A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient

Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an accelerated Schwarz method for radiative transfer equations that makes use of approximate local solution maps constructed offline via a random sampling strategy. Numerical examples demonstrate the accuracy, robustness, and efficiency of the proposed approach.

[1]  G. Plass,et al.  Radiative transfer in an atmosphere-ocean system. , 1969, Applied optics.

[2]  C. Bardos,et al.  DIFFUSION APPROXIMATION AND COMPUTATION OF THE CRITICAL SIZE , 1984 .

[3]  E. Larsen,et al.  Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II , 1989 .

[4]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[5]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[6]  V. Agoshkov.  Boundary Value Problems for Transport Equations , 1998 .

[7]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[8]  Laurent Dumas,et al.  Homogenization of Transport Equations , 2000, SIAM J. Appl. Math..

[9]  T. Goudon,et al.  Diffusion approximation in heterogeneous media , 2001 .

[10]  A. Klose,et al.  Optical tomography using the time-independent equation of radiative transfer-Part 1: Forward model , 2002 .

[11]  T. Goudon,et al.  Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation , 2003 .

[12]  E. Weinan,et al.  Analysis of the heterogeneous multiscale method for elliptic homogenization problems , 2004 .

[13]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[14]  Mario Bebendorf,et al.  Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems , 2008 .

[15]  Luc Mieussens,et al.  A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit , 2008, SIAM J. Sci. Comput..

[16]  Shi Jin.  ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW , 2010 .

[17]  Houman Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[18]  P. Degond.  Asymptotic-Preserving Schemes for Fluid Models of Plasmas , 2011, 1104.1869.

[19]  Robert Lipton,et al.  Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems , 2010, Multiscale Model. Simul..

[20]  Per-Gunnar Martinsson,et al.  A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix , 2011, SIAM J. Matrix Anal. Appl..

[21]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[22]  N. Crouseilles,et al.  An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits. , 2011 .

[23]  Naoufel Ben Abdallah,et al.  Diffusion and Homogenization Limits with Separate Scales , 2012, Multiscale Model. Simul..

[24]  Jianlin Xia,et al.  Randomized Sparse Direct Solvers , 2013, SIAM J. Matrix Anal. Appl..

[25]  Giacomo Dimarco,et al.  Numerical methods for kinetic equations* , 2014, Acta Numerica.

[26]  W. Hackbusch,et al.  Hierarchical Matrices: Algorithms and Analysis , 2015 .

[27]  Yalchin Efendiev,et al.  Sparse Generalized Multiscale Finite Element Methods and their applications , 2015, 1506.08509.

[28]  Victor M. Calo,et al.  Randomized Oversampling for Generalized Multiscale Finite Element Methods , 2014, Multiscale Model. Simul..

[29]  Lei Wu,et al.  Geometric Correction in Diffusive Limit of Neutron Transport Equation in 2D Convex Domains , 2016, 1605.02362.

[30]  Qin Li,et al.  Implicit Asymptotic Preserving Method for Linear Transport Equations , 2016, 1602.00746.

[31]  Qin Li,et al.  Validity and Regularization of Classical Half-Space Equations , 2016, 1606.01311.

[32]  Robert Lipton,et al.  Uncertain loading and quantifying maximum energy concentration within composite structures , 2016, J. Comput. Phys..

[33]  S. Barcza,et al.  Greenhouse effect from the point of view of radiative transfer , 2016, Acta Geodaetica et Geophysica.

[34]  Yalchin Efendiev,et al.  REITERATED MULTISCALE MODEL REDUCTION USING THE GENERALIZED MULTISCALE FINITE ELEMENT METHOD , 2016 .

[35]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[36]  Qin Li,et al.  An Asymptotic Preserving Method for Transport Equations with Oscillatory Scattering Coefficients , 2017, Multiscale Model. Simul..

[37]  Qin Li,et al.  Asymptotic-Preserving Schemes for Multiscale Hyperbolic and Kinetic Equations , 2017 .

[38]  Kathrin Smetana,et al.  Randomized Local Model Order Reduction , 2017, SIAM J. Sci. Comput..

[39]  Ralf Kornhuber,et al.  An analysis of a class of variational multiscale methods based on subspace decomposition , 2016, Math. Comput..

[40]  Stephen J. Wright,et al.  Random Sampling and Efficient Algorithms for Multiscale PDEs , 2018, SIAM J. Sci. Comput..

[41]  Eric T. Chung,et al.  Generalized Multiscale Finite Element Method for the Steady State Linear Boltzmann Equation , 2019, Multiscale Model. Simul..