Towards Adjoint-Based Inversion of the Lamé Parameter Field for Slender Structures With Cantilever Loading

2016 ◽  
Author(s):  
Soheil Fatehiboroujeni ◽  
Noemi Petra ◽  
Sachin Goyal
Keyword(s):  
Inverse Problem ◽  
Optimization Problem ◽  
Nonlinear Least Squares ◽  
Constitutive Law ◽  
Observation Error ◽  
Unknown Parameters ◽  
Cost Functional ◽  
Parameter Field ◽  
Dynamics Simulations ◽  
Slender Structures

Continuum models of slender structures are effective in simulating the mechanics of nano-scale filaments. However, the accuracy of these simulations strictly depends on the knowledge of the constitutive laws that may in general be non-homogeneous. It necessitates an inverse problem framework that can leverage the data provided by physical experiments and molecular dynamics simulations to estimate the unknown parameters in the constitutive law. In this paper, we formulate a simple but representative inverse problem as a nonlinear least-squares optimization problem whose cost functional is the misfit between synthetic observations of a cantilever displacement field and model predictions. A Tikhonov regularization term is added to the cost functional to render the problem well-posed and account for observational error. We solve this optimization problem with an adjoint-based inexact Newton-conjugate gradient method. We show that the reconstruction of the Lamé parameter field converges to the exact coefficient as the observation error decreases.

scholarly journals An inexact Gauss-Newton method for inversion of basal sliding and rheology parameters in a nonlinear Stokes ice sheet model

2012 ◽  
Vol 58 (211) ◽  
pp. 889-903 ◽  
Author(s):  
Noemi Petrat ◽  
Hongyu Zhu ◽  
Georg Stadler ◽  
Thomas J.R. Hughes ◽  
Omar Ghattas
Keyword(s):  
Inverse Problem ◽  
Newton Method ◽  
Ice Sheet ◽  
Nonlinear Least Squares ◽  
Surface Velocity ◽  
Inexact Newton Method ◽  
Observation Error ◽  
Nonlinear Rheology ◽  
Cost Functional ◽  
Basal Sliding

AbstractWe propose an infinite-dimensional adjoint-based inexact Gauss-Newton method for the solution of inverse problems governed by Stokes models of ice sheet flow with nonlinear rheology and sliding law. The method is applied to infer the basal sliding coefficient and the rheological exponent parameter fields from surface velocities. The inverse problem is formulated as a nonlinear least-squares optimization problem whose cost functional is the misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to the cost functional to render the problem well-posed and account for observational error. Our findings show that the inexact Newton method is significantly more efficient than the nonlinear conjugate gradient method and that the number of Stokes solutions required to solve the inverse problem is insensitive to the number of inversion parameters. The results also show that the reconstructions of the basal sliding coefficient converge to the exact sliding coefficient as the observation error (here, the noise added to synthetic observations) decreases, and that a nonlinear rheology makes the reconstruction of the basal sliding coefficient more difficult. For the inversion of the rheology exponent field, we find that horizontally constant or smoothly varying parameter fields can be reconstructed satisfactorily from noisy observations.

scholarly journals Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

2016 ◽  
Author(s):  
Hongyu Zhu ◽  
Noemi Petra ◽  
Georg Stadler ◽  
Tobin Isaac ◽  
Thomas J. R. Hughes ◽  
...  
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Optimization Problem ◽  
Premature Termination ◽  
Adjoint Equations ◽  
Surface Velocity ◽  
Geothermal Heat ◽  
Ice Flow ◽  
Cost Functional ◽  
The Cost

Abstract. We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using an instantaneous thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection-diffusion equation, which couples to the nonlinear Stokes ice flow equations, which then determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well-posed. We derive adjoint-based gradient and Hessian expressions for the resulting PDE-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov-Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases, and that small wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems — i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian — we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. We show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.

scholarly journals Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

2016 ◽  
Vol 10 (4) ◽  
pp. 1477-1494 ◽  
Author(s):  
Hongyu Zhu ◽  
Noemi Petra ◽  
Georg Stadler ◽  
Tobin Isaac ◽  
Thomas J. R. Hughes ◽  
...  
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Optimization Problem ◽  
Premature Termination ◽  
Adjoint Equations ◽  
Surface Velocity ◽  
Geothermal Heat ◽  
Ice Flow ◽  
Cost Functional ◽  
The Cost

Abstract. We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection–diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov–Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems – i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. We show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.

scholarly journals Optimal models for estimating future infected cases of COVID-19 in Oman

2020 ◽  
Vol 9 (2) ◽  
pp. 53-66
Author(s):  
Ahmed Al-Siyabi ◽  
Mehiddin Al-Baali ◽  
Anton Purnama
Keyword(s):  
Optimization Problem ◽  
Nonlinear Least Squares ◽  
High Rate ◽  
Unknown Parameters ◽  
Numerical Experience ◽  
Fast Spreading ◽  
Modification Technique ◽  
Least Squares Optimization ◽  
Simple Ratio ◽  
Quasi Newton

The recent coronavirus disease 2019 (COVID-19) outbreak is of high importance in research topics due to its fast spreading and high rate of infections across the world. In this paper, we test certain optimal models of forecasting daily new cases of COVID-19 in Oman. It is based on solving a certain nonlinear least-squares optimization problem that determines some unknown parameters in fitting some mathematical models. We also consider extension to these models to predict the future number of infection cases in Oman. The modification technique introduces a simple ratio rate of changes in the daily infected cases. This average ratio is computed by employing the rule of Al-Baali [Numerical experience with a class of self-scaling quasi-Newton algorithms, JOTA, 96 (1998), pp. 533–553], in a sense to be defined, for measuring the infection changes.

scholarly journals Linearized Bayesian inference for Young’s modulus parameter field in an elastic model of slender structures

2020 ◽  
Vol 476 (2238) ◽  
pp. 20190476
Author(s):  
Soheil Fatehiboroujeni ◽  
Noemi Petra ◽  
Sachin Goyal
Keyword(s):  
Inverse Problem ◽  
Bayesian Inference ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Small Scale ◽  
Elastic Model ◽  
Elasticity Parameter ◽  
Posterior Probability Distribution ◽  
Parameter Field ◽  
Slender Structures

The deformations of several slender structures at nano-scale are conceivably sensitive to their non-homogenous elasticity. Owing to their small scale, it is not feasible to discern their elasticity parameter fields accurately using observations from physical experiments. Molecular dynamics simulations can provide an alternative or additional source of data. However, the challenges still lie in developing computationally efficient and robust methods to solve inverse problems to infer the elasticity parameter field from the deformations. In this paper, we formulate an inverse problem governed by a linear elastic model in a Bayesian inference framework. To make the problem tractable, we use a Gaussian approximation of the posterior probability distribution that results from the Bayesian solution of the inverse problem of inferring Young’s modulus parameter fields from available data. The performance of the computational framework is demonstrated using two representative loading scenarios, one involving cantilever bending and the other involving stretching of a helical rod (an intrinsically curved structure). The results show that smoothly varying parameter fields can be reconstructed satisfactorily from noisy data. We also quantify the uncertainty in the inferred parameters and discuss the effect of the quality of the data on the reconstructions.

Computer Simulation for Active Control of Vibration in Beam Structures

1995 ◽  
Author(s):  
Masa. Tanaka ◽  
T. Matsumoto ◽  
L. Huang
Keyword(s):  
Inverse Problem ◽  
Active Control ◽  
Inverse Analysis ◽  
Taylor Series Expansion ◽  
Integral Equation Method ◽  
Unknown Parameters ◽  
Bending Vibration ◽  
Linear System Of Equations ◽  
Elastic Beams ◽  
The Laplace Transform

Abstract This paper is concerned with an inverse problem of the active control of non-steady dynamic vibration in elastic beams. A simulation technique based on the boundary element method and the extended Kalman filter or a new filter theory is successfully applied to the inverse problem. The Laplace-transform integral equation method is used for the solution of dynamic bending vibration in elastic beams. Through a Taylor series expansion, the linear system of equations is derived for modification of the unknown parameters, and it is solved iteratively so that an appropriate norm is minimized. The usefulness of the proposed method of inverse analysis is demonstrated through numerical computation of a few examples.

Lipschitz stability for a semi-linear inverse stochastic transport problem

2020 ◽  
Vol 28 (2) ◽  
pp. 185-193
Author(s):  
Zhongqi Yin
Keyword(s):  
Inverse Problem ◽  
Main Tool ◽  
Terminal State ◽  
Lipschitz Stability ◽  
Unknown Parameters ◽  
Initial Displacement ◽  
Stochastic Transport ◽  
Random Term ◽  
Global Carleman Estimate ◽  
Stochastic Transport Equation

AbstractThis paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.

scholarly journals Parameterized Nonlinear Least Squares for Unsupervised Nonlinear Spectral Unmixing

2019 ◽  
Vol 11 (2) ◽  
pp. 148 ◽  
Author(s):  
Risheng Huang ◽  
Xiaorun Li ◽  
Haiqiang Lu ◽  
Jing Li ◽  
Liaoying Zhao
Keyword(s):  
Least Squares ◽  
Optimization Problem ◽  
Nonnegative Matrix Factorization ◽  
Optimization Problems ◽  
State Of The Art ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Nonlinear Least Squares ◽  
Hyperspectral Data ◽  
Spectral Unmixing

This paper presents a new parameterized nonlinear least squares (PNLS) algorithm for unsupervised nonlinear spectral unmixing (UNSU). The PNLS-based algorithms transform the original optimization problem with respect to the endmembers, abundances, and nonlinearity coefficients estimation into separate alternate parameterized nonlinear least squares problems. Owing to the Sigmoid parameterization, the PNLS-based algorithms are able to thoroughly relax the additional nonnegative constraint and the nonnegative constraint in the original optimization problems, which facilitates finding a solution to the optimization problems . Subsequently, we propose to solve the PNLS problems based on the Gauss–Newton method. Compared to the existing nonnegative matrix factorization (NMF)-based algorithms for UNSU, the well-designed PNLS-based algorithms have faster convergence speed and better unmixing accuracy. To verify the performance of the proposed algorithms, the PNLS-based algorithms and other state-of-the-art algorithms are applied to synthetic data generated by the Fan model and the generalized bilinear model (GBM), as well as real hyperspectral data. The results demonstrate the superiority of the PNLS-based algorithms.

scholarly journals Generalized Nonlinear Least Squares Method for the Calibration of Complex Computer Code Using a Gaussian Process Surrogate

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 985
Author(s):  
Youngsaeng Lee ◽  
Jeong-Soo Park
Keyword(s):  
Gaussian Process ◽  
Least Squares ◽  
Covariance Matrix ◽  
Process Model ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Computer Code ◽  
Test Functions ◽  
Unknown Parameters ◽  
Complex Computer

The approximated nonlinear least squares (ALS) method has been used for the estimation of unknown parameters in the complex computer code which is very time-consuming to execute. The ALS calibrates or tunes the computer code by minimizing the squared difference between real observations and computer output using a surrogate such as a Gaussian process model. When the differences (residuals) are correlated or heteroscedastic, the ALS may result in a distorted code tuning with a large variance of estimation. Another potential drawback of the ALS is that it does not take into account the uncertainty in the approximation of the computer model by a surrogate. To address these problems, we propose a generalized ALS (GALS) by constructing the covariance matrix of residuals. The inverse of the covariance matrix is multiplied to the residuals, and it is minimized with respect to the tuning parameters. In addition, we consider an iterative version for the GALS, which is called as the max-minG algorithm. In this algorithm, the parameters are re-estimated and updated by the maximum likelihood estimation and the GALS, by using both computer and experimental data repeatedly until convergence. Moreover, the iteratively re-weighted ALS method (IRWALS) was considered for a comparison purpose. Five test functions in different conditions are examined for a comparative analysis of the four methods. Based on the test function study, we find that both the bias and variance of estimates obtained from the proposed methods (the GALS and the max-minG) are smaller than those from the ALS and the IRWALS methods. Especially, the max-minG works better than others including the GALS for the relatively complex test functions. Lastly, an application to a nuclear fusion simulator is illustrated and it is shown that the abnormal pattern of residuals in the ALS can be resolved by the proposed methods.

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