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Partial differential equations (PDEs) are a fundamental tool in the modelingof many real world phenomena. In a number of such real world phenomena the PDEsunder consideration contain gradient-dependent nonlinearities and arehigh-dimensional. Such high-dimensional nonlinear PDEs can in nearly all casesnot be solved explicitly and it is one of the most challenging tasks in appliedmathematics to solve high-dimensional nonlinear PDEs approximately. It isespecially very challenging to design approximation algorithms for nonlinearPDEs for which one can rigorously prove that they do overcome the so-calledcurse of dimensionality in the sense that the number of computationaloperations of the approximation algorithm needed to achieve an approximationprecision of size $\varepsilon$ > 0 grows at most polynomially in both the PDEdimension $d \in \mathbb{N}$ and the reciprocal of the prescribed approximationaccuracy $\varepsilon$. In particular, to the best of our knowledge thereexists no approximation algorithm in the scientific literature which has beenproven to overcome the curse of dimensionality in the case of a class ofnonlinear PDEs with general time horizons and gradient-dependentnonlinearities. It is the key contribution of this article to overcome thisdifficulty. More specifically, it is the key contribution of this article (i)to propose a new full-history recursive multilevel Picard approximationalgorithm for high-dimensional nonlinear heat equations with general timehorizons and gradient-dependent nonlinearities and (ii) to rigorously provethat this full-history recursive multilevel Picard approximation algorithm doesindeed overcome the curse of dimensionality in the case of such nonlinear heatequations with gradient-dependent nonlinearities.

foundations of computational mathematics

Overcoming the curse of dimensionality in the numerical approximation of  parabolic partial differential equations with gradient-dependent  nonlinearities