相关论文: Local Multilevel Preconditioners for Elliptic Equa…

In this paper, we study a class of bilevel optimization problems, also known as simple bilevel optimization, where we minimize a smooth objective function over the optimal solution set of another convex constrained optimization problem.…

We design and analyze a novel accelerated gradient-based algorithm for a class of bilevel optimization problems. These problems have various applications arising from machine learning and image processing, where optimal solutions of the two…

In this paper, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our…

This paper is concerned with the convergence analysis of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which admits some extended…

We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to…

A novel and scalable geometric multi-level algorithm is presented for the numerical solution of elliptic partial differential equations, specially designed to run with high occupancy of streaming processors inside Graphics Processing…

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is…

Bilevel optimization enjoys a wide range of applications in emerging machine learning and signal processing problems such as hyper-parameter optimization, image reconstruction, meta-learning, adversarial training, and reinforcement…

We present a multiscale mixed finite element method for solving second order elliptic equations with general $L^{\infty}$-coefficients arising from flow in highly heterogeneous porous media. Our approach is based on a multiscale spectral…

This paper introduces a novel double regularization scheme for bilevel optimization problems whose lower-level problem is composite and convex, but not necessarily strongly convex, in the lower-level variable. The analysis focuses on the…

The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or adaptive refinement. We…

A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…

Based on the work of Xu and Zhou [Math.Comput., 69(2000), pp.881-909], we establish new three-level and multilevel finite element discretizations by local defect-correction technique. Theoretical analysis and numerical experiments show that…

In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented…

Bilevel optimization has gained significant attention in recent years due to its broad applications in machine learning. This paper focuses on bilevel optimization in decentralized networks and proposes a novel single-loop algorithm for…

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 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 present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We…

This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in…

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure…