%0 Journal Article
%T Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework
%A Dan Gabriel Cacuci
%J American Journal of Computational Mathematics
%P 329-354
%@ 2161-1211
%D 2020
%I Scientific Research Publishing
%R 10.4236/ajcm.2020.103018
%X This work presents the ¡°Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)¡± for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model¡¯s response (<em>i.e.</em>, model result of interest) is a generic nonlinear function of the model¡¯s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2<sup>nd</sup>-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1<sup>st</sup>-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2<sup>nd</sup>-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (<em>N</em>2/2 + 3 <em>N</em>/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2<sup>nd</sup>-LASS requires very little additional effort beyond the construction of the 1<sup>st</sup>-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1<sup>st</sup>-LASS and 2<sup>nd</sup>-LASS requires the same computational solvers as needed for solving (<em>i.e.</em>, ¡°inverting¡±) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and ¡°solvers¡± used for solving the original system of equations can also be used for solving the 1<sup>st</sup>-LASS and the 2<sup>nd</sup>-LASS. Since neither the 1<sup>st</sup>-LASS nor the 2<sup>nd</sup>-LASS involves any differentials of the operators underlying the original system, the 1<sup>st</sup>-LASS is designated as a ¡°<u>first-level</u>¡± (as opposed to a ¡°first-order¡±) adjoint sensitivity system, while the 2<sup>nd</sup>-LASS is designated as a ¡°<u>second-level</u>¡± (rather than a ¡°second-order¡±) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2<sup>nd</sup>-LASS that involve the imprecisely known boundary parameters. Notably, the 2<sup>nd</sup>-LASS encompasses an automatic, inherent, and independent ¡°solution verification¡±
%K Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2&lt
%K sup&gt
%K nd&lt
%K /sup&gt
%K -CASAM)
%K First-Level Adjoint Sensitivity System (1st&lt
%K /sup&gt
%K -LASS)
%K Second-Level Adjoint Sensitivity System (2&lt
%K sup&gt
%K nd&lt
%K /sup&gt
%K -LASS)
%K Operator-Type  Response
%K Second-Order Sensitivities to Uncertain Model Boundaries
%K Second-Order Sensitivities to Uncertain Model Parameters
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=101646