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Sloan Foundation","doi-asserted-by":"publisher","award":["1R01HG008383-01A1"],"award-info":[{"award-number":["1R01HG008383-01A1"]}],"id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher","award":["T32GM007205"],"award-info":[{"award-number":["T32GM007205"]}],"id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher","award":["1763179"],"award-info":[{"award-number":["1763179"]}],"id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G equals left-parenthesis upper V comma upper E comma w right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>w<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G=(V,E,w)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W subset-of upper V\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>W<\/mml:mi>\n                            <mml:mo>\n                              \u2282\n                              \n                            <\/mml:mo>\n                            <mml:mi>V<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">W \\subset V<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of vertices and weights\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a Subscript w\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mi>w<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">a_w<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction 1 Over StartAbsoluteValue upper V EndAbsoluteValue EndFraction sigma-summation Underscript v element-of upper V Overscript Endscripts f left-parenthesis v right-parenthesis tilde sigma-summation Underscript w element-of upper W Endscripts a Subscript w Baseline f left-parenthesis w right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mfrac>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mrow>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo stretchy=\"false\">|<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mi>V<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo stretchy=\"false\">|<\/mml:mo>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>v<\/mml:mi>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:mi>V<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\"\/>\n                            <\/mml:munderover>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>v<\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u223c\n                              \n                            <\/mml:mo>\n                            <mml:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>w<\/mml:mi>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:mi>W<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>a<\/mml:mi>\n                                <mml:mi>w<\/mml:mi>\n                              <\/mml:msub>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\frac {1}{|V|}\\sum _{v \\in V}^{}{f(v)} \\sim \\sum _{w \\in W}{a_w f(w)} \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    for functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f colon upper V right-arrow double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f:V \\rightarrow \\mathbb {R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that are \u201csmooth\u201d with respect to the geometry of the graph; here\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tilde\">\n                        <mml:semantics>\n                          <mml:mo>\n                            \u223c\n                            \n                          <\/mml:mo>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sim<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    indicates that we want the right-hand side to be as close to the left-hand side as possible. The main application comprises problems where\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                    is known to vary smoothly over the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem (\u201cthe optimal packing of heat balls\u201d). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.\n                  <\/p>","DOI":"10.1090\/mcom\/3515","type":"journal-article","created":{"date-parts":[[2019,12,11]],"date-time":"2019-12-11T10:26:09Z","timestamp":1576059969000},"page":"1933-1952","source":"Crossref","is-referenced-by-count":14,"title":["Numerical integration on graphs: Where to sample and how to weigh"],"prefix":"10.1090","volume":"89","author":[{"given":"George","family":"Linderman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Steinerberger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2020,1,29]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s00365-017-9412-4","article-title":"The Stolarsky principle and energy optimization on the sphere","volume":"48","author":"Bilyk, Dmitriy","year":"2018","journal-title":"Constr. 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