相关论文: Preconditioned normal equations for solving discre…
Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential equations and second extend the previous results for linear…
We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element…
Solving systems of linear equations is a problem occuring frequently in water engineering applications. Usually the size of the problem is too large to be solved via direct factorization. One can resort to iterative approaches, in…
Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer…
Despite hundreds of papers on preconditioned linear systems of equations, there remains a significant lack of comprehensive performance benchmarks comparing various preconditioners for solving symmetric positive definite (SPD) systems. In…
This work considers the iterative solution of large-scale problems subject to non-symmetric matrices or operators arising in discretizations of (port-)Hamiltonian partial differential equations. We consider problems governed by an operator…
This work introduces nodal auxiliary space preconditioners for discretizations of mixed-dimensional partial differential equations. We first consider the continuous setting and generalize the regular decomposition to this setting. With the…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
The numerical simulation and optimization of technical systems described by partial differential equations is expensive, especially in multi-query scenarios in which the underlying equations have to be solved for different parameters. A…
A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized…
Motivated by a wide range of real-world problems whose solutions exhibit boundary and interior layers, the numerical analysis of discretizations of singularly perturbed differential equations is an established sub-discipline within the…
Least squares method is one of the simplest and most popular techniques applied in data fitting, imaging processing and high dimension data analysis. The classic methods like QR and SVD decomposition for solving least squares problems has a…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
We present a computational study of several preconditioning techniques for the GMRES algorithm applied to the stochastic diffusion equation with a lognormal coefficient discretized with the stochastic Galerkin method. The clear block…
Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes…
The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational…
This work is concerned with the quantification of the epistemic uncertainties induced the discretization of partial differential equations. Following the paradigm of probabilistic numerics, we quantify this uncertainty probabilistically.…
In this study, a new coupled Partial Differential Equation (CPDE) based image denoising model incorporating space-time regularization into non-linear diffusion is proposed. This proposed model is fitted with additive Gaussian noise which…
In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…
We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point…