Related papers: Polynomial Preconditioning for Indefinite Matrices
Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited-memory methods for the approximation of the action of…
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…
This paper is concerned with robust preconditioning of wave equations constrained linear inverse problems from boundary observation data. The main result of this paper is a concept for regularization parameter robust preconditioning.…
Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…
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…
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
This article presents a method for solving large-scale linear inverse problems regular- ized with a nonlinear, edge-preserving penalty term such as the total variation or Perona-Malik. In the proposed scheme, the nonlinearity is handled…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
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…
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric…
Preconditioning is the most widely used and effective way for treating ill-conditioned linear systems in the context of classical iterative linear system solvers. We introduce a quantum primitive called fast inversion, which can be used as…
The Interior-Point Methods are a class for solving linear programming problems that rely upon the solution of linear systems. At each iteration, it becomes important to determine how to solve these linear systems when the constraint matrix…
This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient…
Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the…
Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The…
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…
In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We consider parameterized variational inverse problems that are constrained by partial differential equations (PDEs). We seek to efficiently compute the solution of the inverse problem when auxiliary model parameters, which appear in the…
We apply preconditioning, which is widely used in classical solvers for linear systems $A\textbf{x}=\textbf{b}$, to the variational quantum linear solver. By utilizing incomplete LU factorization as a preconditioner for linear equations…