相关论文: Computing the B\'ezier Control Points of the Lagra…
The famous Bernstein conjecture about optimal node systems in classical polynomial Lagrange interpolation, standing unresolved for about half a century, was solved by T. Kilgore in 1978. Immediately following him, also the additional…
This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are…
Grouped data are commonly encountered in applications. The Bernstein polynomial model is proposed as an approximate model in this paper for estimating a univariate density function based on grouped data. The coefficients of the Bernstein…
We develop a new kind of nonnegativity certificate for univariate polynomials on an interval. In many applications, nonnegative Bernstein coefficients are often used as a simple way of certifying polynomial nonnegativity. Our proposed…
Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution,…
In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of…
We present a stable and convergent method for solving initial value problems based on the use of differentiation matrices obtained by Lagrange interpolation. This implicit multistep-like method is easy-to-use and performs pretty well in the…
Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node…
We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract…
The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After surveying algorithms for computing the global b-function,…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties…
The montecarlo method, which is quite commonly used to solve maximum entropy problems in statistical physics, can actually be used to solve inverse problems in a much wider context. The probability distribution which maximizes entropy can…
Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of…
An effective numerical method is presented for optimizing model parameters that can be applied to any type of system of non-linear equations and any number of data-points, which does not require explicit formulation of the objective…
The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the…
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent…
Second order stochastic optimization methods, such as the linear method, couple the updates of different parameters and, in so doing, allow statistical uncertainty in one parameter to affect the update of other parameters. In simple tests,…
In a probabilistic reformulation of a combinatorial problem, we often face an optimization over a hypercube, which corresponds to the Bernoulli probability parameter for each binary variable in the primal problem. The combinatorial nature…
We present a novel efficient theoretical and numerical framework for solving global non-convex polynomial optimization problems. We analytically demonstrate that such problems can be efficiently reformulated using a non-linear objective…