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Author
Date
2014-01Type
- Report
ETH Bibliography
yes
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Abstract
We develop, analyze, and test a sparse tensor product phase space Galerkin discretization framework for the stationary monochromatic radiative transfer problem with scattering. The mathematical model describes the transport of radiation on a phase space of the Cartesian product of a typically three-dimensional physical domain and two-dimensional angular domain. Known solution methods such as the discrete ordinates method and a spherical harmonics method are derived from the presented Galerkin framework. We construct sparse versions of these well-established methods from the framework and prove that these sparse tensor discretizations break the “curse of dimensionality”: essentially (up to logarithmic factors in the total number of degrees of freedom) the solution complexity increases only as in a problem posed in the physical domain alone, while asymptotic convergence orders in terms of the discretization parameters remain essentially equal to those of a full tensor phase space Galerkin discretization. Algorithmically we compute the sparse tensor approximations by the combination technique. In numerical experiments on 2 + 1 and 3 + 2 dimensional phase spaces we demonstrate that the advantages of sparse tensorization can be leveraged in applications. Show more
Permanent link
https://doi.org/10.3929/ethz-a-010386299Publication status
publishedJournal / series
Research ReportVolume
Publisher
ETH ZürichSubject
GALERKIN METHOD (NUMERICAL MATHEMATICS); STRAHLUNGSTRANSPORT; Sparse grids; BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS OF HIGHER ORDER (NUMERICAL MATHEMATICS); RADIATION TRANSPORT; Discrete ordinates method; Spherical harmonics method; Combination technique; MEHRGITTERVERFAHREN + GITTERERZEUGUNG (NUMERISCHE MATHEMATIK); GALERKIN-VERFAHREN (NUMERISCHE MATHEMATIK); Radiative transfer; RANDWERTPROBLEME BEI PARTIELLEN DIFFERENTIALGLEICHUNGEN HÖHERER ORDNUNG (NUMERISCHE MATHEMATIK); MULTIGRID METHODS + GRID GENERATION (NUMERICAL MATHEMATICS)Organisational unit
03435 - Schwab, Christoph / Schwab, Christoph
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
Funding
247277 - Automated Urban Parking and Driving (EC)
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ETH Bibliography
yes
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