{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:47:18Z","timestamp":1776836838348,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"342","license":[{"start":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T00:00:00Z","timestamp":1707436800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"minus d i v left-parenthesis a nabla u right-parenthesis equals f\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>div<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">\n                              \u2207\n                              \n                            <\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-\\operatorname {div}(a\\nabla u)=f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a equals exp left-parenthesis b right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>exp<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a=\\exp (b)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a Gaussian random field. The approximant of the solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u\">\n                        <mml:semantics>\n                          <mml:mi>u<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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>\n                    -term polynomial expansion in the scalar Gaussian random variables that parametrize\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We present a general convergence analysis of weighted least-squares approximants for smooth and arbitrarily rough random field, using a suitable random design, for which we prove optimality in the following sense: their convergence rate matches exactly or closely the rate that has been established in Bachmayr, Cohen, DeVore, and Migliorati [ESAIM Math. Model. Numer. Anal. 51 (2017), pp. 341\u2013363] for best\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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>\n                    -term approximation by Hermite polynomials, under the same minimial assumptions on the Gaussian random field. This is in contrast with the current state of the art results for the stochastic Galerkin method that suffers the lack of coercivity due to the lognormal nature of the diffusion field. Numerical tests with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as the Brownian bridge confirm our theoretical findings.\n                  <\/p>","DOI":"10.1090\/mcom\/3825","type":"journal-article","created":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T13:50:03Z","timestamp":1674049803000},"page":"1665-1691","source":"Crossref","is-referenced-by-count":3,"title":["Near-optimal approximation methods for elliptic PDEs with lognormal coefficients"],"prefix":"10.1090","volume":"92","author":[{"given":"Albert","family":"Cohen","sequence":"first","affiliation":[]},{"given":"Giovanni","family":"Migliorati","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,2,9]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s00041-003-0022-0","article-title":"Rate optimality of wavelet series approximations of fractional Brownian motion","volume":"9","author":"Ayache, Antoine","year":"2003","journal-title":"J. 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