Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function

Abstract

This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random non-autonomous second order linear differential equations with analytic data processes. The Fröbenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Fröbenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, we rely on the law of total probability to express it in closed-form as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a symbolic Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper.

Variance Reduction Methods and Multilevel Monte Carlo Strategy for Estimating Densities of Solutions to Random Second-Order Linear Differential Equations

“…If these variances are large or infinite, the convergence rate of the MC simulation deteriorates severely and noisy estimates of f X N (t) (x) are produced, thus invalidating the results. This phenomenon was observed in the numerical experiments from [25,Example 5.3]. It is highly related to having small denominators |S N 0 (t)| and |S N 1 (t)| at certain realizable paths, as this may produce higher dispersion of 1/|S N 0 (t)| and…”

“…Several theoretical results from [25] justify that f X N (t) (x) tends to the target density function f X(t) (x) as N → ∞ in a neighborhood of t 0 . The convergence is pointwise and in L p (R), for 1 ≤ p < ∞.…”

“…This approach was used in [24] for the autonomous version of (1), when A(t) = A and B(t) = B are time-independent random variables. We extended the computation of the density function to the general problem (1) in [25].…”

“…The procedure presented in [25] to derive the probability density function f X(t) (x) is briefly summarized. Let {S 0 (t), S 1 (t)} be the fundamental set of solutions to (1) satisfying the deterministic initial conditions S 0 (t 0 ) = 1, S 0 (t 0 ) = 0, S 1 (t 0 ) = 0 andṠ 1 (t 0 ) = 1.…”

“…In [25], the expectation from ( 8) and ( 9) is approximated using Monte Carlo (MC) simulation, via an explicit algorithm. Either (8) or (9) are selected, and realizations of the involved random variables are generated to compute the sample average.…”