Related papers: Hierarchical Interpolative Factorization Precondit…
A structured preconditioned conjugate gradient (PCG) solver is developed for the Newton steps in second-order methods for a class of constrained network optimal control problems. Of specific interest are problems with discrete-time dynamics…
The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves…
When solving linear systems arising from PDE discretizations, iterative methods (such as Conjugate Gradient, GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be…
We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in.…
This paper investigates using the conjugate gradient iterative solver for ill-posed problems. We show that preconditioner and Tikhonov-regularization work in conjunction. In particular when they employ the same symmetric positive…
The solution of systems of linear(ized) equations lies at the heart of many problems in Scientific Computing. In particular for systems of large dimension, iterative methods are a primary approach. Stationary iterative methods are generally…
There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems.…
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 paper presents a general method for applying hierarchical matrix skeletonization factorizations to the numerical solution of boundary integral equations with possibly rank-deficient integral operators. Rank-deficient operators arise in…
We present a fast direct algorithm for computing symmetric factorizations, i.e. $A = WW^T$, of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization…
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…
We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the…
The increasing size of transformer-based models in NLP makes the question of compressing them important. In this work, we present a comprehensive analysis of factorization based model compression techniques. Specifically, we focus on…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
We consider the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Gr\"unwald difference in space leads to dense…
For the solution of discrete ill-posed problems, in this paper a novel preconditioned iterative method based on the Arnoldi algorithm for matrix functions is presented. The method is also extended to work in connection with Tikhonov…
Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…
In this paper we introduce an algebraic recursive multilevel incomplete factorization preconditioner, based on a distributed Schur complement formulation, for solving general linear systems. The novelty of the proposed method is to combine…
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…
Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner…