Related papers: A Cascadic Multigrid Method for Eigenvalue Problem
Hyperparameter tuning is an active area of research in machine learning, where the aim is to identify the optimal hyperparameters that provide the best performance on the validation set. Hyperparameter tuning is often achieved using naive…
We solve Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method. We discuss the analytic properties of the discretisation, outline the implementation, and showcase numerical examples.
A new method of solution to the local spin density approximation to the electronic Schr\"{o}dinger equation is presented. The method is based on an efficient, parallel, adaptive multigrid eigenvalue solver. It is shown that adaptivity is…
We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical model is multiscale due to the existence of…
It is significant and challenging to solve eigenvalue problems of partial differential operators when many highly accurate eigenpair approximations are required. The adaptive finite element discretization based parallel orbital-updating…
This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on $\ell_2$, the space of square summable…
In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in…
The paper presents a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other…
The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are…
In this paper, we propose an efficient extrapolation cascadic multigrid (EXCMG) method combined with 25-point difference approximation to solve the three-dimensional biharmonic equation. First, through applying Richardson extrapolation and…
Stochastic gradient methods are scalable for solving large-scale optimization problems that involve empirical expectations of loss functions. Existing results mainly apply to optimization problems where the objectives are one- or two-level…
A multiscale optimization framework for problems over a space of Lipschitz continuous functions is developed. The method solves a coarse-grid discretization followed by linear interpolation to warm-start project gradient descent on…
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of…
In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem, and prove some convergence results for these algorithms and their…
In this paper, we propose an efficient two-level additive Schwarz method for solving large-scale eigenvalue problems arising from the finite element discretization of symmetric elliptic operators, which may compute efficiently more interior…
We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control…
Bilevel optimization has gained prominence in various applications. In this study, we introduce a framework for solving bilevel optimization problems, where the variables in both the lower and upper levels are constrained on Riemannian…
In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and…
In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence…
A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization,…