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Integral Transform Solution of Random Coupled Parabolic Partial Differential Models

M. -C. Casabán, R. Company, V. N. Egorova, L. Jódar

TL;DR

The work targets random coupled parabolic PDEs with uncertain coefficients and initial/boundary data. It introduces an integral-transform framework based on a random cosine Fourier transform $F_c$, combined with truncation of oscillatory integrals and a midpoint Riemann quadrature, to efficiently compute the stochastic solution $u(z,t)$ and its moments. A spectral-type condition on the random matrices guarantees convergence of the approximate solution and its moments, and a Monte Carlo-enhanced Mid_N scheme yields accurate estimates of $\mathbb{E}[u]$ and $\sqrt{\mathrm{Var}[u]}$ with favorable computational costs. The method is validated through deterministic reductions and a random-coefficient example (including truncated Gaussian and gamma distributions), demonstrating accurate moment estimates and practical efficiency, with applicability to other integral transforms and oscillatory problems.

Abstract

Random coupled parabolic partial differential models are solved numerically using random cosine Fourier transform together with non Gaussian random numerical integration that capture the highly oscillatory behavior of the involved integrands. Sufficient condition of spectral type imposed on the random matrices of the system are given so that the approximated stochastic process solution and its statistical moments are numerically convergent. Numerical experiments illustrate the results.

Integral Transform Solution of Random Coupled Parabolic Partial Differential Models

TL;DR

The work targets random coupled parabolic PDEs with uncertain coefficients and initial/boundary data. It introduces an integral-transform framework based on a random cosine Fourier transform , combined with truncation of oscillatory integrals and a midpoint Riemann quadrature, to efficiently compute the stochastic solution and its moments. A spectral-type condition on the random matrices guarantees convergence of the approximate solution and its moments, and a Monte Carlo-enhanced Mid_N scheme yields accurate estimates of and with favorable computational costs. The method is validated through deterministic reductions and a random-coefficient example (including truncated Gaussian and gamma distributions), demonstrating accurate moment estimates and practical efficiency, with applicability to other integral transforms and oscillatory problems.

Abstract

Random coupled parabolic partial differential models are solved numerically using random cosine Fourier transform together with non Gaussian random numerical integration that capture the highly oscillatory behavior of the involved integrands. Sufficient condition of spectral type imposed on the random matrices of the system are given so that the approximated stochastic process solution and its statistical moments are numerically convergent. Numerical experiments illustrate the results.

Paper Structure

This paper contains 4 sections, 2 theorems, 50 equations, 6 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Solving the Sampled Deterministic Case
  3. The Random Case: Numerical Solution and Numerical Convergence
  4. Conclusions

Key Result

lemma 1

Let $B \in \mathbb{R}^{N \times N}$ be a matrix such that $B + B^T$ is positive definite and satisfies (eq:lambda_min). Then

Figures (6)

  • Figure 1: RMSE of numerical approximations of (\ref{['eq:uR']}) by the midpoint Riemann sum with fixed $h=0.05$ for various values of $R$ in the domain $(z,t)\in{[0,5]\times[0,1]}$ for the step-sizes $\Delta z=0.05$ and $\Delta t=0.01$.
  • Figure 2: Solution $u_1(z,t)$ calculated by (\ref{['eq:exact']}) for $a=\nu = 1, \; g(t)=1$.
  • Figure 3: Solution $u_2(z,t)$ calculated by (\ref{['eq:exact']}) for $a=\nu = 1, \; g(t)=1$.
  • Figure 4: Expectation and the standard deviation at the time instant $t=1$ of the exact solution s.p, $[u_1(z,t),u_2(z,t)]$, for the random coupled parabolic problem (\ref{['eq:problem1']})--(\ref{['eq:problem4']}), \ref{['eq:Boundary1Ex']}--\ref{['eq:Boundary2Ex']}, considering the r.v.'s $a(\xi) \sim N_{[0.8, 1.2]}{(2,0.1)}$ and $\nu(\xi) \sim Ga_{[0.5, 1.5]}{(4;2)}$, and the spatial domain $z\in[0,5]$ with step size $\Delta z=0.1$.
  • Figure 5: Absolute errors of the expectation and the standard deviation for both components of the approximate solution s.p. \ref{['eq:Mid_u']} at $t=1$ fixing $R=20$ and $h=0.05$ ($N=400$) in \ref{['eq:EMid_N_R']}--\ref{['eq:VarMid_N_R']} but varing the number of simulations $\xi_i=\{ 400, 1600, 12800\}$. The spatial domain is $z\in[0,5]$ with $\Delta z =0.1$.
  • ...and 1 more figures

Theorems & Definitions (5)

  • lemma 1
  • proof
  • theorem 1
  • Example 2.1
  • Example 3.1