Related papers: A note on using the mass matrix as a preconditione…
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…
Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given…
Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be…
This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that…
In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the…
We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems,…
Radial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires…
A class of preconditioners is introduced to enhance geometry optimisation and transition state search of molecular systems. We start from the Hessian of molecular mechanical terms, decompose it and retain only its positive definite part to…
We discuss the ill conditioning of the matrix for the discretised Poisson equation in the small aspect ratio limit, and motivate this problem in the context of nonhydrostatic ocean modelling. Efficient iterative solvers for the Poisson…
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…
Compressive sampling has been widely used for sparse polynomial chaos (PC) approximation of stochastic functions. The recovery accuracy of compressive sampling highly depends on the incoherence properties of the measurement matrix. In this…
We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on…
Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems,…
Random matrices tend to be well conditioned, and we employ this well known property to advance matrix computations. We prove that our algorithms employing Gaussian random matrices are efficient, but in our tests the algorithms have…
We propose a preconditioner to accelerate the convergence of the GMRES iterative method for solving the system of linear equations obtained from discretize-then-optimize approach applied to optimal control problems constrained by a partial…
When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can…
Several preconditioned AOR methods have been proposed to solve system of linear equations $Ax=b$, where $A \in \mathbb{R}^{n \times n}$ is a unit Z-matrix. The aim of this paper is to give a comparison result for a class of preconditioners…
We derive formulas for the matrix elements of the lattice Green function for the discrete Poisson equation on an infinite square lattice. The partial difference equation for the matrix elements is solved by reducing it to a series of first…
This paper presents a new stochastic preconditioning approach. For symmetric diagonally-dominant M-matrices, we prove that an incomplete LDL factorization can be obtained from random walks, and used as a preconditioner for an iterative…
In this paper, we propose a descent method for composite optimization problems with linear operators. Specifically, we first design a structure-exploiting preconditioner tailored to the linear operator so that the resulting preconditioned…