TITLE:
Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): I. Mathematical Framework
AUTHORS:
Dan Gabriel Cacuci
KEYWORDS:
Computation of High-Order Sensitivities, Sensitivities to Features of Model Parameters, Sensitivities to Domain Boundaries, Adjoint Sensitivity Systems
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.14 No.1,
March
27,
2024
ABSTRACT: This work presents the “nth-Order Feature
Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as
“nth-FASAM-N”), which will be shown to be the most efficient
methodology for computing exact expressions of sensitivities, of any order, of
model responses with respect to features of model parameters and, subsequently,
with respect to the model’s uncertain parameters, boundaries, and internal
interfaces. The unparalleled efficiency and accuracy of the nth-FASAM-N
methodology stems from the maximal reduction of the number of adjoint
computations (which are considered to be “large-scale” computations) for
computing high-order sensitivities. When applying the nth-FASAM-N
methodology to
compute the second- and higher-order sensitivities, the number of large-scale
computations is proportional to the number of “model features” as opposed to
being proportional to the number of model parameters (which are considerably
more than the number of features).When a model has no “feature” functions of
parameters, but only comprises primary parameters, the nth-FASAM-N
methodology becomes identical to the extant nth CASAM-N (“nth-Order
Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”)
methodology. Both the nth-FASAM-N and the nth-CASAM-N
methodologies are formulated in linearly increasing higher-dimensional Hilbert
spaces as opposed to exponentially increasing parameter-dimensional spaces thus
overcoming the curse of dimensionality in sensitivity analysis of nonlinear
systems. Both the nth-FASAM-N and the nth-CASAM-N are
incomparably more efficient and more accurate than any other methods
(statistical, finite differences, etc.) for computing exact expressions of
response sensitivities of any order with respect to the model’s features and/or
primary uncertain parameters, boundaries, and internal interfaces.