Applied and Computational Mathematics Seminars
Oct 06, 2023 11:00 AM
328 Parker Hall
Speaker: Habib Najm, Sandia National Laboratory
Title: TBA
Sep 22, 2023 02:00 PM
328 Parker Hall
Abstract: In recent years, there has been an increasing interest in applying deep learning to geophysical/medical data inversion. However, the direct application of end-to-end data-driven approaches to inversion has quickly shown limitations in practical implementation. Indeed, due to the lack of prior knowledge of the objects of interest, the trained deep-learning neural networks very often have limited generalization. In this talk, we introduce a new methodology of coupling model-based inverse algorithms with deep learning for two typical types of inversion problems. In the first part, we present an offline-online computational strategy of coupling classical least-squares-based computational inversion with modern deep learning-based approaches for full waveform inversion to achieve advantages that cannot be achieved with only one of the components. In the second part, we present an integrated data-driven and model-based iterative reconstruction framework for joint inversion problems. The proposed method couples the supplementary data with the partial differential equation model to make the data-driven modeling process consistent with the model-based reconstruction procedure. We also characterize the impact of learning uncertainty on the joint inversion results for one typical inverse problem.
DMS Applied and Computational Mathematics Seminar
Sep 15, 2023 02:00 PM
328 Parker Hall
Speaker: Yuming Paul Zhang, Auburn University
DMS Applied and Computational Mathematics Seminar
Sep 08, 2023 02:00 PM
328 Parker Hall
Speaker: Yimin Zhong, Auburn University
Title: Implicit boundary integral method for linearized Poisson Boltzmann equation
Abstract: In this talk, I will give an introduction to the so-called implicit boundary integral method based on the co-area formula and it provides a simple quadrature rule for boundary integral on general surfaces. Then, I will focus on the application of solving the linearized Poisson Boltzmann equation, which is used to model the electric potential of protein molecules in a solvent. Near the singularity, I will briefly discuss the choices of regularization/correction and illustrate the effect of both cases. In the end, I will show the numerical error estimate based on the harmonic analysis tools.
DMS Applied and Computational Math
Sep 01, 2023 02:00 PM
328 Parker Hall
Applied and Computational Mathematics
Apr 28, 2023 02:00 PM
328 Parker Hall
Speaker: Yiran Wang, Emory University
Title: Analysis and reduction of metal artifacts in X-ray tomography
Abstract: Due to beam-hardening effects, metal objects in X-ray CT often produce streaking artifacts which cause degradation in image reconstruction. It is known from the work of Seo et al in 2017 that the nature of the phenomena is nonlinear. An outstanding inverse problem is to identify the nonlinearity which is crucial for reduction of the artifacts. In this talk, we show how to use microlocal techniques to analyze the artifacts and extract information of the nonlinearity. In particular, we discuss the interesting connection between the artifact generation and geometry of metal objects.
DMS Applied and Computational Mathematics
Apr 21, 2023 02:00 PM
ZOOM
DMS Special Seminar in Analysis
Apr 17, 2023 02:00 PM
250 Parker Hall
DMS Applied and Computational Mathematics
Apr 14, 2023 02:00 PM
328 Parker Hall
Speaker: Evdokiya (Eva) Kostadinova, Auburn University, Department of Physics
Title: Fractional Laplacian Spectral Approach (FLS) to Anomalous Diffusion of Energetic Particles in Magnetized Plasma
Abstract: Fractional-power operators have received much attention due to their wide application in modeling anomalous diffusion. For the fractional Laplacian \((−Δ)^{\it s}\), values \(𝑠∈(0,1)\) correspond to a superdiffusive process, while \(𝑠∈(1,2)\) defines the subdiffusion regime. Physically, subdiffusion can be thought of as superposition of classical diffusion and a small superdiffusive part. Such a process results in overall trapping of the particle ensemble, while also allowing for nonlocal jumps. It can also be shown that for \(𝑠∈(3/2,2)\), the transport process is bounded leading to probability distribution functions with truncated tails. We also argue that in the superdiffusive regime, for \(𝑠∈(2/3,1)\), particles exhibit nonlocal jumps, but transport is described by a true Lévy process only for values \(𝑠∈(0,2/3)\). Thus, we expect that there are at least four regimes of anomalous diffusion: strong trapping, trapping with nonlocal jumps, nonlocal jumps without trapping, and Lévy flights.
To investigate the proposed sub-regimes of anomalous diffusion, we use a Fractional Laplacian Spectral (FLS) method, where the existence of extended states (interpreted as probability for transport) is determined from the existence of a continuous part in the spectrum of a Hamiltonian. Here we are interested in a Hamiltonian with a fractional Laplacian term and a stochastic disorder term. The range of nonlocal interactions and the amount of stochasticity for the Hamiltonian are informed from experiments where energetic electrons (EEs) were detected in the presence of magnetic islands and stochastic magnetic fields. The spectral properties of each Hamiltonian are investigated for fractions representative of each diffusion regime. Comparison of these calculations to experimental data reveals the presence of at least two types of EEs: runaway electrons, best described by a Lévy process, and suprathermal, but non-relativistic, electrons, best described by trapping with nonlocal jumps.
This is joint work with B. Andrew and D. M. Orlov.
Work supported by DE-FC02-04ER54698, DE- FG02-05ER54809, and DE-SC0023061.
DMS Applied and Computational Mathematics
Apr 14, 2023 11:00 AM
328 Parker Hall
PLEASE NOTE TIME: 11:00AM
Speaker: Amnon J Meir, Southern Methodist University
Title: On the Equations of Electroporoelasticity
Abstract: Complex physical phenomena and systems frequently involve multiple components, complex physics or multi-physics, as well as complex or coupled domains and multiple scales. Such phenomena are usually modeled by systems of coupled partial differential equations, often nonlinear. One such phenomenon is electroporoelasticity.
After introducing the equations of electroporoelasticity (the equations of poroelasticity coupled to Maxwell's equations) which have applications in geoscience, hydrology, and petroleum exploration, as well as various areas of science and technology, I will describe some recent results (well posedness), the numerical analysis of a finite-element based method for approximating solutions, and some interesting challenges.
DMS Applied and Computational Mathematics
Apr 07, 2023 02:00 PM
328 Parker Hall
Speaker: Erik Hiltunen, Yale University
Title: Scattering phenomena and spectral convergence of subwavelength resonators
Abstract: We study wave propagation inside metamaterials consisting of high-contrast subwavelength resonators. In the subwavelength limit, resonant states are described by the eigenstates of the generalized capacitance matrix, which describes a long-range, fully-coupled resonator model. We achieve this by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In this setting, we survey a range of subwavelength resonance phenomena such as Anderson localization, topologically protected edge modes, exceptional points, and Fano resonance. Additionally, we discuss the spectral convergence of finite structures. As the size of the structure increases, we show that defect modes induced by compact perturbations converge to localized modes of the infinite structure. Using Toeplitz eigenvalue distribution results, we additionally demonstrate the distributional convergence of the density of states as the size of the structure increases.
Short Bio: Dr. Erik O. Hiltunen is a Gibbs Assistant Professor at Yale University, USA. Dr. Hiltunen's research focuses on developing the mathematical understanding of wave propagation in materials governed by local or non-local PDEs, using tools from PDE theory, harmonic analysis and solid-state physics. Before moving to Yale, Hiltunen earned his PhD from ETH Zurich under the supervision of prof. Habib Ammari, where his thesis dissertation was awarded the ECCOMAS award for best PhD theses on Computational Methods in Applied Sciences.