Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Two main challenges to design efficient iterative solvers for the frequency-domain Maxwell equations are the indefinite nature of the underlying system and the high resolution requirements. Scalable parallel frequency-domain Maxwell solvers are highly desired.
This talk will introduce the EM-WaveHoltz method which is an extension of the recently developed WaveHoltz method for the Helmholtz equation to the time-harmonic Maxwell equations. Three main advantages of the proposed method are as follows. (1) It always results in a positive definite linear system. (2) Based on the framework of EM-WaveHoltz, it is flexible and simple to build efficient frequency-domain solvers from current scalable time-domain solvers. (3) It is possible to obtain solutions for multiple frequencies in one solve. The formulation of the EM-WaveHoltz and analysis in the continuous setting for the energy conserving case will be discussed. The performance of the proposed method will be demonstrated through numerical experiments.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 19th at 4 PM -- meeting ID: 937 2431 1192
The BGK model was introduced by Bhatnagar, Gross and Krook as a relaxation model for the fundamental Boltzmann equation, which describes the kinetic dynamic of rarefied gases with a probability distribution function. We propose an efficient, high order accurate and asymptotic-preserving (AP) semi-Lagrangian (SL) method for the BGK model with constant or spatially dependent Knudsen number. Our numerical scheme is composed of a mass conservative SL nodal discontinuous Galerkin (NDG) method as spatial discretization; together with third order AP and asymptotically accurate (AA) diagonally implicit Runge-Kutta (DIRK) methods for the stiff relaxation term along characteristics. The AA time discretization methods are constructed based on an accuracy analysis of the SL scheme for stiff hyperbolic relaxation systems and kinetic BGK model in the limiting fluid regime when the Knudsen number approaches 0. An extra order condition for the asymptotic third order accuracy in the limiting regime is derived. Linear von Neumann stability analysis of the proposed third order DIRK methods are performed to a simplified two-velocity linear kinetic model. Extensive numerical tests are presented to demonstrate the AA, AP and stability properties of our proposed schemes
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 2nd at 4 PM -- meeting ID: 937 2431 1192
In realistic flow problems described by partial differential equations (PDEs), where the dynamics are not known, or in which the variables are changing rapidly, the robust, adaptive time-stepping is central to accurately and efficiently predict the long-term behavior of the solution. This is especially important in the coupled flow problems, such as the fluid-structure interaction (FSI), which often exhibit complex dynamic behavior. While the adaptive spatial mesh refinement techniques are well established and widely used, less attention has been given to the adaptive time-stepping methods for PDEs. We will discuss novel, adaptive, partitioned numerical methods for FSI problems with thick and thin structures. The time integration in the proposed methods is based on the refactorized Cauchy's one-legged 'theta-like' method, which consists of a backward Euler method, where the fluid and structure sub-problems are sub-iterated until convergence, followed by a forward Euler method. The bulk of the computation is done by the backward Euler method, as the forward Euler step is equivalent to (and implemented as) a linear extrapolation. We will present the numerical analysis of the proposed methods showing linear convergence of the sub-iterative process and unconditional stability. The time adaptation strategies will be discussed. The properties of the methods, as well as the selection of the parameters used in the adaptive process, will be explored in numerical examples.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 9th at 4 PM -- meeting ID: 937 2431 1192
Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 16th at 4 PM -- meeting ID: 937 2431 1192
This is an expository/introductory talk concerning transport equation that appears in various problems of mathematical physics such as fluid mechanics and kinetic theory. The unknown of this equation is density that is transported by a given vector field. Often, the velocity that transports the density is too rough to admit the uniqueness of the solution. One of the most important works is due to DiPerna and Lions in 1989 in which they introduced a notion of ``renormalized solution'' and proved the uniqueness with a surprisingly rough vector field. While the competition to reduce the smoothness of the given vector field and still retain uniqueness was continued by many, via a new breakthrough technique called ```convex integration,'' some researchers very recently proved non-uniqueness with a surprisingly smooth vector field (even with arbitrarily strong diffusion).
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 23rd at 4 PM -- meeting ID: 937 2431 1192
An abstract indefinite least squares problem with a quadratic constraint is considered. This is a quadratic programming problem with one quadratic equality constraint, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are found for the existence of solutions.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 2nd at 4 PM (UT-6) -- meeting ID: 937 2431 1192
In order to treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. Based on the framework of strong stability-preserving Runge-Kutta approach, we propose two high-order multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) for the resulting split system in this paper. The proposed schemes allow for a large time step to be used for the three-dimensional baroclinic (slow) mode and a small time step for the two-dimensional barotropic (fast) mode, in which each of the two mode solves just need to satisfy their respective CFL conditions for numerical stability. Specifically, at each time step, the baroclinic velocity is first computed by advancing the baroclinic mode and fluid thickness of the system with the large time-step and the assistance of some intermediate approximations of the baroctropic mode obtained by substepping with the small time step; then the barotropic velocity is corrected by using the small time step to re-advance the barotropic mode under an improved barotropic forcing produced by interpolation of the forcing terms from the preceding baroclinic mode solves; lastly, the fluid thickness is updated by coupling the baroclinic and barotropic velocities. Additionally, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are relieved via a reconciliation process with carefully calculated flux deficits. Two benchmark tests from the ``MPAS-Ocean" platform are carried out to numerically demonstrate the performance and parallel scalability of the proposed SSPRK-SE schemes.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 2nd at 4 PM (UT-5) -- meeting ID: 937 2431 1192
Discontinuous Galerkin (DG) finite element methods are widely used in discretizing hyperbolic conservation laws. Analysis of the linear problems will help understand the properties of the DG methods and shed lights on their further development. In the first part of the talk, we use the two-way wave equations to study how the choice of numerical fluxes can affect the convergence rate of the DG methods. By constructing appropriate global projection operators, we prove optimal error estimates for a family of numerical fluxes on unstructured simplex meshes. In the second part of the talk, we consider the stability of local timestepping schemes with spacetime tents. We analyze a class of the so-called structure-aware Taylor methods and prove their weak stability under a slightly more restrictive time step constraint. Improved stability results are also obtained when coupled with low order spatial polynomials. For both topics we cover, our analysis is sharp, which can be confirmed with numerically.
Please virtually attend this week's Applied Math seminar at 4:00 PM Wednesday the 30th via this zoom link
Meeting ID: 930 3148 4740
no passcode
The regulation and interpretation of transcription factor levels is critical in spatiotemporal regulation of gene expression in development biology. However, concentration-dependent transcriptional regulation, and the spatial regulation of transcription factor levels are poorly studied in plants. WUSCHEL, a stem cell-promoting homeodomain transcription factor was found to activate and repress transcription at lower and higher levels respectively. The differential accumulation of WUSCHEL in adjacent cells is critical for spatial regulation on the level of CLAVATA3, a negative regulator of WUSCHEL transcription, to establish the overall gradient. However, the roles of extrinsic spatial cues in maintaining differential accumulation of WUSCHEL are not well understood. We have developed a 3D cell-based computational model which integrates sub-cellular partition with cellular concentration across the spatial domain to analyze the regulation of WUS. By using this model, we investigate the machinery of the maintenance of WUS gradient within the tissue. We also developed a stochastic model to study the binding and unbinding of WUS to cis-elements regulating CLV3 expression to understand the concentration dependent manner mechanistically. The robustness mechanism and the concentration-dependent machinery discovered by the modeling analysis can be general principles for stem cell homeostasis in different biological systems.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 20th at 4 PM -- meeting ID: 937 2431 1192
The regulation and interpretation of transcription factor levels is critical in spatiotemporal regulation of gene expression in development biology. However, concentration-dependent transcriptional regulation, and the spatial regulation of transcription factor levels are poorly studied in plants. WUSCHEL, a stem cell-promoting homeodomain transcription factor was found to activate and repress transcription at lower and higher levels respectively. The differential accumulation of WUSCHEL in adjacent cells is critical for spatial regulation on the level of CLAVATA3, a negative regulator of WUSCHEL transcription, to establish the overall gradient. However, the roles of extrinsic spatial cues in maintaining differential accumulation of WUSCHEL are not well understood. We have developed a 3D cell-based computational model which integrates sub-cellular partition with cellular concentration across the spatial domain to analyze the regulation of WUS. By using this model, we investigate the machinery of the maintenance of WUS gradient within the tissue. We also developed a stochastic model to study the binding and unbinding of WUS to cis-elements regulating CLV3 expression to understand the concentration dependent manner mechanistically. The robustness mechanism and the concentration-dependent machinery discovered by the modeling analysis can be general principles for stem cell homeostasis in different biological systems.
Please virtually attend the Applied Math seminar via this Zoom link Wednesday the 20th at 4 PM -- meeting ID: 937 2431 1192