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In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on…

We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure.…

Integer programming with block structures has received considerable attention recently and is widely used in many practical applications such as train timetabling and vehicle routing problems. It is known to be NP-hard due to the presence…

We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal…

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the…

We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of $H^1$ into the image and the kernel of some novel…

Low-lying shell model states may be approximated accurately by a sum over products of proton and neutron states. The optimal factors are determined by a variational principle and result from the solution of rather low-dimensional eigenvalue…

Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization…

In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization. Then, we construct multi-level finite element schemes by…

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of…

For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its…

Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are…

In this paper, we propose an unconstrained framework for eigenvalue problems in both discrete and continuous settings. We begin our discussion to solve a generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ with two $N\times N$ real…

Topology optimization problems often support multiple local minima due to a lack of convexity. Typically, gradient-based techniques combined with continuation in model parameters are used to promote convergence to more optimal solutions;…

We present a simple spectral approach to the well-studied constrained clustering problem. It captures constrained clustering as a generalized eigenvalue problem with graph Laplacians. The algorithm works in nearly-linear time and provides…

This paper is devoted to the construction of order reduced method of fourth order problems. A framework is presented such that a problem on a high-regularity space can be deduced in a constructive way to an equivalent problem on three…

Inverse Vandermonde matrix calculation is a long-standing problem to solve nonsingular linear system $Vc=b$ where the rows of a square matrix $V$ are constructed by progression of the power polynomials. It has many applications in…

The solution of matrices with $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained optimization problems, or the implicit or steady…