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Face-to-face partition of 3D space with identical well-centered tetrahedra

Radim Hošek (2015)

Applications of Mathematics

The motivation for this paper comes from physical problems defined on bounded smooth domains Ω in 3D. Numerical schemes for these problems are usually defined on some polyhedral domains Ω h and if there is some additional compactness result available, then the method may converge even if Ω h → Ω only in the sense of compacts. Hence, we use the idea of meshing the whole space and defining the approximative domains as a subset of this partition. Numerical schemes for which quantities are defined on dual partitions...

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

P. E. Vincent, A. Jameson (2011)

Mathematical Modelling of Natural Phenomena

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order...

Factorization of CP-rank- 3 completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = B B ⊤ . If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A . In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3 . Failure of this algorithm implies that A does not have cp-rank 3 . Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that...

Fast and correctly rounded logarithms in double-precision

Florent de Dinechin, Christoph Lauter, Jean-Michel Muller (2007)

RAIRO - Theoretical Informatics and Applications

This article is a case study in the implementation of a portable, proven and efficient correctly rounded elementary function in double-precision. We describe the methodology used to achieve these goals in the crlibm library. There are two novel aspects to this approach. The first is the proof framework, and in general the techniques used to balance performance and provability. The second is the introduction of processor-specific optimization to get performance equivalent to the best current...

Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Hani Akbari (2013)

International Journal of Applied Mathematics and Computer Science

We consider a general elliptic Robin boundary value problem. Using orthogonal Coifman wavelets (Coiflets) as basis functions in the Galerkin method, we prove that the rate of convergence of an approximate solution to the exact one is O(2−nN ) in the H 1 norm, where n is the level of approximation and N is the Coiflet degree. The Galerkin method needs to evaluate a lot of complicated integrals. We present a structured approach for fast and effective evaluation of these integrals via trivariate connection...

Fast deterministic pricing of options on Lévy driven assets

Ana-Maria Matache, Tobias Von Petersdorff, Christoph Schwab (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) ∂ t u + 𝒜 [ u ] = 0 . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ -scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for 𝒜 can be replaced by a sparse matrix in the wavelet basis, and the linear...

Fast deterministic pricing of options on Lévy driven assets

Ana-Maria Matache, Tobias von Petersdorff, Christoph Schwab (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) ∂ t u + 𝒜 [ u ] = 0 . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for 𝒜 can be replaced by a sparse matrix in the wavelet basis, and the...

Fast evaluation of thin-plate splines on fine square grids

Petr Luner, Jan Flusser (2005)

Kybernetika

The paper deals with effective calculation of Thin-Plate Splines (TPS). We present a new modification of hierarchical approximation scheme. Unlike 2-D schemes published earlier, we propose an 1-D approximation. The new method yields lower computing complexity while it preserves the approximation accuracy.

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