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Tensor approximation of functional differential equations (2403.04946v1)

Published 7 Mar 2024 in math.NA, cs.NA, math-ph, math.MP, and physics.comp-ph

Abstract: Functional Differential Equations (FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equation), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing new approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations (PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance parallel tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equation evolving from a random initial state.

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References (78)
  1. D. Venturi, The numerical approximation of nonlinear functionals and functional differential equations, Physics Reports 732, 1 (2018).
  2. D. Venturi and A. Dektor, Spectral methods for nonlinear functionals and functional differential equations, Res. Math. Sci. 8, 1 (2021).
  3. E. Hopf, Statistical hydromechanics and functional calculus, J. Rat. Mech. Anal. 1, 87 (1952).
  4. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence (Dover, 2007).
  5. K. Ohkitani, Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling, Phys. Rev. E 101, 013104 (2020).
  6. G. Rosen, Functional calculus theory for incompressible fluid turbulence, J. Math. Phys. 12, 812 (1971).
  7. N. Vakhania, V. Tarieladze, and S. Chobanyan, Probability distributions on Banach spaces, 1st ed. (Springer, 1987).
  8. C. Foias, Statistical study of Navier-Stokes equations, part I, Rend. Sem. Mat. Univ. Padova 48, 219 (1973).
  9. M. E. Peskin and D. V. Schroede, An introduction to quantum field theory (CRC Press, 2018).
  10. J. Zinn-Justin, Quantum field theory and critical phenomena, 4th ed. (Oxford Univ. Press, 2002).
  11. P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical dynamics of classical systems, Phys. Rev. A 8, 423 (1973).
  12. R. Phythian, The functional formalism of classical statistical dynamics, J. Phys A: Math. Gen. 10, 777 (1977).
  13. R. V. Jensen, Functional integral approach to classical statistical dynamics, J. Stat. Phys. 25, 183 (1981).
  14. R. Phythian, The operator formalism of classical statistical dynamics, J. Phys A: Math. Gen. 8, 1423 (1975).
  15. B. Jouvet and R. Phythian, Quantum aspects of classical and statistical fields, Phys. Rev. A 19, 1350 (1979).
  16. W. E, J. Han, and Q. Li, A mean-field optimal control formulation of deep learning, Res. Math. Sci. 6, 10 (2019).
  17. W. Gangbo, S. Mayorga, and A. Swiech, Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures, SIMA. J. Math. Anal. 53, 1320 (2021).
  18. A. Rodgers, A. Dektor, and D. Venturi, Adaptive integration of nonlinear evolution equations on tensor manifolds, J. Sci. Comput. 92, 1 (2022).
  19. A. Rodgers and D. Venturi, Implicit integration of nonlinear evolution equations on tensor manifolds, J. Sci. Comput 97, 1 (2023).
  20. L. Berselli and S. Spirito, On the existence of Leray-Hopf weak solutions to the Navier-Stokes equations, Fluids 6, 6010042 (2021).
  21. G. Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Annali di Matematica 48, 173 (1959).
  22. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63, 193 (1934).
  23. E. Fabes, B. Jones, and N. Rivière, The initial value problem for the navier-stokes equations with data in Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT, Arch. Rational Mech. Anal. 45, 222 (1972).
  24. R. Temam, Navier-Stokes equations: theory and numerical analysis (AMS Chelsea Publishing, 1984).
  25. M. J. Vishik and A. V. Fursikov, Mathematical problems of statistical hydromechanics, 2nd ed. (Kluwer Academic Publishers, 1988).
  26. V. I. Gishlarkaev, Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case, Journal of Mathematical Sciences 169, 64 (2010).
  27. T. Barker, Uniqueness results for weak Leray-Hopf solutions of the Navier-Stokes system with initial values in critical spaces, J. Math. Fluid Mech. 20, 133 (2018).
  28. T. Barker, About local continuity with respect to L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT initial data for energy solutions of the Navier-Stokes equations, Mathematische Annalen , https://doi.org/10.1007/s00208 (2020).
  29. T. Kato, Strong Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-solutions of the Navier-Stokes equation in ℝmsuperscriptℝ𝑚\mathbb{R}^{m}blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT, with applications to weak solutions, Math. Z. 187, 471 (1984).
  30. G. P. Galdi, On the relation between very weak and Leray-Hopf solutions to Navier–Stokes equations, Proc. Amer. Math. Soc. 147, 5349 (2019).
  31. S. Dubois, Uniqueness for some Leray–Hopf solutions to the Navier–Stokes, Journal of Differential Equations 189, 99 (2003).
  32. H. Fujita and T. Kato, On the Navier–Stokes initial value problem. I, Arch. Rational Mech. Anal. 16, 269 (1964).
  33. T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to the Navier–Stokes equation, Annals of Mathematics 189, 101 (2019).
  34. D. Kang and B. Protas, Searching for singularities in Navier–Stokes flows based on the Ladyzhenskaya–Prodi–Serrin conditions, J Nonlinear Sci 32, 1 (2022).
  35. E. Deriaz and V. Perrier, Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows, Journal of Turbulence 7, 1 (2006).
  36. E. Deriaz and V. Perrier, Direct numerical simulation of turbulence using divergence-free wavelets, Multiscale Model. Simul. 7, 1101 (2008).
  37. E. J. Fuselier and G. B. Wright, A radial basis function method for computing Helmholtz–Hodge decompositions, IMA J. Numer. Anal. 37, 774 (2017).
  38. G. Sacchi-Landriani and H. Vandeven, Polynomial approximation of divergence-free functions, Mathematics of Computation 185, 103 (1989).
  39. K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Vol. 194 (Springer, 1999).
  40. D. Guidetti, B. Karasozen, and S. Piskarev, Approximation of abstract differential equations, Journal of Mathematical Sciences 122, 3013 (2004).
  41. A. M. P. Boelens, D. Venturi, and D. M. Tartakovsky, Tensor methods for the Boltzmann-BGK equation, J. Comput. Phys. 421, 109744 (2020).
  42. G. di Marco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica 23, 369 (2014).
  43. H. J. Bungartz and M. Griebel, Sparse grids, Acta Numerica 13, 147 (2004).
  44. V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Advances in Computational Mechanics 12, 273 (2000).
  45. A. Narayan and J. Jakeman, Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation, SIAM J. Sci. Comput. 36, A2952 (2014).
  46. M. Raissi and G. E. Karniadakis, Hidden physics models: Machine learning of nonlinear partial differential equations, J. Comput. Phys. 357, 125 (2018).
  47. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 606 (2019).
  48. M. Bachmayr, R. Schneider, and A. Uschmajew, Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations, Foundations of Computational Mathematics 16, 1423 (2016).
  49. A. M. P. Boelens, D. Venturi, and D. M. Tartakovsky, Parallel tensor methods for high-dimensional linear PDEs, J. Comput. Phys. 375, 519 (2018).
  50. A. Dektor and D. Venturi, Tensor rank reduction via coordinate flows, J. Comp. Phys. 491, 112378 (2023).
  51. A. Dektor and D. Venturi, Coordinate-adaptive integration of PDEs on tensor manifolds, Comm. Appl. Math. Comput. , https://doi.org/10.1007/s42967 (2024).
  52. H. Cho, D. Venturi, and G. E. Karniadakis, Numerical methods for high-dimensional kinetic equations, in Uncertainty quantification for kinetic and hyperbolic equations, edited by S. Jin and L. Pareschi (Springer, 2017) pp. 93–125.
  53. S. V. Dolgov, TT-GMRES: solution to a linear system in the structured tensor format, Russian Journal of Numerical Analysis and Mathematical Modelling 28, 149 (2013).
  54. G. Beylkin and M. J. Mohlenkamp, Numerical operator calculus in higher dimensions, PNAS 99, 10246 (2002).
  55. A. Dektor and D. Venturi, Dynamic tensor approximation of high-dimensional nonlinear PDEs, J. Comput. Phys. 437, 110295 (2021a).
  56. D. Bigoni, A. P. Engsig-Karup, and Y. M. Marzouk, Spectral tensor-train decomposition, SIAM J. Sci. Comput. 38, A2405 (2016).
  57. I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33, 2295– (2011a).
  58. R. Schneider and A. Uschmajew, Approximation rates for the hierarchical tensor format in periodic Sobolev spaces, J. Complexity 30, 56 (2014).
  59. L. Grasedyck and C. Löbbert, Distributed hierarchical SVD in the hierarchical Tucker format, Numer. Linear Algebra Appl. 25, e2174 (2018).
  60. A. Etter, Parallel ALS algorithm for solving linear systems in the hierarchical Tucker representation, SIAM J. Sci. Comput. 38, A2585–A2609 (2016).
  61. A. Uschmajew and B. Vandereycken, The geometry of algorithms using hierarchical tensors, Linear Algebra Appl. 439, 133 (2013a).
  62. A. Dektor, A. Rodgers, and D. Venturi, Rank-adaptive tensor methods for high-dimensional nonlinear PDEs, J. Sci. Comput. 88, 1 (2021).
  63. M. Griebel and G. Li, On the decay rate of the singular values of bivariate functions, SIAM J. Numer. Anal. 56, 974 (2019).
  64. A. Rodgers and D. Venturi, Stability analysis of hierarchical tensor methods for time-dependent PDEs, J. Comput. Phys. 409, 109341 (2020).
  65. T. Kato, Perturbation theory for linear operators, Classics in Mathematics (Springer-Verlag, Berlin, 1995) pp. xxii+619, reprint of the 1980 edition.
  66. A. Uschmajew and B. Vandereycken, The geometry of algorithms using hierarchical tensors, Linear Algebra Appl. 439, 133 (2013b).
  67. D. Kressner and C. Tobler, Algorithm 941: htucker – a Matlab toolbox for tensors in hierarchical Tucker format, ACM Transactions on Mathematical Software 40, 1 (2014).
  68. H. Cho, D. Venturi, and G. E. Karniadakis, Statistical analysis and simulation of random shocks in Burgers equation, Proc. R. Soc. A 2171, 1 (2014).
  69. J. S. Hesthaven, Numerical Methods for Conservation Laws: From Analysis to Algorithm, 1st ed. (SIAM, 2018).
  70. J. S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral methods for time-dependent problems (Cambridge Univ. Press, 2007).
  71. Z. Botev, J. Grotowski, and D. Kroese, Kernel density estimation via diffusion, Annals of Statistics , 2916 (2010).
  72. L. Einkemmer and C. Lubich, A low-rank projector-splitting integrator for the Vlasov-Poisson equation, SIAM J. Sci. Comput. 40, B1330 (2018).
  73. L. Einkemmer and C. Lubich, A quasi-conservative dynamical low-rank algorithm for the Vlasov equation, SIAM J. Sci. Comput. 41, B1061 (2019).
  74. I. Oseledets and E. Tyrtyshnikov, TT-cross approximation for multidimensional arrays, Linear Algebra and its Applications 432, 70 (2010).
  75. A. Dektor and D. Venturi, Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs, J. Comput. Phys. 404, 109125 (2020).
  76. A. Dektor and D. Venturi, Dynamic tensor approximation of high-dimensional nonlinear PDEs, J. Comput. Phys. 437, 110295 (2021b).
  77. I. V. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing 33, 2295 (2011b), https://doi.org/10.1137/090752286 .
  78. H. A. Daas, G. Ballard, and P. Benner, Parallel algorithms for tensor train arithmetic, SIAM J. Sci. Comput. 44, C25 (2022).
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