A semi-analytical scheme for highly oscillatory integrals over tetrahedra

This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and Díez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.This paper details a semi-analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the Partition of Unity Method (PUM) enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non-oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared to standard Gauss-Legendre quadratures.Peer ReviewedPostprint (author's final draft

Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations

In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomials to be real-valued. As applications of our results, we discuss the relation between nonnegative polynomials and sums of squares in the context of complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for complex tensors are introduced, extending properties from the Hermitian matrices. Finally, we discuss an important property for symmetric tensors, which states that the largest absolute value of eigenvalue of a symmetric real tensor is equal to its largest singular value; the result is known as Banach's theorem. We show that a similar result holds in the complex case as well