Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs (Communications in Mathematical Research, 2024)
This work presents a stochastic Chebyshev-Picard iteration methodto efficiently solve nonlinear differential equations with random inputs. If thenonlinear problem involves uncertainty, we need to characterize the uncertainty by using a few random variables. The nonlinear stochastic problemsrequire solving the nonlinear system for a large number of samples in thestochastic space to quantify the statistics of the system of response and explorethe uncertainty quantification. The computational cost is very expensive. Toovercome the difficulty, a low rank approximation is introduced to the solutionof the corresponding nonlinear problem and admits a variable-separation formin terms of stochastic basis functions and deterministic basis functions. No iteration is performed at each enrichment step. These basis functions are model-oriented and involve offline computation. To efficiently identify the stochasticbasis functions, we utilize the greedy algorithm to select some optimal samples. Then the modified Chebyshev-Picard iteration method is used to solvethe nonlinear system at the selected optimal samples, the solutions of whichare used to train the deterministic basis functions. With the deterministic basisfunctions, we can obtain the corresponding stochastic basis functions by solving linear differential systems. The computation of the stochastic Chebyshev-Picard method decomposes into an offline phase and an online phase. This is very desirable for scientific computation. Several examples are presented to illustrate the efficacy of the proposed method for different nonlinear differentialequations.
Keywords: Nonlinear ordinary differential equation; Chebyshev-Picard iteration method; variable-separation method; random inputs.