Multilevel Picard approximations overcome the curse of dimensionality when approximating semilinear heat equations with gradient-dependent nonlinearities in L^p-sense

2024/09/30 by Tuan Anh Nguyen, Nguyen, Tuan Anh
Computer Science · Engineering · Mathematics · #65C99 #68T05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Thermoelastic and Magnetoelastic Phenomena

paper · pdf · doi:10.48550/arxiv.2410.00203

openalex publication_date 2024/09/30 · openalex created_date 2024/10/29 · openalex updated_date 2026/07/28

Abstract

We prove that multilevel Picard approximations are capable of approximating solutions of semilinear heat equations in L^p-sense, p∈ [2,∞), in the case of gradient-dependent, Lipschitz-continuous nonlinearities, in the sense that the computational effort of the multilevel Picard approximations grow at most polynomially in both the dimension d and the reciprocal 1/ε of the prescribed accuracy ε.

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