Browsing by Subject "Inverse problem"

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Advanced techniques for multi-source, multi-parameter, and multi-physics inverse problems
(2017-09-14) Crestel, Benjamin; Ghattas, Omar N.; Engquist, Bjorn; Bui-Thanh, Tan; Fomel, Sergey; Ren, Kui; Stadler, Georg
With the increase in compute power and the advent of the big data era, inverse problems have grown more complex, attempting to extract more information and to use more data. While this evolution manifests itself in multiple forms, we focus in this dissertation on three specific aspects: multi-source, multi-parameter, and multi-physics inverse problems. The computational cost of solving a multi-source inverse problem in- creases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We propose to replace the random weights traditionally used in the linear combinations of the experiments, with deterministic weights (or, encoding weights). We approach the computation of these weights as an optimal experimental design problem, and develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint- based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost. We define a multi-parameter inverse problem, also called joint inverse problem, as the simultaneous inference of multiple parameter fields. In this dissertation, we concentrate on two types of multi-parameter inverse problems. In the first case, we have at our disposal a single type of observations, generated by a single physical phenomenon which depends on multiple parameters. In the second case, we utilize multiple datasets generated from physical phenomena that depend on different parameters; when the data are generated from different physics, this is a multi-physics inverse problem. The regularization of a multi-parameter inverse problem plays a critical role. It not only acts as a regularizer to the inverse problem, but can also be used to impose coupling between the inversion parameters when they are known to share similar structures. We compare four joint regularizations terms: the cross-gradient, the normalized cross-gradient, the vectorial total variation, and a novel regular- izer based on the nuclear norm of a gradient matrix. Following comprehensive numerical investigations, we concluded that vectorial total variation leads to the best reconstructions. We next devoted our attention to develop an efficient primal-dual Newton solver for joint inverse problems regularized with vecto- rial total variation. Introducing an auxiliary dual variable in the first-order optimality condition, which we then solve using Newton method, we were able to reduce the nonlinearity in the inverse problem. Through an extensive nu- merical investigation, we showed that this solver is scalable with respect to the mesh size, the hyperparameter, and the number of inversion parameters. We also observed that it significantly outperforms the classical Newton method and the popular lagged diffusivity method when fine convergence tolerances are needed. Multi-physics inverse problems are becoming more popular as a way to enhance the quality of the reconstructions by combining the strengths of multiple imaging modalities. In this dissertation, we specialize to the case of full-waveform inversion, and the presence of local minima in its objective function when using high-frequency data. The most practical workaround to- day remains a continuation scheme over the frequency of the source term. However, in a seismic exploration setting, modern equipment does not allow to generate data of sufficiently low frequencies. One potential application of multi-physics inverse problems is to allow an auxiliary physical phenomenon, e.g., electromagnetic waves, to provide the missing low-frequency information for full-waveform inversion. In this dissertation, we provide supporting evi- dence for this approach when using the vectorial total variation functional as a regularization.
hIPPYfire : an inexact Newton-CG method for solving inverse problems governed by PDE forward models
(2023-04-26) Hiranandani, Karan Prakash; Ghattas, Omar N.; Villa, Umberto
This study presents the implementation of hIPPYfire, a library for solving large-scale deterministic inverse problems. These inverse problems are governed by partial differential equations (PDEs) with infinite-dimensional parameter fields that become high-dimensional after discretization. It utilizes the inexact Newton Conjugate Gradient (Newton-CG) method for the computation of the maximum a posteriori (MAP) point. This algorithm exploits the fact that several PDE models of physical systems have a low-dimensional solution manifold. hIPPYfire computes the solution of the inverse problem at a cost independent of the parameter dimension, when measured in terms of the number of linearized PDE solves. However, unlike hIPPYlib (which is built on FEniCS), hIPPYfire uses Firedrake to solve the PDE governing the forward problem. Firedrake presents a unique modular structure that clearly distinguishes between the programming and mathematical aspects of the library—thereby enabling contributions from programmers and mathematicians alike and ensuring its consistent development. The functionalities of hIPPYfire are illustrated by solving an inverse problem that is governed by an elliptic PDE. The major components of the inverse problem, namely the forward problem, misfit, and prior functionals, are clearly defined and used to compute the MAP point using the inexact Newton-CG method. The design of hIPPYfire follows that of hIPPYlib, an extensible Python library for the solution of deterministic and Bayesian inverse problems governed by PDEs.
The inverse medium problem for Timoshenko beams and frames : damage detection and profile reconstruction in the time-domain
(2009-12) Karve, Pranav Madhav; Kallivokas, Loukas F.; Manuel, Lance
We discuss a systematic methodology that leads to the reconstruction of the material profile of either single, or assemblies of one-dimensional flexural components endowed with Timoshenko-theory assumptions. The probed structures are subjected to user-specified transient excitations: we use the complete waveforms, recorded directly in the time-domain at only a few measurement stations, to drive the profile reconstruction using a partial-differential-equation-constrained optimization approach. We discuss the solution of the ensuing state, adjoint, and control problems, and the alleviation of profile multiplicity by means of either Tikhonov or Total Variation regularization. We report on numerical experiments using synthetic data that show satisfactory reconstruction of a variety of profiles, including smoothly and sharply varying profiles, as well as profiles exhibiting localized discontinuities. The method is well suited for imaging structures for condition assessment purposes, and can handle either diffusive or localized damage without need for a reference undamaged state.
Inverse problems for basal properties in a thermomechanically coupled ice sheet model
(2017-08-10) Zhu, Hongyu, Ph. D.; Ghattas, Omar N.; Hughes, Thomas J. R.; Stadler, Georg; Dawson, Clint; Heimbach, Patrick
Polar ice sheets are losing mass at a growing rate, and are likely to make a dominant contribution to 21st century sea-level rise. Thus, modeling the dynamics of polar ice sheets is critical for projections of future sea level rise. Yet, this is difficult due to the complexity of accurately modeling ice sheet dynamics and because of the unobservable boundary conditions at the base of the ice sheet. In this work, we address the inverse problem of inferring basal properties--in particular, the geothermal heat flux--from surface velocity observations and a forward model in the form of thermomechanically coupled nonlinear Stokes and energy equations with complementarity conditions representing melting of basal ice. This inverse problem is made even more challenging due to the severely nonlinear and non-smooth nature of the forward problem. The inverse problem is formulated as a nonlinear least squares optimization that minimizes the misfit between the model prediction and the observation. A Tikhonov regularization term is added to render the problem well-posed. To solve the inverse problem for large-scale ice sheets, the use of adjoint-based Newton methods is critical, which requires a smoothly differentiable forward problem. Thus, we regularize the complementarity conditions of the forward problem by a penalty-like method, such that the solution of the regularized problem approaches that of the original forward problem as the penalty approaches infinity. As a consequence of the Petrov-Galerkin discretization of the energy equation, discretization and differentiation do not commute, i.e., the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Here, we employ the discretize-then-optimize approach to guarantee the consistency between the discrete cost function and its 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 we can approximately locate the melting region of the ice sheet and reconstruct the geothermal heat flux in the cold region. The reconstruction improves as the noise level in the observations decreases but 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 leads 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. For large-scale simulations, we extend the parallel solver "ymir" for the nonlinear Stokes model by incorporating the coupled nonlinear advection-diffusion energy equation with complementarity conditions. An inexact Newton-Krylov method with block preconditioning is designed for the thermomechanically coupled ice sheet model. The inexact Newton-Krylov method exhibits near optimal algorithmic scalability, i.e., the numbers of both Newton and Krylov iterations depend only weakly on problem size. We use the parallel solver to simulate the flow of the Antarctic ice sheet and the Pine Island ice stream, with a geometry and boundary conditions from the ALBMAP dataset.
Inverse problems in photoacoustic imaging : analysis and computation
(2017-08) Zhong, Yimin; Ren, Kui; Biros, George; Gonzalez, Oscar; Tsai, Richard Yen-Hsi
Inverse problems in photoacoustic imaging (PAT) have been extensively studied in recent years due to their importance in applications. This thesis addresses three important aspects of PAT inverse problems mathematically and computationally. First, we present a detailed mathematical and numerical analysis of quantitative fluorescence PAT, a variant of PAT for applications in molecular imaging. We develop uniqueness and stability theory on quantitative reconstructions based on the radiative transport model of light propagation and present numerical simulations to validate the mathematical theory. Second, we develop a fast numerical algorithm for solving the radiative transport equation, the model of light propagation in PAT applications on tissue imaging, in isotropic media. Our method is based on an integral equation formulation of the radiative transport equation and a fast multipole method for accelerating matrix-vector multiplications for the discretized system. Third, we perform mathematical analysis on PAT reconstruction problem with unknown ultrasound speed. We prove local uniqueness and stability results on the simultaneous reconstruction of the ultrasound speed, the acoustic attenuation coefficient as well as the optical absorption coefficients.
A mixed unsplit-field PML-based scheme for full waveform inversion in the time-domain using scalar waves
(2010-05) Kang, Jun Won, 1975-; Kallivokas, Loukas F.; Stokoe, Kenneth H.; Tonon, Fulvio; Ghattas, Omar; Gonzalez, Oscar
We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems.
Recovery of the logical gravity field by spherical regularization wavelets approximation and its numerical implementation
(2009-05) Shuler, Harrey Jeong; Tapley, Byron D.
As an alternative to spherical harmonics in modeling the gravity field of the Earth, we built a multiresolution gravity model by employing spherical regularization wavelets in solving the inverse problem, i.e. downward propagation of the gravity signal to the Earth.s surface. Scale discrete Tikhonov spherical regularization scaling function and wavelet packets were used to decompose and reconstruct the signal. We recovered the local gravity anomaly using only localized gravity measurements at the observing satellite.s altitude of 300 km. When the upward continued gravity anomaly to the satellite altitude with a resolution 0.5° was used as simulated measurement inputs, our model could recover the local surface gravity anomaly at a spatial resolution of 1° with an RMS error between 1 and 10 mGal, depending on the topography of the gravity field. Our study of the effect of varying the data volume and altering the maximum degree of Legendre polynomials on the accuracy of the recovered gravity solution suggests that the short wavelength signals and the regions with high magnitude gravity gradients respond more strongly to such changes. When tested with simulated SGG measurements, i.e. the second order radial derivative of the gravity anomaly, at an altitude of 300 km with a 0.7° spatial resolution as input data, our model could obtain the gravity anomaly with an RMS error of 1 ~ 7 mGal at a surface resolution of 0.7° (< 80 km). The study of the impact of measurement noise on the recovered gravity anomaly implies that the solutions from SGG measurements are less susceptible to measurement errors than those recovered from the upward continued gravity anomaly, indicating that the SGG type mission such as GOCE would be an ideal choice for implementing our model. Our simulation results demonstrate the model.s potential in determining the local gravity field at a finer scale than could be achieved through spherical harmonics, i.e. less than 100 km, with excellent performance in edge detection.