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Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By…

The Nystr\"om method is a popular low-rank approximation technique for large matrices that arise in kernel methods and convex optimization. Yet, when the data exhibits heavy-tailed spectral decay, the effective dimension of the problem…

We present the Tensor Train Multiplication (TTM) algorithm for the elementwise multiplication of two tensor trains with bond dimension $\chi$. The computational complexity and memory requirements of the TTM algorithm scale as $\chi^3$ and…

Chebyshev pseudospectral (PS) methods are reported to provide highly accurate solution using polynomial approximation. Use of polynomial basis functions in PS algorithms limits the formulation to univariate systems constraining it to tensor…

Orthogonal and quasi-orthogonal matrices have a long history of use in digital image processing, digital and wireless communications, cryptography and many other areas of computer science and coding theory. The practical benefits of using…

We analyze the recently developed folding algorithm [Phys. Rev. Lett. 102, 240603 (2009)] to simulate the dynamics of infinite quantum spin chains, and relate its performance to the kind of entanglement produced under the evolution of…

Krylov subspace methods are an essential building block in numerical simulation software. The efficient utilization of modern hardware is a challenging problem in the development of these methods. In this work, we develop Krylov subspace…

The convergence behaviour of first-order methods can be severely slowed down when applied to high-dimensional non-convex functions due to the presence of saddle points. If, additionally, the saddles are surrounded by large plateaus, it is…

We develop a simple two-step algorithm for enclosing Chebyshev expansions whose cost is linear in terms of the polynomial degree. The algorithm first transforms the expansion from Chebyshev to the Laurent basis and then applies the interval…

A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete…

We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical…

Tensor contraction operations in computational chemistry consume significant fractions of computing time on large-scale computing platforms. The widespread use of tensor contractions between large multi-dimensional tensors in describing…

Gaussian Mixture Models (GMMs) are one of the most potent parametric density models used extensively in many applications. Flexibly-tied factorization of the covariance matrices in GMMs is a powerful approach for coping with the challenges…

The finite difference time domain method is one of the simplest and most popular methods in computational electromagnetics. This work considers two possible ways of generalising it to a meshless setting by employing local radial basis…

We herein propose a variant of the projected inexact Levenberg--Marquardt method (ILMM) for solving constrained nonsmooth equations. Since the orthogonal projection onto the feasible set may be computationally expensive, we propose a local…

We propose algorithms for efficient time integration of large systems of oscillatory second order ordinary differential equations (ODEs) whose solution can be expressed in terms of trigonometric matrix functions. Our algorithms are based on…

Motivated by gradient methods in optimization theory, we give methods based on $\psi$-fractional derivatives of order $\alpha$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail.…

The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving nonsmooth optimization problems. In this paper, we present the first variant of the PBM for smooth objectives, achieving an accelerated…

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a…

Estimating the Hessian matrix, especially for neural network training, is a challenging problem due to high dimensionality and cost. In this work, we compare the classical Sherman-Morrison update used in the popular BFGS method…