Abstract
This work presents the mathematical framework of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “nth-CASAM-L”), which is conceived for obtaining the exact expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Since many of the most important responses for linear systems involve the solutions of both the forward and the adjoint linear models that correspond to the respective physical system, the sensitivity analysis of such responses makes it necessary to treat linear systems in their own right, rather than treating them as particular cases of nonlinear systems. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward functions, since nonlinear operators do not admit bona-fide adjoint operators (only a linearized form of a nonlinear operator admits an adjoint operator). The nth-CASAM-L determines the exact expression of arbitrarily-high order sensitivities of responses to the parameters underlying both the forward and adjoint models of a nonlinear system, thus enable the most efficient and accurate computation of such sensitivities. The mathematical framework underlying the nth-CASAM is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing “parameter-dimensional” spaces in which response sensitivities are computed by other methods, thus providing the basis for overcoming the “curse of dimensionality” in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. In particular, for a scalar-valued valued response associated with a nonlinear model comprising TP parameters, the 1st-−CASAM-L requires 1 additional large-scale adjoint computation (as opposed to TP large-scale computations, as required by other methods) for computing exactly all of the 1st-−order response sensitivities. All of the (mixed) 2nd-order sensitivities are computed exactly by the 2nd-CASAM-L in at most TP computations, as opposed to TP(TP + 1)/2 computations required by all other methods, and so on. For every lower-order sensitivity of interest, the nth-CASAM-L computes the “TP next-higher-order” sensitivities in one adjoint computation performed in a linearly increasing higher-dimensional Hilbert space. Very importantly, the nth-CASAM-L computes the higher-level adjoint functions using the same forward and adjoint solvers (i.e., computer codes) as used for solving the original forward and adjoint systems, thus requiring relatively minor additional software development for computing the various-order sensitivities.
Highlights
IntroductionNon-linear operators do not admit adjoint operators; only linear operators admit corresponding adjoint operators
Since the model parameters α and the state functions are related to each other through the forward and adjoint systems, it follows that variations δα, in the model parameters will cause corresponding variations δφ, δφi, φi − φ0i, i = 1, . . . , TD and δψ, δψi, ψi − ψi0, i = 1, . . . , TD in the forward and, respectively, adjoint state functions
The 2nd -CASAM-L: Summary hFor each value ofi the index j1 = 1, . . . , TP, the first-order sensitivity j1 ; u(1) (x); a(1) (x); α will be considered to play the role of a “model response.”
Summary
Non-linear operators do not admit adjoint operators; only linear operators admit corresponding adjoint operators For this reason, many of the most important responses for linear systems involve the solutions of both the forward and the adjoint linear models that correspond to the respective physical system. Included among the widest used system responses that involve both the forward and adjoint functions are the various forms of Lagrangian functionals, the Raleigh quotient for computing eigenvalues and/or separation constants when solving partial differential equations, the Schwinger functional for first-order “normalization-free” solutions [1,2,3,4,5] These functionals play a fundamental role in optimization and control procedures, derivation of numerical methods for solving equations (differential, integral, integro-differential), etc. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward functions, since nonlinear operators do not admit bona-fide adjoint operators—only a linearized form of a nonlinear operator admits an adjoint operator
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