Related papers: Hierarchical Interpolative Factorization Precondit…
The main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems. We focus on efficiently solving these linear systems, arising…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied…
We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. The proposed method is an interpolation of the matrix inverse based on a…
There are some types of ill-conditioned algebraic equations that have difficulty in obtaining accurate roots and coefficients that must be expressed with a multiple precision floating-point number. When all their roots are simple, the…
In this paper we consider linear systems with dense-matrices which arise from numerical solution of boundary integral equations. Such matrices can be well-approximated with $\mathcal{H}^2$-matrices. We propose several new preconditioners…
We consider interpolation from the viewpoint of fully automated theorem proving in first-order logic as a general core technique for mechanized knowledge processing. For Craig interpolation, our focus is on the two-stage approach, where…
In this paper, a fast solver is studied for saddle point system arising from a second-order Crank-Nicolson discretization of an initial-valued parabolic PDE constrained optimal control problem, which is indefinite and ill-conditioned.…
Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to…
Iterative methods for solving large sparse systems of linear equations are widely used in many HPC applications. Extreme scaling of these methods can be difficult, however, since global communication to form dot products is typically…
Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically…
We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024, DOI: 10.1007/s10208-024-09681-8), a recently introduced mesh-independent algorithm, to obtain a high-order finite element…
The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned…
We propose a new family of high-order explicit generalized-$\alpha$ methods for hyperbolic problems with the feature of dissipation control. Our approach delivers $2k,\, \left(k \in \mathbb{N}\right)$ accuracy order in time by solving $k$…
In this article we construct and analyze multigrid preconditioners for discretizations of operators of the form D+K* K, where D is the multiplication with a relatively smooth positive function and K is a compact linear operator. These…
Periodic structures can be simulated using the periodic Method of Moments. The quasi-periodicity, i.e. periodicity within a linear phase-shift, is implemented through the use of the periodic Green's function. In this paper, we propose a…
We develop foundations for computing Craig interpolants and similar intermediates of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient…
The construction of robust solvers for linear systems obtained from the discretization of partial differential equations using Isogeometric Analysis is challenging since the condition number of the system matrix not only grows with the…
The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to…
A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the…