{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:44:08Z","timestamp":1778496248429,"version":"3.51.4"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"339","license":[{"start":{"date-parts":[[2023,8,12]],"date-time":"2023-08-12T00:00:00Z","timestamp":1691798400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["G091920N"],"award-info":[{"award-number":["G091920N"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["81617985"],"award-info":[{"award-number":["81617985"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009123","name":"Norges Teknisk-Naturvitenskapelige Universitet","doi-asserted-by":"publisher","award":["G091920N"],"award-info":[{"award-number":["G091920N"]}],"id":[{"id":"10.13039\/100009123","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009123","name":"Norges Teknisk-Naturvitenskapelige Universitet","doi-asserted-by":"publisher","award":["81617985"],"award-info":[{"award-number":["81617985"]}],"id":[{"id":"10.13039\/100009123","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We introduce a new method to approximate integrals\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"integral Underscript double-struck upper R Superscript d Endscripts f left-parenthesis bold-italic x right-parenthesis normal d bold-italic x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mo>\n                                \u222b\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msup>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mi>d<\/mml:mi>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"bold-italic\">x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\int _{\\mathbb {R}^d} f(\\boldsymbol {x}) \\,\\mathrm {d}\\boldsymbol {x}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which simply scales lattice rules from the unit cube\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 0 comma 1 right-bracket Superscript d\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">]<\/mml:mo>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[0,1]^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to properly sized boxes on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript d\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , hereby achieving higher-order convergence that matches the smoothness of the integrand function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and does not assume a particular density nor the ability to sample from it. In particular, for the theoretical analysis we show a new result that the method of adding Bernoulli polynomials to a function to make it \u201cperiodic\u201d on a box without changing its integral value over the box is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its excessive computational complexity and also here we only make use of it in our theoretical analysis. We show that our new method of applying scaled lattice rules to increasing boxes can be interpreted as orthogonal projections with decreasing projection error. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings.\n                  <\/p>","DOI":"10.1090\/mcom\/3754","type":"journal-article","created":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T10:12:20Z","timestamp":1652263940000},"page":"307-347","source":"Crossref","is-referenced-by-count":9,"title":["Scaled lattice rules for integration on \u211d^{\ud835\udd55} achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces"],"prefix":"10.1090","volume":"92","author":[{"given":"Dirk","family":"Nuyens","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuya","family":"Suzuki","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2022,8,12]]},"reference":[{"key":"1","series-title":"Lecture Notes in Computational Science and Engineering","volume-title":"Stochastic spectral {G}alerkin and collocation methods for {PDE}s with random coefficients: a numerical comparison","volume":"76","author":"B\u00e4ck, Joakim","year":"2011"},{"key":"2","isbn-type":"print","first-page":"329","article-title":"Transformation of integrands for lattice rules","author":"Beckers, Marc","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0792315839"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"2162","DOI":"10.1137\/06065074X","article-title":"Constructing embedded lattice rules for multivariable integration","volume":"28","author":"Cools, Ronald","year":"2006","journal-title":"SIAM J. 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