More Like this

1. Adjoint Based Hessians for Optimization Problems in System Identification

   Nandi, Souransu; Singh, Tarunraj (August 2017, 2017 IEEE Conference on Control Technology and Applications)

   An adjoint sensitivity based approach to determine the gradient and Hessian of cost functions for system identification is presented. The motivation is the development of a computationally efficient approach relative to the direct differentiation technique and which overcomes the challenges of the step size selection in finite difference approaches. The discrete time measurements result in discontinuities in the Lagrange multipliers. The proposed approach is illustrated on the Lorenz 63 model where part of the initial conditions and model parameters are estimated. 

   more » « less
2. Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

   https://doi.org/10.1007/978-3-030-95157-3_18  

   Heinkenschloss, Matthias; Magruder, Caleb (July 2022, Realization and Model Reduction of Dynamical Systems - A Festschrift in Honor of the 70th Birthday of Thanos Antoulas)

   Beattie, C.A.; Benner, P.; Embree, M.; Gugercin, S.; Lefteriu, S. (Ed.)

   This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in terms of overall PDE solves required. 

   more » « less
3. Efficient identification for modeling high-dimensional brain dynamics

   https://doi.org/10.23919/ACC53348.2022.9867232  

   Singh, Matthew F.; Wang, Michael; Cole, Michael W.; Ching, ShiNung (June 2022, 2022 American Control Conference (ACC))

   System identification poses a significant bottleneck to characterizing and controlling complex systems. This challenge is greatest when both the system states and parameters are not directly accessible, leading to a dual-estimation problem. Current approaches to such problems are limited in their ability to scale with many-parameter systems, as often occurs in networks. In the current work, we present a new, computationally efficient approach to treat large dual-estimation problems. In this work, we derive analytic back-propagated gradients for the Prediction Error Method which enables efficient and accurate identification of large systems. The PEM approach consists of directly integrating state estimation into a dual-optimization objective, leaving a differentiable cost/error function only in terms of the unknown system parameters, which we solve using numerical gradient/Hessian methods. Intuitively, this approach consists of solving for the parameters that generate the most accurate state estimator (Extended/Cubature Kalman Filter). We demonstrate that this approach is at least as accurate in state and parameter estimation as joint Kalman Filters (Extended/Unscented/Cubature) and Expectation-Maximization, despite lower complexity. We demonstrate the utility of our approach by inverting anatomically-detailed individualized brain models from human magnetoencephalography (MEG) data. 

   more » « less
4. Adjoint-based inversion for stress and frictional parameters in earthquake modeling

   https://doi.org/10.1016/j.jcp.2024.113447  

   Stiernström, Vidar; Almquist, Martin; Dunham, Eric M (December 2024, Journal of Computational Physics)

   We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction and, for forward problems involving fault normal stress changes, nonzero fault opening, with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge–Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments. 

   more » « less
5. A mixed, unified forward/inverse framework for earthquake problems: fault implementation and coseismic slip estimate

   https://doi.org/10.1093/gji/ggac050  

   Puel, S.; Khattatov, E.; Villa, U.; Liu, D.; Ghattas, O.; Becker, T. W. (February 2022, Geophysical Journal International)

   SUMMARY We introduce a new finite-element (FE) based computational framework to solve forward and inverse elastic deformation problems for earthquake faulting via the adjoint method. Based on two advanced computational libraries, FEniCS and hIPPYlib for the forward and inverse problems, respectively, this framework is flexible, transparent and easily extensible. We represent a fault discontinuity through a mixed FE elasticity formulation, which approximates the stress with higher order accuracy and exposes the prescribed slip explicitly in the variational form without using conventional split node and decomposition discrete approaches. This also allows the first order optimality condition, that is the vanishing of the gradient, to be expressed in continuous form, which leads to consistent discretizations of all field variables, including the slip. We show comparisons with the standard, pure displacement formulation and a model containing an in-plane mode II crack, whose slip is prescribed via the split node technique. We demonstrate the potential of this new computational framework by performing a linear coseismic slip inversion through adjoint-based optimization methods, without requiring computation of elastic Green’s functions. Specifically, we consider a penalized least squares formulation, which in a Bayesian setting—under the assumption of Gaussian noise and prior—reflects the negative log of the posterior distribution. The comparison of the inversion results with a standard, linear inverse theory approach based on Okada’s solutions shows analogous results. Preliminary uncertainties are estimated via eigenvalue analysis of the Hessian of the penalized least squares objective function. Our implementation is fully open-source and Jupyter notebooks to reproduce our results are provided. The extension to a fully Bayesian framework for detailed uncertainty quantification and non-linear inversions, including for heterogeneous media earthquake problems, will be analysed in a forthcoming paper. 

   more » « less