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Monte Carlo method and the random isentropic Euler system

Eduard Feireisl, Mária Lukáčová-Medvid'ová, Hana Mizerová, Changsheng Yu

TL;DR

This work rigorously analyzes the convergence of the Monte Carlo method when applied to the isentropic Euler system with uncertain initial data. It develops a robust framework built on dissipative weak solutions and a set-valued Strong Law of Large Numbers, augmented by Komlós-type convergence, to handle nonuniqueness and randomness. The authors prove weak and strong convergence results for Monte Carlo estimates built on consistent numerical approximations, and establish that, when smooth solutions exist, MC averages converge to the unique strong solution. They instantiate the theory with an unconditionally convergent viscosity finite volume method and demonstrate numerical convergence on a Kelvin–Helmholtz test with random interfaces. Overall, the paper provides a rigorous bridge between stochastic data, universal closure solutions, and practical, structure-preserving numerical schemes for multidimensional compressible flows.

Abstract

We show several results on convergence of the Monte Carlo method applied to consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our method is based on combination of several new concepts. We work with the dissipative weak solutions that can be seen as a universal closure of consistent approximations. Further, we apply the set-valued version of the Strong law of large numbers for general multivalued mapping with closed range and the Komlós theorem on strong converge of empirical averages of integrable functions. Theoretical results are illustrated by a series of numerical simulations obtained by an unconditionally convergent viscosity finite volume method combined with the Monte Carlo method.

Monte Carlo method and the random isentropic Euler system

TL;DR

This work rigorously analyzes the convergence of the Monte Carlo method when applied to the isentropic Euler system with uncertain initial data. It develops a robust framework built on dissipative weak solutions and a set-valued Strong Law of Large Numbers, augmented by Komlós-type convergence, to handle nonuniqueness and randomness. The authors prove weak and strong convergence results for Monte Carlo estimates built on consistent numerical approximations, and establish that, when smooth solutions exist, MC averages converge to the unique strong solution. They instantiate the theory with an unconditionally convergent viscosity finite volume method and demonstrate numerical convergence on a Kelvin–Helmholtz test with random interfaces. Overall, the paper provides a rigorous bridge between stochastic data, universal closure solutions, and practical, structure-preserving numerical schemes for multidimensional compressible flows.

Abstract

We show several results on convergence of the Monte Carlo method applied to consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our method is based on combination of several new concepts. We work with the dissipative weak solutions that can be seen as a universal closure of consistent approximations. Further, we apply the set-valued version of the Strong law of large numbers for general multivalued mapping with closed range and the Komlós theorem on strong converge of empirical averages of integrable functions. Theoretical results are illustrated by a series of numerical simulations obtained by an unconditionally convergent viscosity finite volume method combined with the Monte Carlo method.
Paper Structure (25 sections, 18 theorems, 96 equations, 4 figures, 4 tables)

This paper contains 25 sections, 18 theorems, 96 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Well posedness in the class of smooth solutions
  3. Weak solutions
  4. Dissipative weak solutions
  5. Euler system with uncertain data
  6. Data and trajectory spaces
  7. Metrics on the data space
  8. Topology of the trajectory space
  9. Properties of solution sets
  10. Topological properties of the solution set
  11. Measurability of the solution mapping in the weak topology
  12. Measurability in the strong $L^q$-topology
  13. Euler system with random initial data
  14. Strong law of large numbers
  15. Convergence for smooth data
  16. ...and 10 more sections

Key Result

Proposition 1.1

Suppose Then there exists $0 < T_{\rm max} \leq \infty$ and a classical solution $(\varrho, {\bm u})$ of the Euler system E1--E4 defined in $[0, T_{\rm max})$, unique in the class In addition,

Figures (4)

  • Figure 1: The statistical errors $E_1$ (left) and $E_2$(right) computed by the Monte Carlo VFV method on the mesh with $128 \times 128$ cells. The black solid lines without any marker denote the reference slope of $N^{-1/2}$.
  • Figure 2: The statistical errors $E_1$ (left) and $E_2$(right) computed by the Monte Carlo VFV method on the mesh with $512 \times 512$ cells. The black solid lines without any marker denote the reference slope of $N^{-1/2}$.
  • Figure 3: The statistical errors $E_1$ (left) and $E_2$(right) computed by the Monte Carlo VFV method on the mesh with $2048 \times 2048$ cells. The black solid lines without any marker denote the reference slope of $N^{-1/2}$.
  • Figure 4: The errors $E_1$ and $E_2$ in $L^1-$norm with parmaters $h=1/(2^{\ell+5}), N(h)=5\cdot 2^{\ell-1}, \ell=1,2,\dots,5.$ The black solid lines without any marker denote the reference slope of $N^{-1/2}$. The black dash lines without any marker denote the reference slope of $N^{-1}$.

Theorems & Definitions (29)

  • Proposition 1.1: Local existence of smooth solutions
  • Definition 1.2: Admissible weak solution
  • Remark 1.3
  • Definition 1.4: Wild data
  • Proposition 1.5: Density of wild data
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8: Dissipative weak solution
  • Definition 1.9: Maximal (DW) solution
  • Proposition 1.10: Global existence of maximal (DW) solution
  • ...and 19 more