{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:39:09Z","timestamp":1776724749869,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider numerical methods for a \u201cquasi-boundary value\u201d regularization of the backward parabolic problem given by\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout Enlarged left-brace 1st Row 1st Column u Subscript t Baseline plus upper A u equals 0 comma 2nd Column a m p semicolon 0 greater-than t greater-than upper T 2nd Row 1st Column u left-parenthesis upper T right-parenthesis equals f comma 2nd Column a m p semicolon EndLayout\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\">\n                              <mml:mtr>\n                                <mml:mtd>\n                                  <mml:msub>\n                                    <mml:mi>u<\/mml:mi>\n                                    <mml:mi>t<\/mml:mi>\n                                  <\/mml:msub>\n                                  <mml:mo>+<\/mml:mo>\n                                  <mml:mi>A<\/mml:mi>\n                                  <mml:mi>u<\/mml:mi>\n                                  <mml:mo>=<\/mml:mo>\n                                  <mml:mn>0<\/mml:mn>\n                                  <mml:mo>,<\/mml:mo>\n                                <\/mml:mtd>\n                                <mml:mtd>\n                                  <mml:mi>a<\/mml:mi>\n                                  <mml:mi>m<\/mml:mi>\n                                  <mml:mi>p<\/mml:mi>\n                                  <mml:mo>;<\/mml:mo>\n                                  <mml:mn>0<\/mml:mn>\n                                  <mml:mo>&gt;<\/mml:mo>\n                                  <mml:mi>t<\/mml:mi>\n                                  <mml:mo>&gt;<\/mml:mo>\n                                  <mml:mi>T<\/mml:mi>\n                                <\/mml:mtd>\n                              <\/mml:mtr>\n                              <mml:mtr>\n                                <mml:mtd>\n                                  <mml:mi>u<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                  <mml:mi>T<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <mml:mo>=<\/mml:mo>\n                                  <mml:mi>f<\/mml:mi>\n                                  <mml:mo>,<\/mml:mo>\n                                <\/mml:mtd>\n                                <mml:mtd>\n                                  <mml:mi>a<\/mml:mi>\n                                  <mml:mi>m<\/mml:mi>\n                                  <mml:mi>p<\/mml:mi>\n                                  <mml:mo>;<\/mml:mo>\n                                <\/mml:mtd>\n                              <\/mml:mtr>\n                            <\/mml:mtable>\n                            <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\"\/>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin {cases} u_t+Au=0, &amp; 0&gt;t&gt;T \\\\ u(T)=f, &amp; \\end {cases}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u left-parenthesis upper T right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u(T)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by adding\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha u left-parenthesis 0 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha u(0)<\/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=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a small parameter. We show how this leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson. Error estimates and examples are provided. We also compare the regularization used here with that from Ames and Epperson.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-01014-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1451-1471","source":"Crossref","is-referenced-by-count":63,"title":["A comparison of regularizations for an ill-posed problem"],"prefix":"10.1090","volume":"67","author":[{"given":"Karen","family":"Ames","sequence":"first","affiliation":[]},{"given":"Gordon","family":"Clark","sequence":"additional","affiliation":[]},{"given":"James","family":"Epperson","sequence":"additional","affiliation":[]},{"given":"Seth","family":"Oppenheimer","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"Ames, K.A., and Epperson, J.F., A Kernel-based Method for the Approximate Solution of Backward Parabolic Problems, SIAM J. 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Differential Equations"},{"key":"5","series-title":"Johns Hopkins Series in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","volume":"3","author":"Golub, Gene H.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0801830109"},{"key":"6","series-title":"Research Notes in Mathematics","isbn-type":"print","volume-title":"The theory of Tikhonov regularization for Fredholm equations of the first kind","volume":"105","author":"Groetsch, C. W.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086421"},{"key":"7","series-title":"Frontiers in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970944","volume-title":"Iterative methods for linear and nonlinear equations","volume":"16","author":"Kelley, C. 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