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It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…
We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network…
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
We introduce an algorithm of joint approximation of a function and its first derivative by alternative orthogonal polynomials on the interval [0,1].The algorithm exhibits properties of shape preserving approximation for the function. A weak…
We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by…
Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is…
Hardware implementations of complex functions regularly deploy piecewise polynomial approximations. This work determines the complete design space of piecewise polynomial approximations meeting a given accuracy specification. Knowledge of…
Iterated Bernstein polynomial approximations of degree n for continuous function which also use the values of the function at i/n, i=0,1,...,n, are proposed. The rate of convergence of the classic Bernstein polynomial approximations is…
We display four approximation theorems for manifold-valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\Bbb C^n$ with holomorphic embeddings with dense images. The second theorem approximates…
We propose a new algorithm to the problem of polygonal curve approximation based on a multiresolution approach. This algorithm is suboptimal but still maintains some optimality between successive levels of resolution using dynamic…
Smeared link fermionic actions can be straightforwardly simulated with partial-global updating. The efficiency of this simulation is greatly increased if the fermionic matrix is written as a product of several near-identical terms. Such a…
Range aggregate queries find frequent application in data analytics. In some use cases, approximate results are preferred over accurate results if they can be computed rapidly and satisfy approximation guarantees. Inspired by a recent…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a…
We consider problems with multiple linear objectives and linear constraints and use Adjustable Robust Optimization and Polynomial Optimization as tools to approximate the Pareto set with polynomials of arbitrarily large degree. The main…
Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by…