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The motivation for this paper comes from physical problems defined on bounded smooth domains in 3D. Numerical schemes for these problems are usually defined on some polyhedral domains and if there is some additional compactness result available, then the method may converge even if 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...
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...
A symmetric positive semi-definite matrix is called completely positive if there exists a matrix with nonnegative entries such that . If is such a matrix with a minimal number of columns, then is called the cp-rank of . In this paper we develop a finite and exact algorithm to factorize any matrix of cp-rank . Failure of this algorithm implies that does not have cp-rank . Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that...
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...
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...
Arbitrage-free prices of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the -scheme in time and a wavelet Galerkin method with degrees of freedom in log-price space. The dense matrix for can be replaced by a sparse matrix in the wavelet basis, and the linear...
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)
.
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 can be replaced by a sparse
matrix in the wavelet basis, and the...
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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