{"title":"随机微观结构的有限维模型","authors":"M. Grigoriu","doi":"10.1090/tpms/1168","DOIUrl":null,"url":null,"abstract":"<p>Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and construct approximations of solutions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">U(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of ordinary or partial differential equations whose random coefficients depend on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. FD models of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">U(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">U(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for two types of random fields <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and a simple stochastic equation. Some of these conditions are illustrated by numerical examples.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Finite dimensional models for random microstructures\",\"authors\":\"M. Grigoriu\",\"doi\":\"10.1090/tpms/1168\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Z(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and construct approximations of solutions <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>U</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">U(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of ordinary or partial differential equations whose random coefficients depend on <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Z(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. FD models of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Z(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>U</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">U(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Z(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>U</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">U(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for two types of random fields <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>Z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Z(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and a simple stochastic equation. Some of these conditions are illustrated by numerical examples.</p>\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tpms/1168\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1168","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Finite dimensional models for random microstructures
Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields Z(x)Z(x) and construct approximations of solutions U(x)U(x) of ordinary or partial differential equations whose random coefficients depend on Z(x)Z(x). FD models of Z(x)Z(x) and U(x)U(x) constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of Z(x)Z(x) and U(x)U(x) for two types of random fields Z(x)Z(x) and a simple stochastic equation. Some of these conditions are illustrated by numerical examples.