We introduce in this paper a new tool to prove the convergence of the Overlapping Optimized Schwarz Methods with multisubdomains. The technique is based on some estimates of the errors on the boundaries of the overlapping strips. Our guiding example is an n-Dimensional Linear Parabolic Equation.
In this paper, we design and analyze two new methods based on additive average Schwarz -- AAS method introduced in \cite{MR1943457}. The new methods design for elliptic problems with highly heterogeneous coefficients. The methods are of the non-overlapping type, and the subdomain interactions obtain via the coarse space. The first method is the minimum energy Schwarz -- MES method. MES has the minimum energy for the coarse space with constant extension inside each subdomain. The condition number of the MES method is always smaller than in the AAS method. The second class of methods is the non-…
Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources
Domain decomposition methods are widely used for the numerical solution of partial differential equations on high performance computers. We develop an adjoint-based a posteriori error analysis for both multiplicative and additive overlapping Schwarz domain decomposition methods. The numerical error in a user-specified functional of the solution (quantity of interest) is decomposed into contributions that arise as a result of the finite iteration between the subdomains and from the spatial discretization. The spatial discretization contribution is further decomposed into contributions arising f…
In this paper we discuss the convergence of state-of-the-art optimized Schwarz transmission conditions for Helmholtz problems defined on closed domains (i.e. setups which do not exhibit an outgoing wave condition), as commonly encountered when modeling cavities. In particular, the impact of back-propagating waves on the Dirichlet-to-Neumann map is analyzed. Afterwards, the performance of the well-established optimized 0th-order, evanescent modes damping, optimized 2nd-order and Padé-localized square-root transmission conditions is discussed.
Overlapping Schwarz Preconditioners for Pose-Graph SLAM in Robotics
Faculty of Mathematics and Computer Science, Technische Universität Bergakademie Freiberg（弗莱贝格矿业工业大学数学与计算机科学学院）；Universitätsrechenzentrum (URZ), Technische Universität Bergakademie Freiberg（弗莱贝格矿业大学计算中心）
The Schwarz Waveform Relaxation algorithm (SWR) exchanges the waveform of boundary value between neighbouring sub-domains, which provides a more efficient way than the other Schwarz algorithms to realize distributed computation. However, the convergence speed of the traditional SWR is slow, and various optimization strategies have been brought in to accelerate the convergence. In this paper, we propose a non-iterative overlapping variant of SWR for wave equation, which is named Relative Schwarz Waveform Relaxation algorithm (RSWR). RSWR is inspired by the physical observation that the velocity…
In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on t…
We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components' interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component -- associated with the solution on interior boundaries -- and a bubble component that vanishes at ports: since the bubble…
Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier-Stokes Equations
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；Department of Mathematics and Computer Science, University of Cologne（科隆大学数学与计算机科学系）；Center for Data and Simulation Science, University of Cologne（科隆大学数据与模拟科学中心）
We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction to obtain symmetric matrices, which are typically better suited for iterative solvers. The MPI-parallel implementation uses the software libraries deal.II, p4est and FROSch (Fast of Robust Overlapping Schwarz). FROSch is part of the Trilinos library…
The research of two-level overlapping Schwarz (TL-OS) method based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g. mainly for second order elliptic problem and too heavy computational cost of coarse space construction. In this paper, by introducing appropriate assumptions, we propose more concise coarse basis functions for general Hermitian positive and definite discrete systems, and establish the algorithmic and theoretical frameworks of the corresponding TL-OS methods. Furthermore, to enhance the practicability of the algorithm, we de…
An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）；Department of Geoscience and Engineering, Delft University of Technology（地球科学与工程学院，代尔夫特理工大学）
Non-overlapping Schwarz methods with generalized Robin transmission conditions were originally introduced by B. Després for time-harmonic wave propagation problems and have largely developed over the past thirty years. The aim of the paper is to provide both a review of the available formulations and methods as well as a consistent theory applicable to more general cases than studied until to date. An abstract variational framework is provided reformulating the original problem by the well-known form involving a scattering operator and an interface exchange operator, and the equivalence betwee…
In this paper, we propose an overlapping additive Schwarz method for total variation minimization based on a dual formulation. The $O(1/n)$-energy convergence of the proposed method is proven, where $n$ is the number of iterations. In addition, we introduce an interesting convergence property called pseudo-linear convergence of the proposed method; the energy of the proposed method decreases as fast as linearly convergent algorithms until it reaches a particular value. It is shown that such the particular value depends on the overlapping width $δ$, and the proposed method becomes as efficient …
Convergence of a discrete-in-time Approximation to a Degenerate Parabolic-Hyperbolic System
Joanneum Research Forschungsgesellschaft mbH（约阿内姆研究有限公司）；Faculty of Advanced Science and Technology, Ryukoku University（龙谷大学先进科学技术学部）；Institute of Scientific Computing, Faculty of Mathematics, TUD Dresden University of Technology（德累斯顿工业大学数学学院科学计算研究所）
We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.
We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an app…
Two-level convergence of Algebraic Multigrid with Overlapping Smoothers and Spectral Coarse Grids
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Science and Technology Facilities Council, Scientific Computing Department, Rutherford Appleton Laboratory（英国科学与技术设施委员会，科学计算部，卢瑟福阿普尔顿实验室）
In this study we present a non-overlapping Schwarz waveform relaxation (SWR) method applied to a one dimensional model problem representative of the coupling between the ocean and the atmosphere. This problem includes nonlinear interface conditions analogous to a quadratic friction law. We study the convergence of the corresponding SWR at a semi-discrete level for a linear friction and for a linearized quadratic friction at the interface. Using numerical experiments we show that the convergence properties in the linearized quadratic friction case are very close to the ones obtained with the fu…
Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems
University College London（伦敦大学学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；University of Bath（巴斯大学）；CNRS and Institut de Mathématiques de Toulouse, UMR5219（法国国家科学研究中心与图卢兹数学研究所；图卢兹大学）；Université de Toulouse
In this work, a class of parabolic economic optimal control problems is considered. These problems are characterized by pointwise state constraints regularized by a parameter, which transforms the pure state constraints in mixed control-state ones. However, the convergence of classical (semismooth) Newton methods deteriorates for decreasing values of the regularization parameter. To tackle this problem, a nonlinear preconditioner is introduced. This is based on an overlapping optimized waveform-relaxation method characterized by Robin transmission conditions. Numerical experiments show that ap…
