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This is my dissertation for my PhD in Applied Mathematics at the University of Arizona. My thesis is about deblurring images in the Bayesian framework with spatially varying hyperparameters and mixed regularization types.
We have built build and analyzed fundamental mathematical tools that enable sparse recovery of scientific data for high performance computing, experimental facilities and sensor measurement. The proposed work will impact many areas of research across the DOE. We have designed, developed and analyzed fast scalable algorithms for the sparse recovery of scientific data sets that may be extremely large, generated very quickly and distributed across multiple storage devices. The algorithms we developed will be used to process the copious amounts of scientific data generated by the DOE supercomputer simulations and by DOE experimental and observational facilities. They leverage the fact that while the amount of data produced can be extremely large, the actual quantities of interest (QoI's) are often sparse, either in space time, or in some feature space. Mathematically, notions of compressive sensing, including l1 regularized and total variation regularized optimization have played an important part in this effort.
The phased array is a group of regularly spaced lasers with parallel optical axes. Each laser of the group emits light of the same wavelength and the same phase. Mathematical models of laser transmission systems with regular arrays are presented, taking into account rectangular apertures. It is found that an array of free running lasers has the same normalized expectation value as the single laser, with the actual power a factor of N greater.
Here, we introduce a mesh-adaption framework that employs a multi-physical configurational force and Lie algebra to capture multiphysical responses of fluid-infiltrating geological materials while maintaining the efficiency of the computational models. To resolve sharp gradients of both displacement and pore pressure, we introduce an energy-estimate-free re-meshing criterion by extending the configurational force theory to consider the energy dissipation due to the fluid diffusion and the gradient-dependent plastic flow. To establish new equilibria after remeshing, the local tensorial history-dependent variables at the integration points are first decomposed into spectral forms. Then, the principal values and directions are projected onto smooth fields interpolated by the basis function of the finite element space via the Lie-algebra mapping. Our numerical results indicate that this Lie algebra operator in general leads to a new trial state closer to the equilibrium than the ones obtained from the tensor component mapping approach. A new configurational force for dissipative fluid-infiltrating porous materials that exhibit gradient-dependent plastic flow is introduced such that the remeshing may accommodate the need to resolve the sharp pressure gradient as well as the strain localization. The predicted responses are found to be not influenced by the mesh size due to the micromorphic regularization, while the adaptive meshing enables us to capture the width of deformation bands without the necessity of employing fine mesh everywhere in the domain.
Plug-and-play priors (PnP) is a powerful framework for regularizing imaging inverse problems by using advanced denoisers within an iterative algorithm. Recent experimental evidence suggests that PnP algorithms achieve state-of-the-art performance in a range of imaging applications. Here, we introduce a new online PnP algorithm based on the proximal gradient method (PGM). The proposed algorithm uses only a subset of measurements at every iteration, which makes it scalable to very large datasets. We present a new theoretical convergence analysis, for both batch and online variants of PnP-PGM, for denoisers that do not necessarily correspond to proximal operators. We also present simulations illustrating the applicability of the algorithm to image reconstruction in diffraction tomography. Finally, the results in this paper have the potential to expand the applicability of the PnP framework to very large datasets.
A method of moments is developed for the parametric form the solution of convolution equations of the first kind. Two models, the gamma model and the shifted gamma model, are studied in details and a non-parametric approach based on regularization is presented.
We consider a nonstationary initial boundary-value problem describing complex (radiative-conductive) heat transfer in a system of semitransparent bodies. To describe radiation propagation, we use the radiation transfer equation with boundary-value conditions of diffuse reflection and refraction of radiation. We take into account that the radiation intensity and optical properties of bodies depend on the radiation frequency. The unique solvability of a weak solution is established. The comparison theorem is proved. A priori estimates of a weak solution as well as its regularity are obtained.
Diffuse reflectance spectroscopy has been extensively employed to deliver timely and cost-effective predictions of a number of soil properties. However, although several soil spectral laboratories have been established worldwide, the distinct characteristics of instruments and operations still hamper further integration and interoperability across mid-infrared (MIR) soil spectral libraries. In this study, we conducted a large-scale ring trial experiment to understand the lab-to-lab variability of multiple MIR instruments. By developing a systematic evaluation of different mathematical treatments with modeling algorithms, including regular preprocessing and spectral standardization, we quantified and evaluated instruments' dissimilarity and how this impacts internal and shared model performance. We found that all instruments delivered good predictions when calibrated internally using the same instruments' characteristics and standard operating procedures by solely relying on regular spectral preprocessing that accounts for light scattering and multiplicative/additive effects, e.g., using standard normal variate (SNV). When performing model transfer from a large public library (the USDA NSSCKSSL MIR library) to secondary instruments, good performance was also achieved by regular preprocessing (e. g., SNV) if both instruments shared the same manufacturer. However, significant differences between the KSSL MIR library and contrasting ring trial instruments responses were evident and confirmed by a semi-unsupervised spectral clustering. For heavily contrasting setups, spectral standardization was necessary before transferring prediction models. Non-linear model types like Cubist and memory-based learning delivered more precise estimates because they seemed to be less sensitive to spectral variations than global partial least square regression. In summary, the results from this study can assist new laboratories in building spectroscopy capacity utilizing existing MIR spectral libraries and support the recent global efforts to make soil spectroscopy universally accessible with centralized or shared operating procedures.
The successful navigation of Galileo depends on accurate trajectory correction maneuvers (TCM's) performed during the mission. A set of Inertial Sensor (INS) units, comprised of gyros and accelerometers, mounted on the spacecraft, are utilized to control and monitor the performance of the TCM's. To provide the optimum performance, in-flight calibrations of INS are planned. These calibrations will take place on a regular basis. In this paper, a mathematical description is given of the data reduction technique used in analyzing a typical set of calibration data. The design of the calibration and the inertial sensor error models, necessary for the above analysis, are delineated in detail.
Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. In this work, we consider a class of polyhedral functions that arise particularly as convex lower envelopes of multibang regularizers and show that they have beneficial properties with respect to regularization of relaxations of integer optimal control problems. We extend the algorithmic framework of the combinatorial integral approximation such that a subsequence of the computed discrete-valued controls converges to the infimum of the regularized integer control problem.
We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.
Many scientific experiments such as those found in astronomy, geology, microbiology, and X-ray radiography require the use of high-energy instruments to capture images. Since blur and noise are inevitably present in any imaging system, the images must be \deblurred" to extract the full information content. Mathematically, image deblurring is an ill-posed inverse problem that requires regularization. The regularization, in turn, has a large effect on the deblurred image: different regularization strengths, and types, lead to drastically different reconstructions. Moreover, many images contain a mixture of smooth and sharp features which suggests the use of multi-regularization, i.e., varying the type of regularization (e.g. Tikhonov or total variation) across the image. We address these issues by formulating the image deblurring problem within a hierarchical Bayesian framework in which we spatially adapt the strength of the regularization and also vary the regularization type across the image. In this way, the image itself, along with corresponding regularization strength at each pixel, are described jointly by a posterior distribution which we can sample by Markov chain Monte Carlo (MCMC) methods. We illustrate our techniques on simplified test problems and apply them to high-energy X-ray images taken at the Nevada National Security Site. Numerical tests show that our new method is robustly applicable and increases the quality of the image reconstruction when compared to other (Bayesian) methods.
In this paper, we consider the challenge of reconstructing jointly sparse vectors from linear measurements. Firstly, we show that by utilizing the rank of the output data matrix we can reduce the problem to a full column rank case. This result reveals a reduction in the computational complexity of the original problem and enables a simple implementation of joint sparse recovery algorithms for full-rank setting. Secondly, we propose a new method for joint sparse recovery in the form of a non-convex optimization problem on a non-compact Stiefel manifold. In our numerical experiments our method outperforms the commonly used l 2,1 minimization in the sense that much fewer measurements are required for accurate sparse reconstructions. Here, we postulate this approach possesses the desirable rank aware property, that is, being able to take advantage of the rank of the unknown matrix to improve the recovery.
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in C 0 . Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes H 1 . Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.
This paper deals with a problem the packing polyhex clusters in a regular hexagonal container. It is a common problem in many applications with various cluster shapes used, but symmetric polyhex is the most useful in engineering due to its geometrical properties. Hence, we concentrate on mathematical modeling in such an application, where using the “bee” tetrahex is chosen for the new Compact Muon Solenoid (CMS) design upgrade, which is one of four detectors used in Large Hadron Collider (LHC) experiment at European Laboratory for Particle Physics (CERN). We start from the existing hexagonal containers with hexagonal cells packed inside, and uniform clustering applied. We compare the center-aligned (CA) and vertex-aligned (VA) models, analyzing cluster rotations providing the increased packing efficiency. We formally describe the geometrical properties of clustering approaches and show that cluster sharing is inevitable at the container border with uniform clustering. In addition, we propose a new vertex-aligned model decreasing the number of shared clusters in the uniform scenario, but with a smaller number of clusters contained inside the container. Also, we describe a non-uniform tetrahex cluster packing scheme in the proposed container model. With the proposed cluster packing solution, it is accomplished that all clusters are contained inside the container region. Since cluster-sharing is completely avoided at the container border, the maximal packing efficiency is obtained compared to the existing models.
It is shown that the Dirichlet problem for a degenerating many-dimensional equation of mixed type in a cylindrical domain is uniquely solvable. A criterion for the uniqueness of a regular solution is also established.
In this study, we discuss a novel approach to modeling the near-wall region in the class of k – ω to models. The proposed methodology obviates the need for ad hoc boundary conditions of to omega on the wall as typically required in the k – ω model. The primary motivation of this work is to provide a formulation equivalent to the standard k – ω to and k – ω to SST models, but which at the same time overcomes their limitations in the context of their implementation in high-order methods. This is achieved by subtracting the asymptotically known singular behavior of omega at walls. Imposing a grid-dependent value of omega at walls in high-order codes is not straightforward as is demonstrated below and it causes instability as well as accuracy issues. The mathematical formulation of the two novel approaches, termed as the "regularized k – ω model" and the "regularized k – ω OJ SST model", is discussed in detail. A consistency and verification study for these two approaches is performed by proving that the regularized models recover the results of the standard models in various canonical problems, such as turbulent channel and pipe flows, and a systematic investigation of convergence, using both p- (polynomial order) as well as h- (grid) refinement is reported. Furthermore, comparisons highlighting the performance of the proposed methods in more complex configurations such as flow over a backward facing step and the turbulent mixing of fluid streams of different temperatures in a T-junction are also presented.