(untitled) (Communications in Mathematical Research, 2024)
This work presents a stochastic Chebyshev-Picard iteration method to efficiently solve nonlinear differential equations with random inputs. If the nonlinear problem involves uncertainty, we need to characterize the uncertainty by using a few random variables. The nonlinear stochastic problems require solving the nonlinear system for a large number of samples in the stochastic space to quantify the statistics of the system of response and explore the uncertainty quantification. The computational cost is very expensive. To overcome the difficulty, a low rank approximation is introduced to the solution of the corresponding nonlinear problem and admits a variable-separation form in 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 stochastic basis functions, we utilize the greedy algorithm to select some optimal samples. Then the modified Chebyshev-Picard iteration method is used to solve the nonlinear system at the selected optimal samples, the solutions of which are used to train the deterministic basis functions. With the deterministic basis functions, 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 differential equations.