[Paper Review] On the approximation of the probability density function of the randomized heat equation

Julia Calatayud, J.‐C. Cortés+1 more|arXiv (Cornell University)|Feb 8, 2018

Advanced Mathematical Modeling in Engineering13 references3 citations

TL;DR

This paper proposes two novel approaches to approximate the probability density function (PDF) of the solution to a randomized heat equation with a random diffusion coefficient and stochastic initial condition. Using the method of separation of variables, the solution is expressed as a random series, and the PDF is approximated via Random Variable Transformation (RVT) and Karhunen-Loève expansion (KLE), with theoretical convergence proven under mild Lipschitz conditions. The key contribution is a uniformly Cauchy sequence of approximating PDFs, validated numerically with rapid convergence and robustness to distributional assumptions.

ABSTRACT

In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coeffcient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this randomized partial differential equation problem is a stochastic process, which is given by a random series obtained via the classical method of separation of variables. Any stochastic process is determined by its finite-dimensional joint distributions. In this paper, the goal is to obtain approximations to the probability density function of the solution (the first finite-dimensional distributions) under mild conditions. Since the solution is expressed as a random series, we perform approximations of its probability density function. We use two approaches: broadly speaking, first, dealing with the random Fourier coefficients of the random series, and second, taking advantage of the Karhunen-Loeve expansion of the initial condition stochastic process. Finally, several numerical examples illustrating the potentiality of our findings with regard to both approaches are presented.

Motivation & Objective

• To develop reliable approximations for the first finite-dimensional probability density function of the solution to a randomized heat equation with uncertain parameters.
• To address the challenge of computing the PDF of a stochastic process arising from a random partial differential equation with a random diffusion coefficient and stochastic initial condition.
• To establish theoretical conditions—particularly Lipschitz continuity—under which the approximating PDFs converge uniformly.
• To compare the performance of two distinct approaches: one based on random Fourier coefficients and another leveraging the Karhunen-Loève expansion of the initial condition.
• To validate the theoretical findings through numerical experiments demonstrating fast convergence and robustness across different parameter configurations.

Proposed method

• The solution to the randomized heat equation is derived via separation of variables, resulting in a random series representation involving random Fourier coefficients.
• The Random Variable Transformation (RVT) technique is applied to the random Fourier coefficients to derive the PDF of the solution, assuming joint density and Lipschitz continuity of the transformation.
• The Karhunen-Loève expansion (KLE) is used to represent the initial condition stochastic process as an infinite series of uncorrelated random variables, enabling PDF approximation through transformation of these components.
• Theoretical convergence of the approximating PDFs is established by proving they form a uniformly Cauchy sequence under mild assumptions, including Lipschitz continuity of the density functions involved.
• Numerical experiments are conducted using Monte Carlo simulations and L1-norm comparisons to assess convergence speed and accuracy of the approximated PDFs.
• Two distinct approximation strategies are implemented: one based on truncating the random series and applying RVT to the coefficients (Theorem 2.8), and another using KLE to represent the initial condition and applying transformation to the resulting coefficients (Theorem 3.3).

Experimental results

Research questions

• RQ1 Under what conditions does the probability density function of the solution to the randomized heat equation converge uniformly?
• RQ2 How do the RVT-based and KLE-based approaches compare in approximating the solution's PDF in terms of convergence speed and accuracy?
• RQ3 What is the role of the Lipschitz condition on the density functions of the random parameters in ensuring convergence of the PDF approximations?
• RQ4 Can the proposed methods handle a wide range of probability distributions for the diffusion coefficient and initial condition, including non-Gaussian and bounded distributions?
• RQ5 What numerical evidence supports the necessity of the Lipschitz condition in the theoretical convergence framework?

Key findings

• The approximating PDFs of the solution form a uniformly Cauchy sequence under mild assumptions, including Lipschitz continuity of the density functions of the random parameters, ensuring theoretical convergence.
• Numerical experiments show that both the RVT-based and KLE-based approaches yield very similar results, with rapid convergence observed in the L1-norm between successive approximations.
• For Example 4.1, the L1-norm difference between successive PDF approximations drops to approximately 0.00088 at (x,t) = (0.5,0.1), indicating fast convergence.
• In Example 4.2, the L1-norm differences are below 10^-8 for higher-order approximations, confirming high accuracy and stability of the method.
• Example 4.3 demonstrates numerically that the absence of the Lipschitz condition leads to non-convergent PDF approximations, as shown by L1-norm differences of 0.19156 and 1.86146, respectively, indicating the necessity of this condition.
• The expectation and variance of the solution remain close to zero and 0.100146, respectively, across different approximations, indicating consistency in statistical moments.