Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
The subject of the talk are biological and biomedical problems that we model using systems of chemotactic equations with cross-diffusion terms. I will present recent results related to quasi-periodic behavior of biological system arising from chemotactic framework of Keller-Segel. Model will be derived from Eistein fundamental principal of Brownian motion with new law for expected value of the length of the free jump, depending on the spatial changes in the attractor (food).
In the first part of the talk, model representing traveling wave phenomena of bacteria dynamics in the presence of the diffusion in the media source of attraction (e.g. food, drug, etc.) will be presented. We use celebrated Keller-Segel framework using system of partial differential equation with cross-diffusion and reactive terms in case of traveling wave pattern stability. We use closed form type solution of Keller-Segel system when diffusion coefficient for source of the attractor is equal zero. Stability of the traveling band type solution for complete dynamical system with respect to parameters of equation initial and boundary data is investigated in detail. For this purpose Lyapunov type functional is constructed and non-linear Gronwall differential inequality for Energy functional is derived. Obtained result provide explicit estimate for differences between base-line solution with no diffusion in source term and solution of complete realistic system of equation.
Second one is light driven spatial Algae-Daphinia dynamics as primitive evolutionary model with chemotaxis. First constructed analytically time and spatially dependent solution of system of two equation, which model Algae-Daphinia dynamics and proved using maximum principle machinery that this solution is unique. Then I proved that this solution is stable depending on the relations between chemotactic and diffusion coefficients, and reactive terms, using Sobolev embedding theorems, and Energy functional.
Please virtually attend this seminar via this zoom link Wednesday the 27th at 4 PM.The rapid development of single-cell experimental technologies provides unprecedented resolutions to study the dynamical process in the cell-fate decision. Mathematically, the cell-fate decision can be modeled as a (stochastic) dynamical system with multiscale structure. In this talk, we will introduce some recent efforts to combine the techniques of dynamical system models into the single-cell data analysis.
We begin with exploring energy landscape theory for stochastic dynamics, which serves as the mathematical realization of the famous Waddington metaphor of cell-fate decision in developmental biology. Next, we will introduce the experimental evidence for the “barrier-crossing” process on energy landscape in real biology, based on the single-cell trajectory data analysis of S-phase checkpoint in budding yeast cells. Motivated by such a picture, we then propose a new algorithm to dissect the multiscale structure of state-transitions underlying scRNA-seq datasets. Lastly, in the context of dynamical system analysis, we will provide some theoretical results and mathematical insights about RNA velocity -- a recent and popular proposal in biology, to infer dynamics underlying scRNA-seq datasets with the spliced/unspliced RNA abundance.
Please virtually attend this seminar via this zoom link Wednesday the 3rd at 4 PM.The pKa values are important quantities characterizing the ability of protein active sites to give up protons. pKa can be measured using NMR by tracing chemical-shifts of some special atoms, which is however expensive and time-consuming. Alternatively, pKa can be calculated numerically by electrostatic free energy changes subject to the protonation and deprotonation of titration sites. To this end, the Poisson-Boltzmann (PB) model is an effective approach for the electrostatics. However, numerically solving PB equation is challenging due to the jump conditions across the dielectric interfaces, irregular geometries of the molecular surface, and charge singularities. Our recently developed matched interface and boundary (MIB) method treats these challenges rigorously, resulting in a solid second order MIBPB solver. Since the MIBPB solver uses Green's function based regularization of charge singularities by decomposing the solution into a singular component and a regularized component, it is particularly efficient in treating the accuracy-sensitive, numerous, and complicated charges distribution from the pKa calculation. Our numerical results demonstrate that accurate electrostatics potentials, forces, energies, and pKa values are achieved at coarse grid rapidly. In addition, the resulting software, which pipelines the entire pKa calculation procedure, is available to all potential users from the greater bioscience community.
Please virtually attend Dr. Geng's seminar via this zoom link Wednesday the 10th at 4 PM CST.In this talk, I will present an extended Galerkin framework for the design and analysis of the conforming, nonconforming and discontinuous Galerkin methods for partial differential equations. By introducing appropriate multiple discretization variables, most existing and new Galerkin methods can be derived in a unified framework and the well-posedness and error estimates of these methods can be established within the same framework by using the standard Brezzi theory for mixed method. Examples will be given for elasticity and Hodge Laplacian problems. This talk is based on joint works with Professor Jinchao Xu et al.
Please virtually attend the Applied Math seminar via this zoom link Wednesday the 17th at 4 PM.We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by method-of-line Runge-Kutta (RK) methods. The ELDG framework is designed as a generalization of the semi-Lagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general two-dimensional problems. No quadratic-curved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary Lagrangian-Eulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.
Please virtually attend this week's Applied Math seminar via this zoom link Wednesday the 24th at 4 PM.Erosion is a fluid-mechanical process that is present in many geological phenomena such as groundwater flow. We present a boundary integral equation formulation to simulate two-dimensional erosion of a porous media. A numerical challenge to simulate low porosities is accurately resolving the interactions between nearly touching eroding bodies. We present a Barycentric quadrature method to resolve these interactions and compare its accuracy with the standard trapezoid rule. By using a fast summation method, such as the Fast Multipole Method (FMM), the cost of applying the standard trapezoid rule can be reduced to O(N) operations where N is the total number of source and target points. However, this advantage is gone when applying the Barycentric quadrature rules with FMM which involve several N-body calculations. To reduce the computational time, we develop a hybrid method that combines this quadrature rule and an accelerated version of the trapezoid rule. We compute the velocity, vorticity, and tracer trajectories in the geometries that include dense packings of 20, 50, and 100 eroding bodies. Similar to our previous work (B. D. Quaife et al., J. Comput. Phys. 375, 1 (2018).), we observe quickly expanding channels between close bodies, flat faces developing along the regions of near contact, and bodies eventually vanishing. Finally, having computed tracer trajectories, we characterize the transport inside of eroding geometries by computing and analyzing the tortuosity and anomalous dispersion rates. We present the validation of the tortuosities based on two different definitions and apply a reinsertion method to compute asymptotic anomalous dispersion rates.
Please virtually attend the Applied Math seminar via this zoom link Wednesday the 24th at 4 PM.Details at this pdf
Join us virtually at 4 PM Wednesday (UTC-5) for this week's Applied Math seminar via this zoom linkThe concept of random times does appear in several real world models in the contrast to the Newton time motion usual in classical mechanics. Our aim is to show how a random time will change the behavior of considered systems. We consider two classes of dynamics. At first, random time Markov processes will be analyzed. Secondly, we study random time deterministic dynamical systems which are (in certain sense) special cases of Markov evolution.
Please virtually attend this week's Applied Math seminar via this zoom link on Friday the 9th at 9 AM (UTC-5).
In this talk, we will present a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the Conservation-dissipation Formalism of irreversible thermodynamics. As governing equations for non-equilibrium flows in one dimension, the learned PDEs are parameterized by fully-connected neural networks and satisfy the conservation-dissipation principle automatically. In particular, they are hyperbolic balance laws and Galilean invariant. The training data are generated from a kinetic model with smooth initial data. Numerical results indicate that the learned PDEs can achieve good accuracy in a wide range of Knudsen numbers. Remarkably, the learned dynamics can give satisfactory results with randomly sampled discontinuous initial data and Sod's shock tube problem although it is trained only with smooth initial data.
Please virtually attend the Applied Math seminar on Wednesday the 21st at 4 PM (UTC - 5) via this zoom link.We present a design of nonlinear stabilization techniques for the finite element discretization of steady and transient Euler equations. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by improving the differentiability properties of the method through some regularization. In order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. Numerical experiments show that this method results in sharp and well resolved shocks. The importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. In the case of transient problem, we also observe a reduction in the computational cost.
Please virtually attend the Applied Math seminar via this zoom link Friday the 30th at 2 PM.