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Implicit particle filtering is a sequential Monte Carlo method for data assim- ilation, designed to keep the number of particles manageable by focussing attention on regions of large probability. These regions are found by min- imizing, for…
Quasi-Newton methods employ an update rule that gradually improves the Hessian approximation using the already available gradient evaluations. We propose higher-order secant updates which generalize this idea to higher-order derivatives,…
Finding a \emph{single} best solution is the most common objective in combinatorial optimization problems. However, such a single solution may not be applicable to real-world problems as objective functions and constraints are only…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…
This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD)…
We propose a new class of rigorous methods for derivative-free optimization with the aim of delivering efficient and robust numerical performance for functions of all types, from smooth to non-smooth, and under different noise regimes. To…
In this paper we focus on the linear functionals defining an approximate version of the gradient of a function. These functionals are often used when dealing with optimization problems where the computation of the gradient of the objective…
Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…
We study active learning (AL) based on Gaussian Processes (GPs) for efficiently enumerating all of the local minimum solutions of a black-box function. This problem is challenging due to the fact that local solutions are characterized by…
We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that…
Approximations to the integral $\int_a^b\int_c^d f(x,y)\,dy\,dx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\leq p\leq\infty$. We assume ${\lVert f_{xy}\rVert}_p$ is…
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…
We derive analytical shape derivative formulas of the system matrix representing electric field integral equation discretized with Raviart-Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries…
We propose a new method for simplification of Gaussian process (GP) models by projecting the information contained in the full encompassing model and selecting a reduced number of variables based on their predictive relevance. Our results…
Gaussian process models are commonly used as emulators for computer experiments. However, developing a Gaussian process emulator can be computationally prohibitive when the number of experimental samples is even moderately large. Local…
Nonlinear least-squares problems are a special class of unconstrained optimization problems in which their gradient and Hessian have special structures. In this paper, we exploit these structures and proposed a matrix-free algorithm with a…
Anytime approximation algorithms that compute the probabilities of queries over probabilistic databases can be of great use to statistical learning tasks. Those approaches have been based so far on either (i) sampling or (ii)…
We propose a novel study of the stochastic proximal gradient method for minimizing the sum of two convex functions, one of which is smooth. Under suitable assumptions and without requiring any boundedness or control of the variance of the…
Over the past a few years, research and development has made significant progresses on big data analytics. A fundamental issue for big data analytics is the efficiency. If the optimal solution is unable to attain or not required or has a…