Schwarz Method Research Articles

Additive Schwarz Methods for Fourth-Order Variational Inequalities

Additive Schwarz Methods for Fourth-Order Variational Inequalities

On the linear convergence of additive Schwarz methods for the p-Laplacian

Abstract We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. First, we present a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behaviour akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincaré–Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements we establish the asymptotic linear convergence of additive Schwarz methods for the $p$-Laplacian.

On a sharp estimate of overlapping Schwarz methods in H(curl; Ω) and H(div; Ω)

Abstract The previously proved upper bound is $C\big(1+\frac{H^{2}}{\delta ^{2}}\big)$ for the condition number of the discrete system arising from the classical two-level overlapping Schwarz method for symmetric positive definite problems in $\boldsymbol{H}(\textbf{curl}; \varOmega )$ and $\boldsymbol{H}(\textrm{div}; \varOmega )$, where $H$ and $\delta $ are the sizes of subdomains and overlap, respectively. But all numerical results indicate that a sharper bound is $C\big (1+ \frac{H}{\delta } \big )$. In this work, we prove that the condition number is indeed bounded above by $C\big(1+\frac{H}{\delta }\big)$.

Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

Application of the localized method of fundamental solutions to heat transfer problems

A localized version of the Method of Fundamental Solutions is applied to the 2D steady heat transfer equation with spatially varying thermal conductivity. Though the corresponding fundamental solution cannot be computed in general, the localization splits the original problem into several subproblems defined on small subdomains; in these subdomains, the subproblems are approximately converted to certain convection-diffusion equations with constant coefficients, so that the Method of Fundamental Solutions is applicable. In each subdomain, the corresponding subproblem is solved separately. This results in an iterative method, which mimics the overlapping (alternating) Schwarz method. Due to its advantageous numerical properties, the technique seems a useful generalization of the Method of Fundamental Solutions. The method is illustrated through numerical examples.

Accurate implementation of two-level asynchronous domain decomposition solvers

Recently, asynchronous coarse-grid correction has been achieved within additive Schwarz-type and primal Schur domain decomposition frameworks. Both additive and multiplicative coarse-grid corrections were discussed, however, the implemented asynchronous Schwarz-type solver with additive correction relies on the specific design of the restricted additive Schwarz (RAS) method, and also requires an overlap between the subsets of unknowns. In this paper, we first highlight a gap between the theoretical analysis from the literature and the associated RAS implementation. It turns out that communications delays would actually need to be bounded in order to fit the theory. This has to be stressed since, despite the asynchronous nature of the solver, the coarse-grid correction requires non-blocking global synchronization, which is subject to communications bottleneck. Second, we propose an implementation approach which applies to a wider class of additive Schwarz-type methods while still coping with the bounded delays requirement.

Accelerating multivariate functional approximation computation with domain decomposition techniques

Modeling large datasets through Multivariate Functional Approximations (MFA) provide an elegant way to handle many visualization and scientific analysis workflows. The process necessitates scalable data partitioning methods to compute MFA representations efficiently without compromising the accuracy or continuity of the reconstructed solution. We propose a domain-decomposed method for computing the MFA with B-spline bases, which reduces the total work per task and uses a restricted Additive Schwarz (RAS) method to converge the control point data degrees-of-freedom along subdomain boundaries. We provide an in-depth analysis of the parallel approach with domain decomposition solvers, aiming to minimize local subdomain error residuals and recover high-order continuity at subdomain interfaces with appropriate choices of knot overlaps. The communication cost, determined by the overlap regions in the RAS implementation, is optimized to recover the numerical error profile of the single subdomain case. Our proposed method stands in contrast to previous methods, which typically only recover either C0 or at best C1 continuity for arbitrary B-spline degree expansions, or those that require post-processing to blend discontinuities in the reconstructed data. We demonstrate the effectiveness of our approach using analytical and real-world datasets in 1D, 2D, and 3D through both strong and weak scaling studies. The performance results indicate that the overall cost of computing the approximation is directly proportional to the underlying nearest-neighbor communication implementation, and is only weakly dependent on the overlap region size that determines the size of the messages. This finding underscores the efficiency and scalability of our proposed method, making it a promising solution for handling large datasets in scientific workflows.

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition, an accelerated Schwarz framework, and two-layer neural networks to approximate the boundary-to-boundary map for the subdomains, which is the key step in the Schwarz procedure. Conventionally, the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain. By leveraging the compressibility of multiscale problems, our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map. Our method is applied to a multiscale semilinear elliptic equation and a multiscale $p$-Laplace equation. In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.

A family of Nonoverlapping Spectral Additive Schwarz methods (NOSAS) and their economic versions

NOSAS (Nonoverlapping Spectral Additive Schwarz) methods are nonoverlapping iterative domain decomposition methods for efficiently solving large sparse linear systems arising from elliptic problems with heterogeneous coefficients. NOSAS methods are of additive Schwarz types with local Dirichlet solver on each nonoverlapping subdomain Ωi, plus a coarse problem on the interface of the subdomains. The coarse problem has global and local interaction components which are associated respectively to low-frequency and high-frequency modes obtained locally from a generalized eigenvalue problem defined on each subdomain’s interface. In this paper, we propose several variants of the NOSAS methods by selecting different generalized eigenvalue problems and different coarse space extensions. Additionally, we propose an economic variant to reduce the complexity of these generalized eigenvalue problems. The analysis of all the variants is established, and numerical tests are provided.

On the convergence of overlapping and non-overlapping Schwarz methods for the Cahn–Hilliard equation

In this study, we develop and investigate the Schwarz method with or without overlap for the Cahn–Hilliard (CH) problem. The CH equation has a wide range of applications, hence it is crucial to develop effective numerical techniques. In this article, we provide the formulation of the classical Schwarz methods (CSM) and optimized Schwarz methods (OSM) for the CH equation and present the convergence behaviour for two as well as multiple subdomain settings. We also formulate the nonlinear variant of the Schwarz and quasi-optimized Schwarz techniques for the same. We provide numerical results to support our conclusions.

Cache-optimized and low-overhead implementations of additive Schwarz methods for high-order FEM multigrid computations

This contribution presents data-locality optimizations of the additive Schwarz method (ASM) based on the fast-diagonalization method defined on overlapping cell-centric and vertex-star patches in the context of high-order matrix-free finite-element computations on modern CPU-based hardware. The developments are guided by detailed performance models of the ASM in the context of Chebyshev iterations when used as smoothers for p-multigrid. The proposed efficient implementation of ASM adopts concepts known from cell-loop infrastructures for efficient operator evaluation, in particular, the storage of information per geometric entity and the cache-friendly interleaving of cell loops and vector updates as a means to increase data locality. We use the latter concept for both applying the weights required by ASM and performing the vector updates required by the Chebyshev iteration, which are memory-bound operations with non-negligible costs in comparison to efficient operator evaluation. Furthermore, the solution of a scalar Poisson problem on a highly anisotropic and an unstructured mesh with p-multigrid using the developed smoothers indicates that efficient implementations of the additive Schwarz method can outperform optimized point-Jacobi preconditioners already for simple problems despite being more than twice as expensive per iteration. Even though ASM introduces additional communication steps per smoother application, the reduced number of iterations can lead to improved parallel scalability for intermediate problem sizes. At the scaling limit, the results are inconclusive due to these two opposing effects.

NonLocal Optimized Schwarz Method for the Helmholtz Equation with Physical Boundaries

NonLocal Optimized Schwarz Method for the Helmholtz Equation with Physical Boundaries

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

Nonlinear parallel-in-time simulations of multiphase flow in porous media

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution.

An aneurysm-specific preconditioning technique for the acceleration of Newton-Krylov method with application in the simulation of blood flows.

In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two-level additive Schwarz method with a mixed-dimensional coarse preconditioner. The key components of the preconditioner include (1) a three-dimensional coarse preconditioner covering the aneurysm; (2) a one-dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three-dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier-Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second-order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton-Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient-specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

Optimized Schwarz methods for the time-dependent Stokes–Darcy coupling

Abstract This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results validate the proposed approach in several test cases with physically relevant parameters.

A Unified Theory of Non-overlapping Robin–Schwarz Methods: Continuous and Discrete, Including Cross Points

A Unified Theory of Non-overlapping Robin–Schwarz Methods: Continuous and Discrete, Including Cross Points

Why Fourier mode analysis in time is different when studying Schwarz Waveform Relaxation

Schwarz waveform relaxation methods are Schwarz methods applied to evolution problems. Like for steady problems, they are based on an overlapping domain decomposition of the spatial domain, and an iteration which only requires subdomain solutions, now in space-time, to get better and better approximations of the global, monodomain solution. Fourier analysis has been used to study the convergence of both Schwarz and Schwarz waveform relaxation methods. We show here that their convergence is however quite different: for steady problems of diffusive type, Schwarz methods converge linearly, which is also well predicted by Fourier analysis. For a time dependent heat equation however, the Schwarz waveform relaxation algorithm first has a rapid convergence phase, followed by a slow down, and eventually convergence increases again to become superlinear, none of which is predicted by classical Fourier analysis. Introducing a new Fourier analysis combined with kernel estimates, we can explain this behavior for the heat equation. We then generalize our approach to the case of advection reaction diffusion problems. We illustrate all our results with numerical experiments.

Additive Schwarz Methods for Serendipity Elements

Additive Schwarz Methods for Serendipity Elements