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The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
We investigate superconvergence properties of the spectral interpolation involving fractional derivatives. Our interest in this superconvergence problem is, in fact, twofold: when interpolating function values, we identify the points at…
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…
This paper gives the existence and uniqueness results for solution of fractional differential equations with Hilfer derivative. Using some new techniques and generalizing the restrictive conditions imposed on considered function, the…
This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial…
This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. These approximations are obtained using the techniques called \emph{generalized simplex Hessian} and…
This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform…
Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and…
This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in…
This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper in J.…
Recently, the authors Khalil, R., Al Horani, M., Yousef. A. and Sababheh, M., in " A new Denition Of Fractional Derivative, J. Comput. Appl. Math. 264. pp. 6570, 2014. " introduced a new simple well-behaved definition of the fractional…
This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst…
We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness…
Considering fractional fast diffusion equations on bounded open polyhedral domains in $\mathbb{R}^N$, we give a fully Galerkin approximation of the solutions by $C^0$-piecewise linear finite elements in space and backward Euler…
Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is H{\"o}lder…