{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T03:37:01Z","timestamp":1758080221080,"version":"3.44.0"},"reference-count":45,"publisher":"American Mathematical Society (AMS)","issue":"357","license":[{"start":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:00:00Z","timestamp":1767139200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100005693","name":"Hainan University","doi-asserted-by":"publisher","award":["XJ2400010491"],"award-info":[{"award-number":["XJ2400010491"]}],"id":[{"id":"10.13039\/501100005693","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper, we propose an implicit-explicit Monte Carlo method (IMEX-MCM) for solving semi-linear parabolic partial differential equations (PDEs) driven by <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n  <mml:semantics>\n    <mml:mi>\u03b1<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-stable L\u00e9vy process on bounded domains, also known as semi-linear PDEs involving integral fractional Laplacian operator (i.e., <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha element-of left-parenthesis 0 comma 2 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03b1<\/mml:mi>\n      <mml:mo>\u2208<\/mml:mo>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n      <mml:mo>,<\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\alpha \\in (0,2)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>) and classical Laplacian operator (i.e., <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha equals 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03b1<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\alpha =2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>). The main idea behind our approach is to divide the time interval into <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n  <mml:semantics>\n    <mml:mi>N<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> sub-intervals and derive the Feynman-Kac formula within each time sub-interval. Then, we develop a fully discretized parallel scheme using the implicit-explicit technique for temporal discretization, followed by applying a novel walk-on-sphere method to obtain the numerical solution. Notably, the proposed method takes advantage of the Poisson kernel associated with the stochastic processes to accommodate the full discretization of the classical and fractional semi-linear PDEs in a single framework. Moreover, we rigorously analyze the error bound for the proposed scheme that is totally explicit with respect to the number of simulation paths <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M\">\n  <mml:semantics>\n    <mml:mi>M<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">M<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and the time step size <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta t\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n      <mml:mi>t<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\Delta t<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. As a by-product, we prove the maximum principle of the proposed schemes for fractional Allen Cahn equation with double well potential and provide an error estimate by using the established maximum principle. Extensive numerical experiments in two and three-dimensional spaces are presented to verify our theoretical results and to show the robustness and accuracy of our method.<\/p>","DOI":"10.1090\/mcom\/4044","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T13:36:07Z","timestamp":1731504967000},"page":"263-291","source":"Crossref","is-referenced-by-count":1,"title":["An implicit-explicit Monte Carlo method for semi-linear PDEs driven by \ud835\udefc-stable L\u00e9vy process and its error estimates"],"prefix":"10.1090","volume":"95","author":[{"given":"Changtao","family":"Sheng","sequence":"first","affiliation":[]},{"given":"Bihao","family":"Su","sequence":"additional","affiliation":[]},{"given":"Chenglong","family":"Xu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,12,31]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1137\/15M1033952","article-title":"A fractional Laplace equation: regularity of solutions and finite element approximations","volume":"55","author":"Acosta, Gabriel","year":"2017","journal-title":"SIAM J. 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