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To numerically recover the diffusion coefficient, we employ the standard output least-squares formulation with an\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$H^1(\\varOmega )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>\u03a9<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -seminorm penalty, and discretize the regularized problem by the Galerkin finite element method with continuous piecewise linear finite elements in space and backward Euler convolution quadrature in time. Further, we provide an error analysis of discrete approximations, and prove a convergence rate that matches the stability estimate. The derived\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$L^2(\\varOmega )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>\u03a9<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    error bound depends explicitly on the noise level, regularization parameter and discretization parameters, which gives a useful guideline of the a priori choice of discretization parameters with respect to the noise level in practical implementation. The error analysis is achieved using the conditional stability argument and discrete maximum-norm resolvent estimates. Several numerical experiments are also given to illustrate and complement the theoretical analysis.\n                  <\/jats:p>","DOI":"10.1007\/s00211-025-01495-2","type":"journal-article","created":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T15:26:16Z","timestamp":1760455576000},"page":"2323-2355","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Numerical recovery of the diffusion coefficient in diffusion equations from terminal measurement"],"prefix":"10.1007","volume":"157","author":[{"given":"Bangti","family":"Jin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiliang","family":"Lu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qimeng","family":"Quan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,14]]},"reference":[{"issue":"12","key":"1495_CR1","doi-asserted-by":"publisher","first-page":"3293","DOI":"10.1029\/92WR01757","volume":"28","author":"EE Adams","year":"1992","unstructured":"Adams, E.E., Gelhar, L.W.: Field study of dispersion in a heterogeneous aquifer: 2. 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