# Browsing by Subject "PDE-constrained optimization"

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Item Full-waveform inversion in three-dimensional PML-truncated elastic media : theory, computations, and field experiments(2015-05) Fathi, Arash; Kallivokas, Loukas F.; Dawson, Clinton N; Demkowicz, Leszek F; Ghattas, Omar; Manuel, Lance; Stokoe II, Kenneth HShow more We are concerned with the high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveforms of the medium's response to drive the inverse problem. Specifically, we use a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to the proposed inversion schemes. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). The PML is a buffer zone that surrounds the domain of interest, and enforces the decay of outgoing waves. In order to resolve the forward problem, we present a hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. We discuss several time-integration schemes, including an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers. We report numerical results demonstrating stability and efficacy of the forward wave solver, and also provide examples attesting to the successful reconstruction of the two Lame parameters for both smooth and sharp profiles, using synthetic records. We also report the details of two field experiments, whose records we subsequently used to drive the developed inversion algorithms in order to characterize the sites where the field experiments took place. We contrast the full-waveform-based inverted site profile against a profile obtained using the Spectral-Analysis-of-Surface-Waves (SASW) method, in an attempt to compare our methodology against a widely used concurrent inversion approach. We also compare the inverted profiles, at select locations, with the results of independently performed, invasive, Cone Penetrometer Tests (CPTs). Overall, whether exercised by synthetic or by physical data, the full-waveform inversion method we discuss herein appears quite promising for the robust subsurface imaging of near-surface deposits in support of geotechnical site characterization investigations.Show more Item Inverse source problems for focusing wave energy to targeted subsurface formations: theory and numerical experiments(2016-08) Karve, Pranav Madhav; Kallivokas, Loukas F.; Manuel, Lance; Ghattas, Omar; Fomel, Sergey; Lake, Larry; Huh, Chun; Stokoe II, KennethShow more Economically competitive and reliable methods for the removal of oil or contaminant particles from the pores of geological formations play a crucial role in petroleum engineering, hydro-geology, and environmental engineering. Post-earthquake observations at depleted oil fields as well as limited field experiments suggest that stress wave stimulation of a formation may lead to the release of particles trapped in its interstices. The stimulation can be applied using wave sources placed on or below the ground surface, and, typically, the effectiveness of the stimulation is proportional to the magnitude of the wave motion generated in the geological formation of interest. When wave sources are used to initiate the wave motion, equipment limitations and various sources of attenuation impose restrictions on the magnitude of the wave motion induced in the target formation. Thus, the engineering design of the wave energy delivery systems that are able to produce the wave motion of a required magnitude in the target zone is key to a successful mobilization of trapped interstitial particles. In this work, we discuss an inverse source approach that yields the optimal source time signals and source locations and could be used to design wave energy delivery systems. We cast the underlying forward wave propagation problem in two or three spatial dimensions. We model the target formation as an elastic or poroelastic inclusion embedded within heterogeneous, elastic, semi-infinite hosts. To simulate the semi-infiniteness of the elastic host, we augment the (finite) computational domain with a buffer of perfectly-matched-layers (PMLs). We define a metric of the wave motion generated in the target inclusion to quantify the amount of the delivered wave energy. The inverse source algorithm is based on a systematic framework of constrained optimization, where minimization of a suitably defined objective functional is tantamount to the maximization of the motion metric of the target formation. We demonstrate, via numerical experiments, that the algorithm is capable of converging to the spatial and temporal characteristics of surface loads that maximize energy delivery to the target formation. The numerical-simulation-based methodology is based on the assumption of perfect knowledge of the material properties and of the overall geometry of the geostructure of interest. In practice, however, precise knowledge of the properties of the geological formations is elusive, and quantification of the reliability of a deterministic approach is crucial for evaluating the technical and economical feasibility of the design. To this end, we also discuss a methodology that could be used to quantify the uncertainty in the wave energy delivery. Specifically, we treat the material properties of the layers as random variables, and perform a first-order uncertainty analysis of the elastodynamic system to compute the probabilities of failure to achieve threshold values of the motion metric. We illustrate the uncertainty quantification procedure for the case of two-dimensional, layered, isotropic, elastic host containing an elastic target inclusion. The inverse source and the uncertainty quantification methodologies, in conjunction, can be used for designing the characteristics of the wave sources used to deliver the wave energy to a targeted subsurface formation.Show more Item 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, OscarShow more 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.Show more Item Predictive modeling and uncertainty quantification for diblock copolymer self-assembly(2022-12-02) Cao, Lianghao; Oden, J. Tinsley (John Tinsley), 1936-; Ghattas, Omar N.; Demkowicz, Leszek F; Marzouk, Youssef M; Willcox, Karen EShow more Diblock copolymers (Di-BCP) are polymers consistent of two blocks of distinct monomers. Upon thermal or solvent annealing, the two blocks in a Di-BCP melt spontaneously segregate as a result of their incompatibility. Periodically-ordered nano-scale structures emerge as the melt achieves an equilibrium. This phenomenon is known as the self-assembly of Di-BCPs. The self-assembly can be further influenced by external guidance, such as chemically modified substrates, enabling the design of complex nano-structures. This process is referred to as the directed self-assembly (DSA) of Di-BCPs. The DSA of BCPs has attracted massive attention for its application in patterning for lithography to be used for fabricating nano-scale devices. Recently, there has been a rising interest in making consequential decisions based on computer simulations of Di-BCP self-assembly; most noticeably being the use of model-based simulations to propose DSA procedures for meeting the manufacturing constraints of nanofabrication. With the growing impact of computer simulations in the applications of the self-assembly, it is more imperative than ever to understand as well as systematically control and enhance the quality, validity, and reliability of these simulations. Aiming at these goals, this study focuses on predictive modeling with quantified uncertainties for Di-BCP self-assembly. First, the physical and mathematical origins of models for Di-BCP self-assembly are presented to set the ground for understanding the predictability of continuum models for the self-assembly, such as those based on self-consistent field theory and density functional theory. Then fast algorithms are designed based on approximations of the Hessian operators for solving these models via free energy minimization. These algorithms help accelerate many-query applications, such as design and inverse problems, associated with model-based simulations of Di-BCP self-assembly. A Bayesian framework for predictive modeling of Di-BCP thin film self-assembly via the Occam Plausibility Algorithm is presented. This procedure involves model calibration, selection, and validation centered around Bayes' rule and image data produced by microscopy or X-ray scattering characterizations. The aleatoric uncertainties of Di-BCP self-assembly represented by metastability and defectivity require introducing randomness into model predictions. These uncertainties, however, lead to integrated likelihoods that are generally intractable to evaluate. To tackle this challenge that hampers the execution of the predictive modeling procedure, likelihood-free inference approaches via pseudo-marginal methods and measure transport are considered in this study. To improve the efficiency of these inference methodologies, they are used in adjunct with carefully designed summary statistics that extract features of characterization images. Several Fourier- and energy-based summary statistics are proposed for top-down microscopy characterizations of Di-BCP thin films. Expected information gains are used to quantify the utilities of summary statistics choices, and we show that they can be computed via measure transport with no significant additional computational cost.Show more Item Scalable, adaptive methods for forward and inverse problems in continental-scale ice sheet modeling(2015-08) Isaac, Tobin Gregory; Ghattas, Omar N.; Stadler, Georg, Ph. D.; Arbogast, Todd; Biros, George; Catania, Ginny; Oden, John TinsleyShow more Projecting the ice sheets' contribution to sea-level rise is difficult because of the complexity of accurately modeling ice sheet dynamics for the full polar ice sheets, because of the uncertainty in key, unobservable parameters governing those dynamics, and because quantifying the uncertainty in projections is necessary when determining the confidence to place in them. This work presents the formulation and solution of the Bayesian inverse problem of inferring, from observations, a probability distribution for the basal sliding parameter field beneath the Antarctic ice sheet. The basal sliding parameter is used within a high-fidelity nonlinear Stokes model of ice sheet dynamics. This model maps the parameters "forward" onto a velocity field that is compared against observations. Due to the continental-scale of the model, both the parameter field and the state variables of the forward problem have a large number of degrees of freedom: we consider discretizations in which the parameter has more than 1 million degrees of freedom. The Bayesian inverse problem is thus to characterize an implicitly defined distribution in a high-dimensional space. This is a computationally demanding problem that requires scalable and efficient numerical methods be used throughout: in discretizing the forward model; in solving the resulting nonlinear equations; in solving the Bayesian inverse problem; and in propagating the uncertainty encoded in the posterior distribution of the inverse problem forward onto important quantities of interest. To address discretization, a hybrid parallel adaptive mesh refinement format is designed and implemented for ice sheets that is suited to the large width-to-height aspect ratios of the polar ice sheets. An efficient solver for the nonlinear Stokes equations is designed for high-order, stable, mixed finite-element discretizations on these adaptively refined meshes. A Gaussian approximation of the posterior distribution of parameters is defined, whose mean and covariance can be efficiently and scalably computed using adjoint-based methods from PDE-constrained optimization. Using a low-rank approximation of the covariance of this distribution, the covariance of the parameter is pushed forward onto quantities of interest.Show more