A Node Elimination Algorithm for Cubature of High-Dimensional Polytopes

Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial quadrature rule for various polytopes in several space dimensions is described. High efficiency rules are presented for two, three and four dimensional polytopes. The new rules are compared with rules that are obtained by combining tensor products of one dimensional quadrature rules and domain transformations, as well as with known analytically constructed cubature rules.Comment: 18 pages, 6 figure

On the numerical solution of a time-dependent shape optimization problem for the heat equation

This article is concerned with the solution of a time-dependent shape identification problem. Specifically we consider the heat equation in a domain, which contains a time-dependent inclusion of zero temperature. The objective is to detect this inclusion from the given temperature and heat flux at the exterior boundary of the domain. To this end, for a given temperature at the exterior boundary, the mismatch of the Neumann data is minimized. This time-dependent shape optimization problem is then solved by a gradient-based optimization method. Numerical results are presented which validate the present approach

Boundary integral operators for the heat equation

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the C 2 -diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully

A fast sparse grid based space-time boundary element method for the nonstationary heat equation

This article presents a fast sparse grid based space-time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation. This integral equation is discretized by Galerkin's method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the H -matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

A fast sparse grid based space-time boundary element method for the nonstationary heat equation

This article presents a fast sparse grid based space-time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation.This integral equation is discretized by Galerkin’s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the $\mathcal{H}$-matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

Computational Prototyping Tools and Techniques

Contains reports on five research projects.Industry Consortium (Mobil, Statoil, DNV Software, Shell, OTRC, Petrobras, NorskHydro, Exxon, Chevron, SAGA, NSWC)U.S. Navy - Office of Naval ResearchAnalog DevicesDefense Advanced Research Projects Agency Contract J-FBI-95-215Cadence Design SystemsHarris SemiconductorMAFET ConsortiumMotorola SemiconductorDefense Advanced Research Projects AgencyMultiuniversity Research InitiativeSemiconductor Research CorporationIBM Corporatio

Custom Integrated Circuits

Contains table of contents for Part III, table of contents for Section 1 and reports on eleven research projects.IBM CorporationMIT School of EngineeringNational Science Foundation Grant MIP 94-23221Defense Advanced Research Projects Agency/U.S. Army Intelligence Center Contract DABT63-94-C-0053Mitsubishi CorporationNational Science Foundation Young Investigator Award Fellowship MIP 92-58376Joint Industry Program on Offshore Structure AnalysisAnalog DevicesDefense Advanced Research Projects AgencyCadence Design SystemsMAFET ConsortiumConsortium for Superconducting ElectronicsNational Defense Science and Engineering Graduate FellowshipDigital Equipment CorporationMIT Lincoln LaboratorySemiconductor Research CorporationMultiuniversity Research IntiativeNational Science Foundatio