix as well as its eigenvectors are discussed. Numerical results of the proposed preconditioner are compared with several existing preconditioners to show its efficiency.
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…
Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity
Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions, LJLL（索邦大学，巴黎西岱大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育与研究学院特里凡得琅数学学院）
Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity
Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions（索邦大学，巴黎 Cité 大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method
Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（法国国家信息与自动化研究院南锡-大东部中心）；Inria Paris（法国国家信息与自动化研究院巴黎中心）；Université Paris-Est, CERMICS (ENPC)（巴黎东大学，CERMICS（国立路桥学校））；Laboratoire de Mathématiques Raphaël Salem, Université de Rouen（鲁昂大学拉斐尔·萨莱姆数学实验室）
Preconditioning PDE-constrained optimization with $\rm L^1$-sparsity and control constraints
Università degli Studi di Firenze（佛罗伦萨大学）；Università di Bologna（博洛尼亚大学）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）
This paper presents a novel numerical method for the Newton seismic full-waveform inversion (FWI). The method is based on the full-space approach, where the state, adjoint state, and control variables are optimized simultaneously. Each Newton step is formulated as a PDE-constrained optimization problem, which is cast in the form of the Karush-Kuhn-Tucker (KKT) system of linear algebraic equitations. The KKT system is solved inexactly with a preconditioned Krylov solver. We introduced two preconditioners: the one based on the block-triangular factorization and its variant with an inexact block …
Numerical study in stochastic homogenization for elliptic PDEs: convergence rate in the size of representative volume elements
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）
In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, including the heat equation and the non-steady convection--diffusion equation. After applying an optimize-then-discretize approach, one is faced with continuous first-order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first-order accurate) backward Euler method is used for the discretization of the time…
The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently develo…
We consider parametric families of partial differential equations--PDEs where the parameter $κ$ modifies only the (1,1) block of a saddle point matrix product of a discretization below. The main goal is to develop an algorithm that removes, as much as possible, the dependence of iterative solvers on the parameter $κ$. The algorithm we propose requires only one matrix factorization which does not depend on $κ$, therefore, allows to reuse it for solving very fast a large number of discrete PDEs for different $κ$ and forcing terms. The design of the proposed algorithm is motivated by previous wor…
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…
Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are deve…
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…
Optimal solvers for fourth-order PDEs discretized on unstructured grids
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University（宾夕法尼亚州立大学数学系计算数学与应用中心）
The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the $p$-step BDF ($p\geq 2$) is not selfstarting, when they are exploited to solve time-dependent PDEs. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractio…
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…
