Browsing by Subject "Inverse problems"
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Item Feature modeling and tomographic reconstruction of electron microscopy images(2012-05) Gopinath, Ajay, 1980-; Bovik, Alan C. (Alan Conrad), 1958-; Ress, David Bruce; Xu, Guoliang; Pearce, John; Ghosh, JoydeepThis work introduces a combination of image processing and analysis methods that perform feature extraction, shape analysis and tomographic reconstruction of Electron Microscopy images. These have been implemented on images of the AIDS virus interacting with neutralizing molecules. The AIDS virus spike is the primary target of drug design as it is directly involved in infecting host cells. First, a fully automated technique is introduced that can extract sub-volumes of the AIDS virus spike and be used to build a statistical model without the need for any user supervision. Such an automatic feature extraction method can significantly enhance the overall process of shape analysis of the AIDS virus spike imaged through the electron microscope. Accurate models of the virus spike will help in the development of better drug design strategies. Secondly, a tomographic reconstruction method implemented using a shape based regularization technique is introduced. Spatial models of known features in the structure being reconstructed are integrated into the reconstruction process as regularizers. This regularization scheme is driven locally through shape information obtained from segmentation and compared with a known spatial model. This method shows reduced blurring, and an improvement in the resolution of the reconstructed volume was also measured. It performs better than popular current techniques and can be extended to other tomographic modalities. Improved Electron Tomography reconstructions will provide better structure elucidation and improved feature visualization, which can aid in solving key biological issues.Item Hessian-based response surface approximations for uncertainty quantification in large-scale statistical inverse problems, with applications to groundwater flow(2013-08) Flath, Hannah Pearl; Ghattas, Omar N.Subsurface flow phenomena characterize many important societal issues in energy and the environment. A key feature of these problems is that subsurface properties are uncertain, due to the sparsity of direct observations of the subsurface. The Bayesian formulation of this inverse problem provides a systematic framework for inferring uncertainty in the properties given uncertainties in the data, the forward model, and prior knowledge of the properties. We address the problem: given noisy measurements of the head, the pdf describing the noise, prior information in the form of a pdf of the hydraulic conductivity, and a groundwater flow model relating the head to the hydraulic conductivity, find the posterior probability density function (pdf) of the parameters describing the hydraulic conductivity field. Unfortunately, conventional sampling of this pdf to compute statistical moments is intractable for problems governed by large-scale forward models and high-dimensional parameter spaces. We construct a Gaussian process surrogate of the posterior pdf based on Bayesian interpolation between a set of "training" points. We employ a greedy algorithm to find the training points by solving a sequence of optimization problems where each new training point is placed at the maximizer of the error in the approximation. Scalable Newton optimization methods solve this "optimal" training point problem. We tailor the Gaussian process surrogate to the curvature of the underlying posterior pdf according to the Hessian of the log posterior at a subset of training points, made computationally tractable by a low-rank approximation of the data misfit Hessian. A Gaussian mixture approximation of the posterior is extracted from the Gaussian process surrogate, and used as a proposal in a Markov chain Monte Carlo method for sampling both the surrogate as well as the true posterior. The Gaussian process surrogate is used as a first stage approximation in a two-stage delayed acceptance MCMC method. We provide evidence for the viability of the low-rank approximation of the Hessian through numerical experiments on a large scale atmospheric contaminant transport problem and analysis of an infinite dimensional model problem. We provide similar results for our groundwater problem. We then present results from the proposed MCMC algorithms.Item Multi-material nanoindentation simulations of viral capsids(2010-05) Subramanian, Bharadwaj; Bajaj, Chandrajit; Oden, Tinsley J.An understanding of the mechanical properties of viral capsids (protein assemblies forming shell containers) has become necessary as their perceived use as nano-materials for targeted drug delivery. In this thesis, a heterogeneous, spatially detailed model of the viral capsid is considered. This model takes into account the increased degrees of freedom between the capsomers (capsid sub-structures) and the interactions between them to better reflect their deformation properties. A spatially realistic finite element multi-domain decomposition of viral capsid shells is also generated from atomistic PDB (Protein Data Bank) information, and non-linear continuum elastic simulations are performed. These results are compared to homogeneous shell simulation re- sults to bring out the importance of non-homogenous material properties in determining the deformation of the capsid. Finally, multiscale methods in structural analysis are reviewed to study their potential application to the study of nanoindentation of viral capsids.Item Noninvasive material discrimination using spectral radiography and an inverse problem approach(2014-12) Gilbert, Andrew James; Deinert, Mark; McDonald, Benjamin; Biegalski, Steven; Ghattas, Omar; Schneider, ErichNoninvasive material discrimination of an arbitrary object is applicable to a wide range of fields, including medical scans, security inspections, nuclear safeguards, and nuclear material accountancy. In this work, we present an algorithmic framework to accurately determine material compositions from multi-spectral X-ray and neutron radiography. The algorithm uses an inverse problem approach and regularization, which amounts to adding information to the problem; stabilizing the solution so that accurate material estimations can be made from a problem that would otherwise be intractable. First, we show the utility of the algorithm with simulated inspections of small objects, such as baggage, for small quantities of high-atomic-numbered materials (i.e. plutonium). The algorithm shows excellent sensitivity to shielded plutonium in a scan using an X-ray detector that can bin X-rays by energy. We present here a method to adaptively weight the regularization term, obtaining an optimal solution with minimal user input. Second, we explore material discrimination with high-energy, multiple-energy X-ray. Experimental X-ray data is obtained here and accurate discrimination of steel among lower-atomic-numbered materials is shown. Accurate modeling of the inspection system physics is found to be essential for accurate material estimations with this data, especially the detector response and the scattered flux on the image plane. Third, we explore the use of neutron radiography as complementary to X-ray radiography for the inspection of nuclear material storage containers. Utility of this extra data is shown, especially in detecting a hypothetical attempt to divert material. We present a method to choose inspection system design parameters (i.e. source energy and detector thickness) a priori by using the Cramér-Rao lower bound as a measure of resulting material estimation accuracy. Finally, we present methodology to use tomography data obtained with an energy discriminating detector for direct reconstruction of material attenuation coefficients.Item Numerical algorithms for inverse problems in acoustics and optics(2014-05) Ding, Tian, 1986-; Ren, Kui; Engquist, Bjorn; Gamba, Irene Martínez; Ghattas, Omar; Gonzalez, Oscar; Wheeler, Mary FanettThe objective of this dissertation is to develop computational algorithms for solving inverse coefficient problems for partial differential equations that appear in two medical imaging modalities. The aim of these inverse problems is to reconstruct optical properties of scattering media, such as biological tissues, from measured data collected on the surface of the media. In the first part of the dissertation, we study an inverse boundary value problems for the radiative transport equation. This inverse problem plays important roles in optics-based medical imaging techniques such as diffuse optical tomography and fluorescence optical tomography. We propose a robust reconstruction method that is based on subspace minimization techniques. The method splits the unknowns, both the unknown coefficient and the corresponding transport solutions (or a functional of it) into low-frequency and high-frequency components, and uses singular value decomposition to analytically recover part of low-frequency information. Minimization is then applied to recover part of the high-frequency components of the unknowns. We present some numerical simulations with synthetic data to demonstrate the performance of the proposed algorithm. In the second part of the dissertation, we develop a three-dimensional reconstruction algorithm for photoacoustic tomography in isotropic elastic media. There have been extensive study of photoacoustic tomography in recent years. However, all existing numerical reconstructions are developed for acoustic media in which case the model for wave propagation is the acoustic wave equation. We develop here a two-step reconstruction algorithm to reconstruct quantitatively optical properties, mainly the absorption coefficient and the Gr\"uneisen coefficient using measured elastic wave data. The algorithm consists of an inverse source step where we reconstruct the source function in the elastic wave equation from boundary data and an inverse coefficient step where we reconstruct the coefficients of the diffusion equation using the result of the previous step as interior data. We present some numerical reconstruction results with synthetic data to demonstrate the performance of our algorithm. This is, to the best of our knowledge, the first reconstruction algorithm developed for quantitative photoacoustic imaging in elastic media. Despite the fact that we separate the dissertation into these two different parts to make each part more focused, the algorithms we developed in the two parts are closely related. In fact, if we replace the diffusion model for light propagation in photoacoustic imaging by the radiative transport model, which is often done in the literature, the algorithm we developed in the first part can be integrated into the algorithm in the second part after some minor modifications.Item Numerical methods for multiscale inverse problems(2014-05) Frederick, Christina A; Engquist, Björn, 1945-This dissertation focuses on inverse problems for partial differential equations with multiscale coefficients in which the goal is to determine the coefficients in the equation using solution data. Such problems pose a huge computational challenge, in particular when the coefficients are of multiscale form. When faced with balancing computational cost with accuracy, most approaches only deal with models of large scale behavior and, for example, account for microscopic processes by using effective or empirical equations of state on the continuum scale to simplify computations. Obtaining these models often results in the loss of the desired fine scale details. In this thesis we introduce ways to overcome this issue using a multiscale approach. The first part of the thesis establishes the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory. The theory developed is based on the mathematical analysis of multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Typical examples are two-scale functions f (x, x/[epsilon]), (0 < [epsilon] ≪ 1) that are periodic in the second variable. We prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from nonuniform samples of optimal rate. In the second part, we present a new multiscale approach for inverse homogenization problems. We prove that in certain cases where the specific form of the multiscale coefficients is known a priori, imposing an additional constraint of a microscale parametrization results in a well-posed inverse problem. The mathematical analysis is based on homogenization theory for partial differential equations and classical theory of inverse problems. The numerical analysis involves the design of multiscale methods, such as the heterogeneous multiscale method (HMM). The use of HMM solvers for the forward model has unveiled theoretical and numerical results for microscale parameter recovery, including applications to inverse problems arising in exploration seismology and medical imaging.Item Quantitative PAT with unknown ultrasound speed : uncertainty characterization and reconstruction methods(2015-05) Vallélian, Sarah Catherine; Ren, Kui; Ghattas, Omar; Müller, Peter; Tsai, Yen-Hsi; Ward, RachelQuantitative photoacoustic tomography (QPAT) is a hybrid medical imaging modality that combines high-resolution ultrasound tomography with high-contrast optical tomography. The objective of QPAT is to recover certain optical properties of heterogeneous media from measured ultrasound signals, generated by the photoacoustic effect, on the surfaces of the media. Mathematically, QPAT is an inverse problem where we intend to reconstruct physical parameters in a set of partial differential equations from partial knowledge of the solution of the equations. A rather complete mathematical theory for the QPAT inverse problem has been developed in the literature for the case where the speed of ultrasound inside the underlying medium is known. In practice, however, the ultrasound speed is usually not exactly known for the medium to be imaged. Using an approximated ultrasound speed in the reconstructions often yields images which contain severe artifacts. There is little study as yet to systematically investigate this issue of unknown ultrasound speed in QPAT reconstructions. The objective of this dissertation is exactly to investigate this important issue of QPAT with unknown ultrasound speed. The first part of this dissertation addresses the question of how an incorrect ultrasound speed affects the quality of the reconstructed images in QPAT. We prove stability estimates in certain settings which bound the error in the reconstructions by the uncertainty in the ultrasound speed. We also study the problem numerically by adopting a statistical framework and applying tools in uncertainty quantification to systematically characterize artifacts arising from the parameter mismatch. In the second part of this dissertation, we propose an alternative reconstruction algorithm for QPAT which does not assume knowledge of the ultrasound speed map a priori, but rather reconstructs it alongside the original optical parameters of interest using data from multiple illumination sources. We explain the advantage of this simultaneous reconstruction approach compared to the usual two-step approach to QPAT and demonstrate numerically the feasibility of our algorithm.Item Recovering the payoff structure of a utility maximizing agent(2016-05) Goswami, Pulak; Žitković, Gordan; Sirbu, Mihai; Pavlovic, Natasa; Larsen, KasperAny agent with access to information that is not available to the market at large is considered an ‘insider’. It is possible to interpret the effect of this private information as change in the insider’s probability measure. In the case of exponential utility, logarithm of the Radon-Nikodym derivative for the change in measure will appear as a random endowment in the objective the insider would maximize with respect to the original measure. The goal of this paper is to find conditions under which it is possible to recover the structure of this random endowment given only a single trajectory of his/her wealth. To do this, it is assumed that the random endowment is a function of the terminal value of the state variable and that the market is complete.Item Statistical Inference in Inverse Problems(2012-07-16) Xun, XiaoleiInverse problems have gained popularity in statistical research recently. This dissertation consists of two statistical inverse problems: a Bayesian approach to detection of small low emission sources on a large random background, and parameter estimation methods for partial differential equation (PDE) models. Source detection problem arises, for instance, in some homeland security applications. We address the problem of detecting presence and location of a small low emission source inside an object, when the background noise dominates. The goal is to reach the signal-to-noise ratio levels on the order of 10^-3. We develop a Bayesian approach to this problem in two-dimension. The method allows inference not only about the existence of the source, but also about its location. We derive Bayes factors for model selection and estimation of location based on Markov chain Monte Carlo simulation. A simulation study shows that with sufficiently high total emission level, our method can effectively locate the source. Differential equation (DE) models are widely used to model dynamic processes in many fields. The forward problem of solving equations for given parameters that define the DEs has been extensively studied in the past. However, the inverse problem of estimating parameters based on observed state variables is relatively sparse in the statistical literature, and this is especially the case for PDE models. We propose two joint modeling schemes to solve for constant parameters in PDEs: a parameter cascading method and a Bayesian treatment. In both methods, the unknown functions are expressed via basis function expansion. For the parameter cascading method, we develop the algorithm to estimate the parameters and derive a sandwich estimator of the covariance matrix. For the Bayesian method, we develop the joint model for data and the PDE, and describe how the Markov chain Monte Carlo technique is employed to make posterior inference. A straightforward two-stage method is to first fit the data and then to estimate parameters by the least square principle. The three approaches are illustrated using simulated examples and compared via simulation studies. Simulation results show that the proposed methods outperform the two-stage method.Item Towards the predictive modeling of ductile failure(2015-12) Gross, Andrew Jeffrey; Ravi-Chandar, K.; Kovar, Desiderio; Landis, Chad; Liechti, Kenneth; Kyriakides, SteliosThe ability to predict ductile failure is considered by an experimental examination of the failure process, validation exercises to assess predictive ability, and development of a coupled experimental-numerical strategy to enhance model development. In situ loading of a polycrystalline metal inside a scanning electron microscope is performed on Al 6061-T6 that reveals matrix-dominated response for both deformation and failure. Highly localized deformation fields are found to exist within each grain as slip accumulates preferentially on a small fraction of crystallographic planes. No evidence of damage or material softening is found, implying that a strain-to-failure model is adequate for modeling fracture in this and similar material. This modeling insight is validated through blind predictive simulations performed in response to the 2012 and 2014 Sandia Fracture Challenges. Constitutive and failure models are calibrated and then embedded in highly refined finite element simulations to perform blind predictions of the failure behavior of the challenge geometries. Comparison of prediction to experiment shows that a well-calibrated model that captures the essential elastic-plastic constitutive behavior is necessary to capture confidently the response for structures with complex stress states, and is a prerequisite for a precise prediction of material failure. The validation exercises exposed the need to calibrate sophisticated plasticity models without a large experimental effort. To answer this need, a coupled experimental and numerical method is developed for characterizing the elastic-plastic constitutive properties of ductile materials using local deformation field information to enrich calibration data. The method is applied to a tensile test specimen and the material’s constitutive model, whose parameters are unknown a priori, is determined through an optimization process that compares these experimental measurements with iterative finite element simulations. The final parameters produce a simulation that tracks the local experimental displacement field to within a couple percent of error. Simultaneously, the percent error in the simulation for the load carried by the specimen throughout the test is less than one percent. The enriched calibration data is found to be sufficient to constrain model parameters describing anisotropy that could not be constrained by the global data alone.