3,220 research outputs found
Stochastic methods for solving high-dimensional partial differential equations
We propose algorithms for solving high-dimensional Partial Differential
Equations (PDEs) that combine a probabilistic interpretation of PDEs, through
Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and
time-integration schemes are used to estimate pointwise evaluations of the
solution of a PDE. We use a sequential control variates algorithm, where
control variates are constructed based on successive approximations of the
solution of the PDE. Two different algorithms are proposed, combining in
different ways the sequential control variates algorithm and adaptive sparse
interpolation. Numerical examples will illustrate the behavior of these
algorithms
Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing
The main contents of this paper is two-fold.First, we present a method to approximate multivariate convex functions by piecewise linear upper and lower bounds.We consider a method that is based on function evaluations only.However, to use this method, the data have to be convex.Unfortunately, even if the underlying function is convex, this is not always the case due to (numerical) errors.Therefore, secondly, we present a multivariate data-smoothing method that smooths nonconvex data.We consider both the case that we have only function evaluations and the case that we also have derivative information.Furthermore, we show that our methods are polynomial time methods.We illustrate this methodology by applying it to some examples.approximation theory;convexity;data-smoothing
Subresultants in Multiple Roots
We extend our previous work on Poisson-like formulas for subresultants in
roots to the case of polynomials with multiple roots in both the univariate and
multivariate case, and also explore some closed formulas in roots for
univariate polynomials in this multiple roots setting.Comment: 21 pages, latex file. Revised version accepted for publication in
Linear Algebra and its Application
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