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2012

Volume 34, Issue 3 (partial)

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Arithmetic Operations on Independent Random Variables: A Numerical Approach

Szymon Jaroszewicz and Marcin Korzeń

SIAM J. Sci. Comput. 34, pp. A1241-A1265 (25 pages)

Online Publication Date: May 01, 2012

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Dealing with imprecise quantities is an important problem in scientific computation. Model parameters are known only approximately, typically in the form of probability density functions. Unfortunately there are currently no methods of taking uncertain parameters into account which would at the same time be easy to apply and highly accurate. An important special case is operations on independent random variables which occur frequently in obtaining confidence intervals for physical measurements and statistical estimators. In this paper we investigate the possibility of implementing arithmetic operations on independent random variables numerically. Equivalently, the problem can be viewed as propagating approximated probability density functions through arithmetic operations. We introduce a very broad family of distributions which is closed under the four arithmetic operations and taking powers. Furthermore we show that the densities of the distributions in the family can be effectively approximated using Chebyshev polynomials. A practical implementation is also provided, demonstrating the feasibility and usefulness of the approach. Several examples show applications in physical measurements, statistics, and probability theory, demonstrating very high numerical accuracy. These include an interesting problem related to combining independent measurements of a physical quantity, distributions of sample statistics of the Hill's estimator for tail exponents, generalized $\chi^2$ distribution, and others. The results are usually extremely accurate, in some cases more accurate than specialized solutions available in statistical packages, and can be achieved with very little effort.

A Scalable Nonoverlapping and Nonconformal Domain Decomposition Method for Solving Time-Harmonic Maxwell Equations in $\mathbb{R}^3$

Zhen Peng and Jin-Fa Lee

SIAM J. Sci. Comput. 34, pp. A1266-A1295 (30 pages)

Online Publication Date: May 15, 2012

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We present a novel nonoverlapping and nonconformal domain decomposition method (DDM) for solving the time-harmonic Maxwell equations in $\mathbb{R}^3$. There are three major technical ingredients in the proposed nonconformal DDM: (a) a true second order transmission condition (SOTC) to enforce fields continuities across domain interfaces; (b) a corner edge penalty term to account for corner edges between neighboring subdomains; and (c) a global plane wave deflation technique to further improve the convergence of DDM for electrically large problems. It has been shown previously that a SOTC, which involves two second order transverse derivatives, facilitates convergence in the conformal domain decomposition method for both propagating and evanescent electromagnetic waves across domain interfaces. However, the discontinuous nature of the cement variables across the corner edges between neighboring subdomains remains troublesome. To mitigate the technical difficulty encountered and to enforce the needed divergence-free condition, we introduced a corner edge penalty term into the interior penalty formulation for the nonconformal DDM. The introduction of the corner edge penalty term successfully restored the superior performance of the SOTC. Finally, through an analysis of the DDM with the SOTC, we show that there still exists a weakly convergent region where the convergence in the DDM can still be unbearably slow for electrically large problems. Furthermore, it is found that the weakly convergent region is centered at the cutoff modes, or electromagnetic waves propagate in parallel to the domain interfaces. Subsequently, a global plane wave deflation technique is utilized to derive an effective global-coarse-grid preconditioner to promote fast convergence of the cutoff or near cutoff modes in the vicinity of domain interfaces. Finally, the strength of the proposed method is illustrated by means of three numerical examples.
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A Framework for Mimetic Discretization of the Rotating Shallow-Water Equations on Arbitrary Polygonal Grids

J. Thuburn and C. J. Cotter

SIAM J. Sci. Comput. 34, pp. B203-B225 (23 pages)

Online Publication Date: May 01, 2012

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Accurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties. Although these properties can be achieved relatively straightforwardly on a latitude-longitude grid, they are much more challenging on the various quasi-uniform spherical grids that are now under consideration. A recently developed scheme—called TRiSK—has these desirable properties on grids that have an orthogonal dual. The present work extends the TRiSK scheme into a more general framework suitable for grids that have a nonorthogonal dual, such as the equiangular cubed sphere. We also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus. One key ingredient is the definition of certain mapping operators that are discrete analogues of the Hodge star operator, enabling the definition of a compatible inner product. Discrete Coriolis terms are also included within the mimetic framework, and in such a way as to conserve energy and ensure that discrete geostrophic balance can be maintained; this requires the definition of a further mapping operator, with special properties, that transfers the discrete velocity field from the primal to the dual grid.

Transverse Electric Scattering on Inhomogeneous Objects: Spectrum of Integral Operator and Preconditioning

Grigorios P. Zouros and Neil V. Budko

SIAM J. Sci. Comput. 34, pp. B226-B246 (21 pages)

Online Publication Date: May 03, 2012

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The domain integral equation method with its FFT-based matrix-vector products is a viable alternative to local methods in free-space scattering problems. However, it often suffers from the extremely slow convergence of iterative methods, especially in the transverse electric (TE) case with large or negative permittivity. We identify very dense line segments in the spectrum as being partly responsible for this behavior and the main reason why a normally efficient deflating preconditioner does not work. We solve this problem by applying an explicit multiplicative regularizing operator, which on the operator level transforms the system to the form “identity plus compact.” On the matrix level this regularization reduces the length of the dense spectral segments roughly by a factor of four while preserving the ability to calculate the matrix-vector products using the FFT algorithm. Such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes, which results in a robust acceleration of the restarted GMRES under constraint memory conditions.

Finite Element Methods for Maxwell's Transmission Eigenvalues

Peter Monk and Jiguang Sun

SIAM J. Sci. Comput. 34, pp. B247-B264 (18 pages)

Online Publication Date: May 10, 2012

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The transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory. Efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory, and, more importantly, are parts of inverse algorithms for estimating material properties. In this paper, we propose two finite element methods to compute a few lowest Maxwell's transmission eigenvalues which are of interest in applications. Since the discrete matrix eigenvalue problem is large, sparse, and, in particular, non-Hermitian due to the fact that the problem is neither elliptic nor self-adjoint, we devise an adaptive method which combines the Arnoldi iteration and estimation of transmission eigenvalues. Exact transmission eigenvalues for balls are derived and used as a benchmark. Numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues.
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