The Vlasov–Fokker–Planck equation with high dimensional parametric forcing term

We consider the Vlasov-Fokker-Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs [Cohen, DeVore, 2015], we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in [Chkifa, Cohen, Schwab, 2014], we develop a residual-based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multi-scale linear kinetic equation, when using numerical schemes that are time-dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.

[1]  Esther S. Daus,et al.  On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation , 2019, ESAIM: Mathematical Modelling and Numerical Analysis.

[2]  Ronald DeVore,et al.  Greedy Algorithms for Reduced Bases in Banach Spaces , 2012, Constructive Approximation.

[3]  Albert Cohen,et al.  Approximation of high-dimensional parametric PDEs * , 2015, Acta Numerica.

[4]  R. DeVore,et al.  ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S , 2011 .

[5]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[6]  H. Hwang,et al.  On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian , 2011, 1112.5504.

[7]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[8]  Shi Jin,et al.  Author's Personal Copy a Class of Asymptotic-preserving Schemes for the Fokker–planck–landau Equation , 2011 .

[9]  Angela Kunoth,et al.  Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs , 2013, SIAM J. Control. Optim..

[10]  Shi Jin,et al.  Hypocoercivity and Uniform Regularity for the Vlasov-Poisson-Fokker-Planck System with Uncertainty and Multiple Scales , 2018, SIAM J. Math. Anal..

[11]  Christoph Schwab,et al.  Karhunen-Loève approximation of random fields by generalized fast multipole methods , 2006, J. Comput. Phys..

[12]  A. M. Stuart,et al.  Sparse deterministic approximation of Bayesian inverse problems , 2011, 1103.4522.

[13]  Christoph Schwab,et al.  Sparse, adaptive Smolyak quadratures for Bayesian inverse problems , 2013 .

[14]  Shi Jin,et al.  Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro–macro decomposition-based asymptotic-preserving method , 2017, Research in the Mathematical Sciences.

[15]  Moulay Abdellah Chkifa.  On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection , 2013, J. Approx. Theory.

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

[17]  LIU LIU,et al.  Hypocoercivity Based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs , 2017, Multiscale Model. Simul..

[18]  Frances Y. Kuo,et al.  Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients , 2012, 1208.6349.

[19]  Liu Liu,et al.  An Asymptotic-Preserving Stochastic Galerkin Method for the Semiconductor Boltzmann Equation with Random Inputs and Diffusive Scalings , 2017, Multiscale Model. Simul..

[20]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[21]  Michel Loève,et al.  Probability Theory I , 1977 .

[22]  Frances Y. Kuo,et al.  Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients , 2015, Foundations of Computational Mathematics.

[23]  Fabio Nobile,et al.  An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[24]  Massimo Fornasier,et al.  A Kinetic Flocking Model with Diffusion , 2010 .

[25]  Jingwei Hu,et al.  On stochastic Galerkin approximation of the nonlinear Boltzmann equation with uncertainty in the fluid regime , 2019, J. Comput. Phys..

[26]  Albert Cohen,et al.  High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs , 2013, Foundations of Computational Mathematics.

[27]  Enrique Zuazua,et al.  Greedy controllability of finite dimensional linear systems , 2016, Autom..

[28]  R. Caflisch,et al.  Quasi-Monte Carlo integration , 1995 .

[29]  Albert Cohen,et al.  Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs , 2015 .

[30]  Radu Alexandru Todor,et al.  Robust Eigenvalue Computation for Smoothing Operators , 2006, SIAM J. Numer. Anal..

[31]  C. Schwab,et al.  Sparsity in Bayesian inversion of parametric operator equations , 2014 .

[32]  Shi Jin,et al.  A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources , 2009, J. Comput. Phys..

[33]  Li Wang,et al.  Uniform Regularity for Linear Kinetic Equations with Random Input Based on Hypocoercivity , 2016, SIAM/ASA J. Uncertain. Quantification.

[34]  C. Schwab,et al.  Sparse Adaptive Approximation of High Dimensional Parametric Initial Value Problems , 2013 .

[35]  Shi Jin,et al.  Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[36]  Christoph Schwab,et al.  Multilevel approximation of parametric and stochastic PDES , 2019, Mathematical Models and Methods in Applied Sciences.

[37]  Shi Jin,et al.  An asymptotic preserving scheme for the vlasov-poisson-fokker-planck system in the high field regime , 2011 .

[38]  Enrique Zuazua,et al.  Greedy optimal control for elliptic problems and its application to turnpike problems , 2018, Numerische Mathematik.

[39]  Albert Cohen,et al.  Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , 2010, Found. Comput. Math..

[40]  M. Loève.  Probability Theory II , 1978 .

[41]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

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

[43]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

[44]  Christoph Schwab,et al.  Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs , 2013 .