Related papers: Polynomial and Parallelizable Preconditioning for …
The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag…
We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned…
In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix…
We introduce a parallel algorithm to construct a preconditioner for solving a large, sparse linear system where the coefficient matrix is a Laplacian matrix (a.k.a., graph Laplacian). Such a linear system arises from applications such as…
We develop a robust matrix-free, communication avoiding parallel, high-degree polynomial preconditioner for the Conjugate Gradient method for large and sparse symmetric positive definite linear systems. We discuss the selection of a scaling…
The convergence rates of iterative methods for solving a linear system $\mathbf{A} x = b$ typically depend on the condition number of the matrix $\mathbf{A}$. Preconditioning is a common way of speeding up these methods by reducing that…
We present an optimized algorithm calculating determinant for multivariate polynomial matrix on GPU. The novel algorithm provides precise determinant for input multivariate polynomial matrix in controllable time. Our approach is based on…
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…
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…
To precondition a large and sparse linear system, two direct methods for approximate factoring of the inverse are devised. The algorithms are fully parallelizable and appear to be more robust than the iterative methods suggested for the…
In this article we present a new multigrid preconditioner for the linear systems arising in the semismooth Newton method solution of certain control-constrained, quadratic distributed optimal control problems. Using a piecewise constant…
We consider the numerical solution of large scale time-harmonic Maxwell equations. To this day, this problem remains difficult, in particular because the equations are neither Hermitian nor semi-definite. Our approach is to compare…
McDonald, Pestana and Wathen (SIAM J. Sci. Comput. 40(2), pp. A2012-A1033, 2018) present a method for preconditioning of time-dependent PDEs via approximation by a nearby time-periodic problem, that is, they employ circulant-related…
We consider the problem of finding the optimal diagonal preconditioner for a positive definite matrix. Although this problem has been shown to be solvable and various methods have been proposed, none of the existing approaches are scalable…
We introduce a unified framework for computing approximately-optimal preconditioners for solving linear and non-linear systems of equations. We demonstrate that the condition number minimization problem, under structured transformations…
We address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of PDE-constrained optimal control problems via active-set Newton methods, with control and (regularized) state constraints. We…
While preconditioning is a long-standing concept to accelerate iterative methods for linear systems, generalizations to matrix functions are still in their infancy. We go a further step in this direction, introducing polynomial…
In this work, we propose a class of novel preconditioned Krylov subspace methods for solving an optimal control problem of parabolic equations. Namely, we develop a family of block $\omega$-circulant based preconditioners for the…
Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices…
We present a stationary iteration based upon a block splitting for a class of indefinite least squares problem. Convergence of the proposed method is investigated and optimal value of the involving parameter is used. The induced…