Sensitivity analysis of fractional linear systems based on random walks with negligible memory usage

TL;DR

This paper introduces a random walk-based, memory-efficient method for solving large fractional linear differential systems and estimating their sensitivities with respect to parameters, ensuring unbiasedness and stability.

Contribution

It presents a novel stochastic approach for efficiently solving fractional linear systems and computing sensitivities without additional computational cost.

Findings

Method is unbiased and unconditionally stable

Provides unbiased estimators for solutions and sensitivities

Validated through multiple test cases

Abstract

A random walk-based method is proposed to efficiently compute the solution of a large class of fractional in time linear systems of differential equations (linear F-ODE systems), along with the derivatives with respect to the system parameters. Such a method is unbiased and unconditionally stable, and can therefore be used to provide an unbiased estimation of individual entries of the solution, or the full solution. By using stochastic differentiation techniques, it can be used as well to provide unbiased estimators of the sensitivities of the solution with respect to the problem parameters without any additional computational cost. The time complexity of the algorithm is discussed here, along with suitable variance bounds, which prove in practice the convergence of the algorithm. Finally, several test cases were run to assess the validity of the algorithm.