adjoint sensitivity
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Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 93
Author(s):  
Geovanny Gordillo ◽  
Mario Morales-Hernández ◽  
Pilar García-Navarro

Water quality control and the control of contaminant spill in water in particular are becoming a primary need today. Gradient descent sensitivity methods based on the adjoint formulation have proved to be encouraging techniques in this context for river and channel flows. Taking into account that most channels and rivers include junctions with other branches, the objective of this study is to explore the adjoint technique on a channel network to reconstruct the upstream boundary condition of the convection-reaction equation. For this purpose, the one-dimensional shallow water equations and the transport equation for a reactive solute are considered. The control is formulated through the gradient-descent technique supplied with a first-order iterative process. Both the physical and the adjoint equations are supplied with suitable internal boundary conditions at the junction and are numerically solved using a finite volume upwind scheme. The results reveal that the adjoint technique is capable of reconstructing the inlet solute concentration boundary condition in an acceptable number of iterations for both steady state and transient configurations using a downstream measurement location. It was also observed that the reconstruction of the boundary condition tends to be less effective the further away the measurement station is from the target.


Author(s):  
Wenjia Wang ◽  
Peter M. Clausen ◽  
Kai-Uwe Bletzinger

AbstractIn 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.


Energies ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 8315
Author(s):  
Dan Gabriel Cacuci

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.


Energies ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 8314
Author(s):  
Dan Gabriel Cacuci

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.


Author(s):  
Gregory Wilson

Abstract 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.


2021 ◽  
Author(s):  
Rachel Mester ◽  
Alfonso Landeros ◽  
Christopher Rackauckas ◽  
Kenneth Lange

Differential sensitivity analysis is indispensable in fitting parameters, understanding uncertainty, and forecasting the results of both thought and lab experiments. Although there are many methods currently available for performing differential sensitivity analysis of biological models, it can be difficult to determine which method is best suited for a particular model. In this paper, we explain a variety of differential sensitivity methods and assess their value in some typical biological models. First, we explain the mathematical basis for three numerical methods: adjoint sensitivity analysis, complex-perturbation sensitivity analysis, and forward-mode sensitivity analysis. We then carry out four instructive case studies. (i) The CARRGO model for tumor-immune interaction highlights the additional information that differential sensitivity analysis provides beyond traditional naive sensitivity methods, (ii) the deterministic SIR model demonstrates the value of using second-order sensitivity in refining model predictions, (iii) the stochastic SIR model shows how differential sensitivity can be attacked in stochastic modeling, and (iv) a discrete birth-death-migration model illustrates how the complex-perturbation method of differential sensitivity can be generalized to a broader range of biological models. Finally, we compare the speed, accuracy, and ease of use of these methods. We find that forward-mode automatic differentiation has the quickest computation time, while the complex-perturbation method is the simplest to implement and the most generalizable.


2021 ◽  
Vol 89 (1) ◽  
Author(s):  
Yisong Qiu ◽  
Shuaiqi Zhang ◽  
Weisheng Zhang ◽  
Hongfei Ye ◽  
Hongwu Zhang ◽  
...  

Abstract A coupling of moving morphable void and component approach for the topology optimization of hydrogel structures involving recoverable large deformation is proposed in this paper. In this approach, the geometric parameters of moving morphable voids and components are set as design variables to respectively describe the outline and material distribution of hydrogel structures for the first time. To facilitate the numerical simulation of large deformation behavior of hydrogel structures during the optimization process, the design variables are mapped to the density field of the design domain and the density field is then used to interpolate the strain energy density function of the element. Furthermore, the adjoint sensitivity of the optimization formulation is derived and combined with the gradient-based algorithm to solve the topology optimization problem effectively. Finally, two representative numerical examples of the optimization of isotropic hydrogel structures are used to prove the effectiveness of the proposed method, and the optimization design of an anisotropic bionic hydrogel structure is presented to illustrate the applicability of the method. Experimental results are also presented to demonstrate that the explicit topologies obtained from the method can be directly used in the manufacture of hydrogel-based soft devices.


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