Abstract
This work presents the fourth-order comprehensive sensitivity analysis methodology for nonlinear systems (abbreviated as “4th-CASAM-N”) for exactly and efficiently computing the first-, second-, third-, and fourth-order functional derivatives (customarily called “sensitivities”) of physical system responses (i.e., “system performance parameters”) to the system’s (or model) parameters. The qualifier “comprehensive” indicates that the 4th-CASAM-N methodology enables the exact and efficient computation not only of response sensitivities with respect to the customary model parameters (including computational input data, correlations, initial and/or boundary conditions) but also with respect to imprecisely known material boundaries, caused by manufacturing tolerances, of the system under consideration. The 4th-CASAM-N methodology presented in this work enables the hitherto very difficult, if not intractable, exact computation of all of the first-, second-, third-, and fourth-order response sensitivities for large-scale systems involving many parameters, as usually encountered in practice. Notably, the implementation of the 4th-CASAM-N requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. The application of the principles underlying the 4th-CASAM-N to an illustrative paradigm nonlinear heat conduction model will be presented in an accompanying work.
Highlights
Academic Editor: Brian KiedrowskiThe computational model of a physical system comprises the following conceptual components: (a) a well-posed system of that relate the system’s independent variables and parameters to the system’s state variables; (b) probability distributions, moments thereof, inequality and/or equality constraints that define the range of variations of the system’s parameters; and (c) one or several quantities, customarily referred to as system responses, which are computed using the mathematical model
The foundation for the material presented in this work is provided by the first-order adjoint sensitivity analysis procedure for nonlinear systems that was originally formulated in a general, functional analytic framework by Cacuci [1,2]
Circumvents the need for solving the 2nd-LVSS by deriving an alternative expression for the indirect-effect term defined in Equation (47), in which the function V(2) (2; x) is replaced by a second-level adjoint function which is independent of variations in the model parameter and state functions
Summary
The computational model of a physical system comprises the following conceptual components: (a) a well-posed system of that relate the system’s independent variables and parameters to the system’s state (i.e., dependent) variables; (b) probability distributions, moments thereof, inequality and/or equality constraints that define the range of variations of the system’s parameters; and (c) one or several quantities, customarily referred to as system responses (or objective functions, or indices of performance), which are computed using the mathematical model This works presents a new, general-purpose methodology for computing exactly and efficiently functional derivatives (called “sensitivities”) of results (“system responses”), predicted by nonlinear mathematical models of systems (physical, engineering, biological) involving imprecisely known (i.e., uncertain) parameters, including input data, correlations, initial and/or boundary conditions, as well as manufacturing tolerances that affect domain of the model’s definition in phase space. The application of the principles underlying the 4th-CASAM-N is illustrated in an accompanying work [17] by means of a paradigm nonlinear heat conduction model
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