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In a number of such real-world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly, and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of size <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\varepsilon }&gt; 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> grows at most polynomially in both the PDE dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$d \\in \\mathbb {N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the reciprocal of the prescribed approximation accuracy <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\varepsilon }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent nonlinearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities.<\/jats:p>","DOI":"10.1007\/s10208-021-09514-y","type":"journal-article","created":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T19:02:41Z","timestamp":1626202961000},"page":"905-966","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities"],"prefix":"10.1007","volume":"22","author":[{"given":"Martin","family":"Hutzenthaler","sequence":"first","affiliation":[]},{"given":"Arnulf","family":"Jentzen","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Kruse","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,13]]},"reference":[{"key":"9514_CR1","unstructured":"Beck, C., Becker, S., Cheridito, P., Jentzen, A., and Neufeld, A. 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