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We consider linear ill-conditioned operator equations in a Hilbert space setting. Motivated by the aggregation method, we consider approximate solutions constructed from linear combinations of Tikhonov regularization, which amounts 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…
This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy…
We present a novel preconditioning technique for Krylov subspace algorithms to solve fluid-structure interaction (FSI) linearized systems arising from finite element discretizations. An outer Krylov subspace solver preconditioned with a…
We classify a family of matrices of shift operators that can be factorised in a computationally tractable manner with the Cholesky algorithm. Such matrices arise in the linear quadratic regulator problem, and related areas. We use the…
Randomized neural networks (RaNNs), in which hidden layers remain fixed after random initialization, provide an efficient alternative for parameter optimization compared to fully parameterized networks. In this paper, RaNNs are integrated…
The goal of this paper is to propose preconditioners for the system of linear equations that arises from a discretization of fourth order elliptic problems using spectral element methods. These preconditioners are constructed using…
Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being…
The additive Schwarz method is usually presented as a preconditioner for a PDE linearization based on overlapping subsets of nodes from a global discretization. It has previously been shown how to apply Schwarz preconditioning to a…
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method…
We consider the linear least squares problem with linear equality constraints (LSE problem) formulated as $\min_{x\in\mathbb{R}^{n}}\|Ax-b\|_2 \ \mathrm{s.t.} \ Cx = d$. Although there are some classical methods available to solve this…
Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for…
We demonstrate algorithm-based fault tolerance for silent, transient data corruption in "black-box" preconditioners. We consider both additive Schwarz domain decomposition with an ILU(k) subdomain solver, and algebraic multigrid, both…
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…
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…
We establish necessary and sufficient conditions for the existence of an LU factorization $A=LU$ for an arbitrary square matrix $A$, including singular and rank-deficient cases, without the use of row or column permutations. We prove that…
Matrix-free techniques play an increasingly important role in large-scale simulations. Schur complement techniques and massively parallel multigrid solvers for second-order elliptic partial differential equations can significantly benefit…
Interpolative and CUR decompositions involve "natural bases" of row and column subsets, or skeletons, of a given matrix that approximately span its row and column spaces. These low-rank decompositions preserve properties such as sparsity or…
Advanced Krylov subspace methods are investigated for the solution of large sparse linear systems arising from stiff adjoint-based aerodynamic shape optimization problems. A special attention is paid to the flexible inner-outer GMRES…
A preconditioning strategy is proposed for the iterative solve of large numbers of linear systems with parameter-dependent matrix and right-hand side which arise during the computation of solution statistics of stochastic elliptic partial…