We extend the operator preconditioning framework [R. Hiptmair, Comput. Math. with Appl. 52 (2006), pp.~699--706] to Petrov-Galerkin methods while accounting for parameter-dependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner, our bi-parametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and super-linear convergence estimates for iterative solvers, such as $h$-independent convergence bounds, when preconditioning with low-accuracy or, equivalently, with highly compressed approximations.
Parametric Analytical Preconditioning and its Applications to the Reduced Collocation Methods
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Sorbonne Universités, UPMC Univ Paris 06（索邦大学，巴黎第六大学）；Institut Universitaire de France（法国高等研究院）；Brown University（布朗大学）
Using the framework of operator or Calderón preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) elements. The preconditioners are constructed as the composition of an opposite order operator, discretized on the same ansatz space, and two diagonal scaling operators.
The auxiliary space preconditioning (ASP) technique is applied to the HDG schemes for three different elliptic problems with a parameter dependent low order term, namely, a symmetric interior penalty HDG scheme for the scalar reaction-diffusion equation, a divergence-conforming HDG scheme for a vectorial reaction-diffusion equation, and a C 0 -continuous interior penalty HDG scheme for the generalized biharmonic equation with a low order term. Uniform preconditioners are obtained for each case and the general ASP theory by J. Xu [21] is used to prove the optimality with respect to the mesh siz…
Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator equations with a large number of discrete variables. In this paper we show that the accuracy can be improved significantly if the equation is transformed before discretization, a process we call full operator preconditioning (FOP). It bears many similarities with traditional preconditioning for iterative methods but, crucially, transformations are applied at t…
We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple trace formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes [Adv. Comp. Math. 37 (2012), 37-91] for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a nov…
This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix-vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evide…
Freezing Calderón: Robust Preconditioning for Maxwell Scattering under Shape Uncertainty
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学工程与科学学院）；Department of Mathematics and Digital Futures, KTH Royal Institute of Technology（瑞典皇家理工学院数学与数字未来系）；Inria Chile（法国国家信息与自动化研究所智利分部）
Stable Matrix-Free Q-less QR via Preconditioning and Randomization
University of California, Berkeley (UCB)（加州大学伯克利分校）；International Computer Science Institute (ICSI)（国际计算机科学研究所）；Lawrence Berkeley National Laboratory (LBNL)（劳伦斯伯克利国家实验室）
We investigate a range of techniques for the acceleration of Calderón (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve …
The Electric Field Integral Equation (EFIE) is a well-established tool to solve electromagnetic scattering problems. However, the development of efficient and easy to implement preconditioners remains an active research area. In recent years, operator preconditioning approaches have become popular for the EFIE, where the electric field boundary integral operator is regularised by multiplication with another convenient operator. A particularly intriguing choice is the exact Magnetic-to-Electric (MtE) operator as regulariser. But, evaluating this operator is as expensive as solving the original …
Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator
Institute of Computer Sciences, Czech Academy of Science（捷克科学院计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）；Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory（Simula研究实验室数值分析与科学计算部）；Faculty of Science and Technology, Norwegian University of Life Sciences（挪威生命科技大学理学院）
We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $Ω$, where $Ω$ is either in $\mathbb{R}^n$ or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkin discretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases for test and trial functions. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gi…
We present two minimum residual methods for solving sequences of shifted linear systems, the right-preconditioned shifted GMRES and shifted recycled GMRES algorithms which use a seed projection strategy often employed to solve multiple related problems. These methods are compatible with general preconditioning of all systems, and when restricted to right preconditioning, require no extra applications of the operator or preconditioner. These seed projection methods perform a minimum residual iteration for the base system while improving the approximations for the shifted systems at little addit…
In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a…
We propose a multi-level type operator that can be used in the framework of operator (or Caldéron) preconditioning to construct uniform preconditioners for negative order operators discretized by piecewise polynomials on a family of possibly locally refined partitions. The cost of applying this multi-level operator scales linearly in the number of mesh cells. Therefore, it provides a uniform preconditioner that can be applied in linear complexity when used within the preconditioning framework from our earlier work [Math. of Comp., 322(89) (2020), pp. 645-674].
Robust stability and preconditioning of Darcy-Forchheimer equations
Indian Institute of Technology Bombay（印度理工学院孟买分校）；BITS-Pilani, KK Birla Goa Campus（巴季学院-皮拉尼，KK维尔戈阿校区）；Monash University（蒙纳士大学）；Universidad Adventista de Chile（智利基督复临安息日会大学）
We propose a preconditioner for the Helmholtz exterior problems on multi-screens. For this, we combine quotient-space BEM and operator preconditioning. For a class of multi-screens (which we dub \emph{type A} multi-screens), we show that this approach leads to block diagonal Calderón preconditioners and results in a spectral condition number that grows only logarithmically with $h$, just as in the case of simple screens. Since the resulting scheme contains many more DoFs than strictly required, we also present strategies to remove almost all redundancy without significant loss of effectiveness…
The paper focuses on numerical solution of parametrized diffusion equations with scalar parameter-dependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical…
Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation
Folding grid value vectors of size $2^L$ into $L$th order tensors of mode sizes $2\times \cdots\times 2$, combined with low-rank representation in the tensor train format, has been shown to lead to highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying basis, such as piecewise multilinear finite elements on uniform tensor …
We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the diverge…
A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions
Computer Science Research Institute, Sandia National Laboratories（桑迪亚国家实验室计算机科学研究所）；Scientific Computing and Imaging (SCI) Institute and Mathematics Department, University of Utah（犹他大学科学计算与成像研究所及数学系）；Institute of Computational Mathematics and the Chinese Academy of Sciences（中国科学院计算数学研究所）
In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order-order optimality Karush-Kuhn-Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the sta…
The coupled Darcy-Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolitic solution algorithms of the coupled Darcy-Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers-Joseph-Saffman parameter. Our approach is similar …
Uniform Inf-Sup Norm Equivalence and Robust Operator Preconditioning for Stokes Flow in tight domains with Periodic Pillars
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳大数据研究院工业与应用数学国际中心）；SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学力学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Shenzhen Loop Area Institute（深圳河套学院）；Applied Mathematics and Computational Sciences, CEMSE Division, King Abdullah University of Science and Technology（阿卜杜拉国王科技大学CEMSE学院应用数学与计算科学系）
