{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:23Z","timestamp":1787336903454,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.<\/jats:p>","DOI":"10.1137\/s1064827599352136","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"39-62","source":"Crossref","is-referenced-by-count":35,"title":["On Efficient Solutions to the Continuous Sensitivity Equation Using Automatic Differentiation"],"prefix":"10.1137","volume":"22","author":[{"given":"Jeff","family":"Borggaard","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arun","family":"Verma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":"C. Bischof, A. Carle, A. Griewank, and P. Hovland,\n                      ADIFOR: Generating Derivative Codes From Fortran Programs\n                      , Technical Report MCS\u2010P263\u20100991, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, and Center for Research on Parallel Computation, Rice University, Houston, TX, 1991."},{"key":"R2","unstructured":"Jeff Borggaard, John Burns, Asymptotically consistent gradients in optimal design, SIAM, Philadelphia, PA, 1997, 303\u20133141438320"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5743"},{"key":"R4","unstructured":"Jeff Borggaard, John Burns, Eugene Cliff, Max Gunzburger, Sensitivity calculations for a 2D, inviscid, supersonic forebody problem, SIAM, Philadelphia, PA, 1993, 14\u2013251231084"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"J. Borggaard and D. Pelletier,\n                      Computing design sensitivities using an adaptive finite element method\n                      , in Proceedings of the 27th AIAA Fluid Dynamics Conference, New Orleans, LA, 1996.","DOI":"10.2514\/6.1996-1938"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620361807"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. Burns, L. Stanley, and D. Stewart,\n                      Computational methods for design sensitivities\n                      , in Optimal Control: Theory, Algorithms Appl., Appl. Optim. 15, W. H. Hager and P. M. Pardalos, eds., Kluwer, Dordrecht, 1998, pp. 40\u201366.","DOI":"10.1007\/978-1-4757-6095-8_3"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01743969"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"Thomas Coleman, Arun Verma, Structure and efficient Hessian calculation, Appl. Optim., Vol. 14, Kluwer Acad. Publ., Dordrecht, 1998, 57\u20137299d:90110","DOI":"10.1007\/978-1-4613-3335-7_3"},{"key":"R10","unstructured":"T. Coleman and A. Verma,\n                      Structure and efficient Jacobian calculation\n                      , in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bishof, G. Corliss, and A. Griewank, eds., SIAM, Philadelphia, PA, 1996, pp. 149\u2013159."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"Thomas Coleman, Fadil Santosa, Arun Verma, Semi\u2010automatic differentiation, Progr. Systems Control Theory, Vol. 24, Birkh\u00e4user Boston, Boston, MA, 1998, 113\u201312699e:65031","DOI":"10.1007\/978-1-4612-1780-0_7"},{"key":"R12","unstructured":"A. Verma,\n                      ADMAT: Automatic differentiation in MATLAB using object oriented methods\n                      , in Proceedings of the 1998 SIAM Workshop on Object Oriented Methods for Interoperable Scientific and Engineering Computing, M. E. Henderson, C. R. Anderson, and S. L. Lyons, eds., SIAM, Philadelphia, 1999, pp. 174\u2013183."},{"key":"R13","unstructured":"T. F. Coleman and A. Verma,\n                      ADMIT\n                      \u20101:\n                      Automatic differentiation and MATLAB interface toolbox\n                      , ACM Trans. Math. Software, to appear."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01027-3"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1080\/10556789208805505"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A. Griewank,\n                      Some bounds on the complexity of gradients, Jacobians, and Hessians\n                      , in Complexity in Nonlinear Optimization, P. Pardalos, ed., World Scientific Publishers, River Edge, NJ, 1993, pp. 131\u2013167.","DOI":"10.1142\/9789814354363_0008"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1145\/229473.229474"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0703010"},{"key":"R19","unstructured":"D. Stewart,\n                      Numerical Methods for Accurate Computation of Design Sensitivities\n                      , Ph.D. thesis, Virginia Tech, Blacksburg, VA, 1998."},{"key":"R20","unstructured":"J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin,\n                      Numerical Grid Generation: Foundations and Applications\n                      , North\u2010Holland Publishing Company, New York, 1985."},{"key":"R21","unstructured":"A. Verma,\n                      Structured Automatic Differentiation\n                      , Ph.D. thesis, Cornell University, Ithaca, NY, 1998."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827599352136","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:46Z","timestamp":1787333746000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827599352136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S1064827599352136"],"URL":"https:\/\/doi.org\/10.1137\/s1064827599352136","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}