Load step reduction for adjoint sensitivity analysis of finite strain elastoplasticity

In this paper, load step reduction techniques are investigated for adjoint sensitivity analysis of path-dependent nonlinear finite element systems. In particular, the focus is on finite strain elastoplasticity with typical hardening models. The aim is to reduce the computational cost in the adjoint sensitivity implementation. The adjoint sensitivity formulation is derived with the multiplicative decomposition of deformation gradient, which is applicable to finite strain elastoplasticity. Two properties of adjoint variables are investigated and theoretically proved under certain prerequisites. Based on these properties, load step reduction rules in the sensitivity analysis are discussed. The efficiency of the load step reduction and the applicability to isotropic hardening and kinematic hardening models are numerically demonstrated. Examples include a small-scale cantilever beam structure and a large-scale conrod structure under huge plastic deformations.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): II. Illustrative Application

This work illustrates the application of the nth-order comprehensive adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (abbreviated as “nth-CASAM-L”) to a paradigm model that describes the transmission of particles (neutrons and/or photons) through homogenized materials, as encountered in radiation protection and shielding. The first-, second-, and third-order sensitivities of responses that depend on both the forward and adjoint particle fluxes are obtained exactly, in closed-form, underscoring the principles and methodology underlying the nth-CASAM-L. The results presented in this work underscore the fundamentally important role of the nth-CASAM-L in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): I. Mathematical Framework

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.

Sensitivity of Surface Currents to Bathymetry in a Partially-Mixed Estuary with Applications to Inverse Modeling

An inversion technique was tested for estimating bathymetry from observations of surface currents in a partially-mixed estuary, Mouth of the Columbia River (MCR). The methodology uses an iterative ensemble-based assimilation scheme which is found to have good skill for recovering bathymetry from observations distributed in space and time. However, the inversion skill is highly dependent on the tidal phase, location of the observations, and flow-dependent estuary dynamics. Inversion skill was found to degrade during periods of higher river discharge (up to ~ 12,000m3), or low tidal amplitude, while inversion of depth-averaged velocities instead of surface velocities caused increased skill throughout the domain. These results point to dynamical limits on inversion skill, caused by changes in estuary dynamics that affect the sensitivity of surface velocities to bathymetry. An adjoint sensitivity analysis is used to visualize these effects and is combined with data-denial experiments to explore the flow-dependent inversion skill.

Fourth-Order Comprehensive Adjoint Sensitivity Analysis (4th-CASAM) of Response-Coupled Linear Forward/Adjoint Systems: I. Theoretical Framework

The most general quantities of interest (called “responses”) produced by the computational model of a linear physical system can depend on both the forward and adjoint state functions that describe the respective system. This work presents the Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (4th-CASAM) for linear systems, which enables the efficient computation of the exact expressions of the 1st-, 2nd-, 3rd- and 4th-order sensitivities of a generic system response, which can depend on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward/adjoint systems. Among the best known such system responses are various Lagrangians, including the Schwinger and Roussopoulos functionals, for analyzing ratios of reaction rates, the Rayleigh quotient for analyzing eigenvalues and/or separation constants, etc., which require the simultaneous consideration of both the forward and adjoint systems when computing them and/or their sensitivities (i.e., functional derivatives) with respect to the model parameters. Evidently, such responses encompass, as particular cases, responses that may depend just on the forward or just on the adjoint state functions pertaining to the linear system under consideration. This work also compares the CPU-times needed by the 4th-CASAM versus other deterministic methods (e.g., finite-difference schemes) for computing response sensitivities These comparisons underscore the fact that the 4th-CASAM is the only practically implementable methodology for obtaining and subsequently computing the exact expressions (i.e., free of methodologically-introduced approximations) of the 1st-, 2nd, 3rd- and 4th-order sensitivities (i.e., functional derivatives) of responses to system parameters, for coupled forward/adjoint linear systems. By enabling the practical computation of any and all of the 1st-, 2nd, 3rd- and 4th-order response sensitivities to model parameters, the 4th-CASAM makes it possible to compare the relative values of the sensitivities of various order, in order to assess which sensitivities are important and which may actually be neglected, thus enabling future investigations of the convergence of the (multivariate) Taylor series expansion of the response in terms of parameter variations, as well as investigating the range of validity of other important quantities (e.g., response variances/covariance, skewness, kurtosis, etc.) that are derived from Taylor-expansion of the response as a function of the model’s parameters. The 4th-CASAM presented in this work provides the basis for significant future advances towards overcoming the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

Nonlinear Schrodinger Equation-Based Adjoint Sensitivity Analysis

A general nonlinear adjoint sensitivity analysis (ASA) approach for the time-dependent nonlinear Schrodinger equation (NLSE) is presented. The proposed algorithm estimates the sensitivities of a desired objective function with respect to all design parameters using only one extra adjoint system simulation. The approach efficiency is shown here through a numerical example.