Academic Journal of Mathematical Sciences, 2023, 4(4); doi: 10.25236/AJMS.2023.040405.

## A Method for Solving Global Solutions to Partial Differential Equations Based on the Annealing Algorithm and Polynomial Regression for the Feynman-Kac Formulation

Author(s)

Xin Gui1, Shirui Zheng2

Corresponding Author:
Xin Gui
Affiliation(s)

1School of Civil Engineering, Shandong Jianzhu University, Jinan, 250101, China

2Zhizhen College, Beihang University, Beijing, 100191, China

### Abstract

At present, the global solution of PDE is usually obtained by the Feynman-Kac formula and then constructed by the interpolation method. In this paper, a polynomial fitting method based on the annealing algorithm is proposed to replace the interpolation method to construct the global solution. The simulated annealing algorithm is used to determine the coefficients of the fitting polynomials to obtain higher accuracy. The numerical results show that the improved model has less error than the model obtained by the interpolation method, and the annealing algorithm has a great contribution to improving the accuracy. Then the convergence analysis of the time step, simulation number, and polynomial order is carried out, and the result shows that the convergence is good. Finally, because the Feynman-Kac formula is used to determine the numerical solution, the method is expected to be applied to high-dimensional problems.

### Keywords

Feynman-Kac formula, Polynomial regression, Simulated annealing algorithm

### Cite This Paper

Xin Gui, Shirui Zheng. A Method for Solving Global Solutions to Partial Differential Equations Based on the Annealing Algorithm and Polynomial Regression for the Feynman-Kac Formulation. Academic Journal of Mathematical Sciences (2023) Vol. 4, Issue 4: 27-35. https://doi.org/10.25236/AJMS.2023.040405.

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