Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
In this talk, I give a particular example in which a deterministic approach seems hopeless and yet a stochastic approach successfully allows us to get a glimpse of physics underneath. The celebrated Helmholtz - Kelvin's conservation of circulation from 1858 states that for a smooth solution to the Euler equations, the circulation around a curve moving with the fluid is a constant in time. Its proof completely breaks down once we add diffusion to the Euler equations, which is the Navier-Stokes equations, and no precise statement could be made for the subsequent 150 years. Remarkably, Constantin and Iyer in 2008 (Iyer's Ph.D. thesis in 2006) showed that if we consider random characteristics (essentially characteristics from PDE graduate course but added by noise), then an analog of the circulation theorem for the Navier-Stokes equations holds and states that the circulation on the loop is given by the average over the circulations of the ensemble of loops at the earlier times. Analogous statements can be made for other equations of hydrodynamics such as magnetohydrodynamics, and Boussinesq systems.Explicit time integrators for parabolic PDE are subject to a restrictive time-step limit, so A-stable integrators are essential. It is well known that although there are no A-stable explicit linear multistep methods and implicit multistep methods cannot be A-stable beyond order two, there exist A-stable and L-stable implicit Runge-Kutta (IRK) methods at all orders. IRK methods offer an appealing combination of stability and high order; however, these methods are not widely used for PDE because they lead to large, strongly coupled linear systems. An s-stage IRK system has s-times as many degrees of freedom as the systems resulting from backward Euler or implicit trapezoidal rule discretization applied to the same equation set. In this talk, I will introduce a new block preconditioner for IRK methods, based on a block LDU factorization with algebraic multigrid subsolves for scalability. I will demonstrate the effectiveness of this preconditioner on the heat equation as a simple test problem, and compare in condition number and eigenvalue distribution, and in numerical experiments with other preconditioners currently in the literature. Experiments are run with IRK stages up to s = 7, and it is found that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.The discrete fracture model (DFM) has been widely used in the simulation of fluid flow in fractured porous media. Traditional DFM use the so-called hybrid-dimensional approach to treat fractures explicitly as low-dimensional entries (e.g. line entries in 2D media and face entries in 3D media) on the interfaces of matrix cells to avoid local grid refinements in fractured region and then couple the matrix and fracture flow systems together based on the principle of superposition with the fracture thickness used as the dimensional homogeneity factor. Because of this methodology, DFM is considered to be limited on conforming meshes and thus may raise difficulties in generating high qualified unstructured meshes due to the complexity of fracture’s geometrical morphology. In this talk, we clarify that the discrete fracture model actually can be extended to non-conforming meshes without any essential changes. To show it clearly, we provide another perspective for DFM based on hybrid-dimensional representation of permeability tensor modified from the comb model to describe fractures as one-dimensional line Dirac delta functions contained in permeability tensors. A finite element DFM scheme for single-phase flow on non-conforming meshes is then derived by applying Galerkin finite element method to it. Analytical analysis and numerical experiments show that our DFM scheme automatically degenerates to the classical finite element DFM when the mesh is conforming with fractures. Moreover, the accuracy and efficiency of the model on non-conforming meshes is demonstrated by testing several benchmark problems. This model is also applicable to curved fracture with variable thickness.
We use the Einstein random-walk paradigm for diffusion to derive a degenerate nonlinear parabolic equation and study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein's random walk paradigm, but instead of Taylor expansion we used Caratheodory theorem unlike Einstein's original work leading to a partial differential equation in non-divergence form. On the other hand, using conservation
principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. We investigate a qualitative properties of the solution using maximum principle and energy method, in order to obtain bounds above and below for the time-evolution of the solution to the non-linear diffusion problem. Having thus sandwiched, we prove that, unlike the case of degenerate diffusion, the solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process.We considered the generalization of the Einstein model of Brownian motion when the key parameter of the time interval of free jump degenerates via a solution and its gradient. This phenomena manifests in two scenarios: a) when fluid is highly dispersing like a non-dense gas and b) when distance of the flow w.r.t source is so big that velocity and the gradient of pressure are not subject to the linear Darcy equation. In this work we jointly investigate the question of what feature will exhibit particle flows if the time interval of free jump inverse is proportional to the density of the fluids and its gradient. It was shown that in this scenario, the flow exhibits a localization feature, namely: if at some moment of time t0 in the region gradient of pressure or pressure itself is equal to zero then for some T during time interval [t0, t0 + T] there is no flow as well. This directly links to Barenblatt's finite speed of propagation property for degenerate equations. Method of proof is very different and based on the application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique. PDF available.
Glycans are one of the most widely investigated biomolecules, due to their roles in numerous vital biological processes. This involvement makes it critical to understand their structure-function relationships. Few system-independent, LC-MS/MS (Liquid chromatography tandem mass spectrometry) based studies have been developed with this particular goal, however. When studied, the employed methods generally rely on normalized retention times as well as m/z-mass to charge ratio of an ion value. Due to these limitations, there is need for quantitative characterization methods which can be used independent of m/z values, thus utilizing only normalized retention times. As such, the primary goal of this article is to construct an LC-MS/MS based classification of the permethylated glycans derived from standard glycoproteins and human blood serum, using a Glucose Unit Index (GUI) as the reference frame in the space of compound parameters. For the reference frame we develop a closed-form analytic formula, which is obtained from Green’s function of a relevant convection-diffusion-absorption equation used to model composite material transport. The aforementioned equation is derived from an Einstein-Brownian motion paradigm, which provides a physical interpretation of the time-dependence at the point of observation for molecular transport in the experiment. The necessary coefficients are determined via a data-driven learning procedure. The methodology is presented in an abstract manner which allows for immediate application to related physical and chemical processes. Results employing the proposed classification method are validated via comparison to experimental mass spectrometer data.Capillary surfaces arise when mixing different fluids. In mathematics, they can be considered as critical points of a geometric functional under certain constraints. The stability index measures how far away one is from being a local minimizer. A critical point is stable if the index is zero. Naturally, constraints would play an influential role in applications as they decrease the degree of freedom available. In this talk, we discuss capillary hypersurfaces in a Euclidean ball and how to obtain sharp estimates for indices with respect to varying constraints. In particular, we show that, when fixing the enclosed volume and wetting area, the only stable ones are geodesic disks and spherical caps. At the heart of the proof is a functional analysis theorem which will be discussed in the Elasticity seminar preceding the Applied Math Seminar. This is a joint work with Detang Zhou.This project has developed a class of bound preserving and energy dissipative schemes for the porous medium equation. The schemes are based on a positivity preserving approach and a perturbation technique, and are shown to be uniquely solvable, bound preserving, and in the first-order case, also energy dissipative. We have also conducted ample numerical experiments to validate the theoretical results and demonstrate the new schemes' effectivenessSymplectic integrators are commonly used to integrate Hamiltonian systems due to its desired property of exact energy conservation. However, symplectic integrators can be less efficient than conventional Runge-Kutta integrators. In this work, we consider a different approach based on deep learning to integrate Hamiltonian systems, and propose a novel symplectic neural network (HenonNet) to identify underlying flow maps from a finite dataset of discrete time maps. Our network architecture consists of layers that are symplectic maps due to its construction. Therefore, the network is a structure-preserving approximation to Hamiltonian systems and enjoy the same property symplectic integrators have. We further prove a universal approximation theorem of HenonNet to any C^r symplectic diffeomorphism.
As an example, we consider the Poincare maps for toroidal magnetic fields that are routinely employed to study gross confinement properties in fusion devices. In practice, evaluating a Poincare map requires numerical integration along a magnetic field line, a process that can be slow and that cannot be easily accelerated using parallel computations. We show that our network architecture is capable of accurately learning realistic Poincare maps from observations. After training, such learned Poincare maps evaluate orders of magnitude faster than non-symplectic Runge-Kutta integrators. Moreover, the learned network exactly reproduces the primary physics constraint imposed on field-line Poincare maps: flux preservation, which indicates its long-time stability and accuracy.
Watch online Wednesday the 19th at 4 PM via this Zoom linkThis is a joint zoom presentation with the Biomath seminar group
Watch online Wednesday the 2nd at 4 PM via this Zoom link