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An efficient proximal-gradient-based method, called proximal extrapolated gradient method, is designed for solving monotone variational inequality in Hilbert space. The proposed method extends the acceptable range of parameters to obtain…
We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and…
We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
In this paper we analyze the approximation of multivariate integrals over the Euclidean plane for functions which are analytic. We show explicit upper bounds which attain the exponential rate of convergence. We use an infinite grid with…
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of…
We construct and analyze a projection-free linearly implicit method for the approximation of flows of harmonic maps into spheres. The proposed method is unconditionally energy stable and, under a sharp discrete regularity condition,…
A method is suggested allowing for the improvement of accuracy of self-similar factor and root approximants, constructed from asymptotic series. The method is based on performing a power transform of the given asymptotic series, with the…
Maximum consensus estimation plays a critically important role in robust fitting problems in computer vision. Currently, the most prevalent algorithms for consensus maximization draw from the class of randomized hypothesize-and-verify…
Two-time-scale stochastic approximation algorithms are iterative methods used in applications such as optimization, reinforcement learning, and control. Finite-time analysis of these algorithms has primarily focused on fixed point…
For a continuous function $f$ defined on a closed and bounded domain, there is at least one maximum and one minimum. First, we introduce some preliminaries which are necessary through the paper. We then present an algorithm, which is…
In non-linear estimations, it is common to assess sampling uncertainty by bootstrap inference. For complex models, this can be computationally intensive. This paper combines optimization with resampling: turning stochastic optimization into…
We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator…
In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse…
In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations. We introduce an integral-form objective function and prove its equivalence to the error between consecutive…
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…
In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g.,…
A new method, called the method of self-similar approximants, and its recent developments are described. The method is based on the ideas of renormalization group theory and optimal control theory. It allows for the effective extrapolation…
An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically…