Related papers: B-Splines for Sparse Grids: Algorithms and Applica…
This paper is concerned with developing an efficient numerical algorithm for fast implementation of the sparse grid method for computing the $d$-dimensional integral of a given function. The new algorithm, called the MDI-SG ({\em multilevel…
Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar…
We present a novel method to significantly speed up cosmological parameter sampling. The method relies on constructing an interpolation of the CMB-log-likelihood based on sparse grids, which is used as a shortcut for the…
Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric…
The sparse modeling is an evident manifestation capturing the parsimony principle just described, and sparse models are widespread in statistics, physics, information sciences, neuroscience, computational mathematics, and so on. In…
In the light of GPU accelerations, sequential operations such as solving ordinary differential equations can be bottlenecks for gradient evaluations and hinder potential speed gains. In this work, we focus on growth functions and their time…
This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order…
This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate…
We present a framework for smooth optimization of explicitly regularized objectives for (structured) sparsity. These non-smooth and possibly non-convex problems typically rely on solvers tailored to specific models and regularizers. In…
Uni- and bivariate data smoothing with spline functions is a well established method in nonparametric regression analysis. The extension to multivariate data is straightforward, but suffers from exponentially increasing memory and…
In this work, we develop a novel efficient quadrature and sparse grid based polynomial interpolation method to price American options with multiple underlying assets. The approach is based on first formulating the pricing of American…
Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations,…
This paper approximates simulation models by B-splines with a penalty on high-order finite differences of the coefficients of adjacent B-splines. The penalty prevents overfitting. The simulation output is assumed to be nonnegative. The…
Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes,…
Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree…
We provide a sparse version of the bounded degree SOS hierarchy BSOS [7] for polynomial optimization problems. It permits to treat large scale problems which satisfy a structured sparsity pattern. When the sparsity pattern satisfies the…
Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Mat\'ern kernels…
A systematic construction of higher order splines using two hierarchies of polynomials is presented. Explicit instructions on how to implement one of these hierarchies are given. The results are limited to interpolations on regular,…
In some cases, computational benefit can be gained by exploring the hyper parameter space using a deterministic set of grid points instead of a Markov chain. We view this as a numerical integration problem and make three unique…
Low-rank optimization problems with sparse simplex constraints involve variables that must satisfy nonnegativity, sparsity, and sum-to-1 conditions, making their optimization particularly challenging due to the interplay between low-rank…