Related papers: Nonlinear Preconditioning: How to use a Nonlinear …
A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz…
We investigate non-overlapping Schwarz preconditioners for the algebraic systems stemming from high-order discretizations of the coupled monodomain and Barreto-Cressman models, with applications to brain electrophysiology. The spatial…
The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially…
The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains…
In this paper we introduce a new Schwarz framework and theory, based on the well-known idea of space decomposition, for nonsymmetric and indefinite linear systems arising from continuous and discontinuous Galerkin approximations of general…
The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplace's equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We…
We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a…
Schwarz methods use a decomposition of the computational domain into subdomains and need to put boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and…
This report shows on real data that the direct methods such as LDL decomposition and Gaussian elimination for solving linear systems with ill-conditioned matrices provide inaccurate results due to divisions by very small numbers, which in…
We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to…
Vector extrapolation methods are widely used in large-scale simulation studies, and numerous extrapolation-based acceleration techniques have been developed to enhance the convergence of linear and nonlinear fixed-point iterative methods.…
In this paper we study a monolithic Newton-Krylov solver with exact Jacobian for the solution of incompressible FSI problems. A main focus of this work is on the use of geometric multigrid preconditioners with modified Richardson smoothers…
We study a natural alternating method of Schwarz type (domain decomposition) for certain class of couplings between local and nonlocal operators. We show that our method fits into Lion's framework and prove, as a consequence, convergence in…
The analysis of the acceleration behavior of gradient-based eigensolvers with preconditioning presents a substantial theoretical challenge. In this work, we present a novel framework for preconditioning on Riemannian manifolds and introduce…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
We study accelerated Krasnoselskii-Mann-type methods with preconditioners in both continuous and discrete time. From a continuous-time model, we derive a generalized fast Krasnoselskii-Mann method, providing a new yet simple proof of…
An iterative coupling algorithm based on restricted additive Schwarz domain decomposition is investigated to co-simulate electrical circuits with hybrid electromagnetic (EMT) and transient stability (TS) modeled using dynamic phasors. This…
We derive an extension of the sequential homotopy method that allows for the application of inexact solvers for the linear (double) saddle-point systems arising in the local semismooth Newton method for the homotopy subproblems. For the…
This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling to the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove…
In this paper, we propose an efficient method for solving multi-dimensional Riesz space fractional diffusion equations with variable coefficients. The Crank-Nicolson (CN) method is used for temporal discretization, while the fourth-order…