|
Date
|
Speaker/Institution
|
Title and Abstract
|
|
Sept 2
|
Organizational meeting
|
|
|
Sept 9
|
Weishi Liu
|
Title: Can positive permanent charges enhance the flux of cations while reduce that of anions
Abstract: Under reasonable and general assumptions, we claim that the answer to the question raised in the title is NO.
The claim is established by a simple analysis based on Poisson-Nernst-Planck models for ionic flow.
Combining with other results, one concludes: Positive permanent charges can enhance the fluxes of both cations and anions,
can reduce the fluxes of both cations and anions, can reduce the flux of cations while enhance that of anions,
but cannot enhance the flux of cations while reduce that of anions.
|
|
Sept 16
|
Xianglin Li (KU, ME)
|
Title: Introduction to Computational Fluid Dynamics Based on Finite Volume Method
Abstract:
Computational fluid dynamics (CFD) is used in a wide variety of engineering applications. The general strategy of CFD is to replace the continuous problem domain with a discrete domain using a grid. In the discrete domain, each flow variable is defined only at the grid points or within the control volume. There are three typical discretization methods: finite difference, finite element and finite volume methods. This talk will only introduce finite volume method which is used in many commercial CFD packages such as PHOENICS, FLUENT, FLOW3D and STAR-CD.
This talk will use examples that apply the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations)-like algorithm to couple the pressure and velocity in the Navier-Stokes equation and tri-diagonal matrix algorithm (TDMA) to solved the discretized equations. The examples include the heat and mass transfer of phase change materials (PCMs) in a cavity and the heat and mass transfer coupled with electrochemical reactions in porous media.
|
|
Sept 23
|
Andrew Steyer
|
Title: A Lyapunov exponents based stability theory for one-step ODE initial value problem solvers.
Abstract: We consider the stability of variable step-size one-step methods methods approximating bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to
determine conditions on the maximum allowable step-size that guarantees the numerical solution of an asymptotically decaying time-dependent linear problem also decays. This result is used to justify using a one-dimensional asymptotically contracting real-valued nonautonomous linear test problem to characterize the stability of a one-step method. The linear stability result is applied to show the stability of the numerical solution of stable nonlinear problems.
|
|
Sept 30
|
Shuguan Ji (Jilin University/China)
|
Title: Unidirectional fluxes and flux ratio of charged tracers via a quasi-one-dimensional classical Poisson-Nernst-Planck model
Abstract: In this talk, we will investigate unidirectional fluxes and flux ratio of charged tracers via a quasi-one-dimensional classical Possion-Nernst-Planck (cPNP) model. We first prove the unidirectional flux of the tracer is proportional to the flux of the main species, especially the proportional constant depends ONLY on the boundary conditions, but NOT on the permanent charge and channel geometry. Furthermore, by viewing the cPNP as a singularly perturbed system and using our previous result, we also derive an explicit approximation for the flux ratio, and analyze its properties.
|
|
Oct 7
|
Chen Su
|
Title: Sequential Implicit Sampling Methods for Bayesian Inverse Problems.
Abstract: The solution to the inverse problems, under the Bayesian framework, is given by a posterior probability density. For large scale problems, sampling the posterior can be an extremely challenging task. Traditional methods include MCMC, which generates a chain whose invariance density is the target density, and Gaussian approximations, such as EnKF. In this paper, the implicit sampling method and the newly proposed sequential implicit sampling method are investigated for the inverse problem involving time-dependent partial differential equations (PDEs). The sequential implicit sampling method combines the idea of the EnKF and implicit sampling and it is particularly suitable for time-dependent problems. Moreover, the new method is capable of reducing the computational cost in the optimization, which is the most expensive part. The sequential implicit sampling method has been tested on a seismic wave inversion. The numerical experiments shows its efficiency by comparing with the MCMC and some Gaussian approximations.
|
|
Oct 14
|
Fall break
|
|
|
Oct 21
|
Mary Hill (KU, Geology)
|
Title: Exploring Models: What is Important and Quantifying Uncertainty
Abstract: Numerical models are used to simulate complex systems in many fields,
including environmental systems such as climate and hydrology.
Models allow data, conceptualizations, and conservation principles to be
brought together in ways that, it is hoped, will provide great insight for
system performance and management. Yet often modeling efforts have produced
results that are confusing to resource managers and policy makers, and many
modeling efforts have been more expensive and less useful than was hoped.
As a result, the role played by numerical models in decision making is being
reevaluated. This is an important time for modelers to take a step back and
consider the current status of numerical models, how we use them to quantify
things like prediction, uncertainty, and risk, and how we might proceed in
ways that produce more useful results. In this talk we proceed from the
perspective that while much has been accomplished, from a long-term
perspective we are still at the beginning of modeling for resource management.
This is because numerical modeling is a very complicated tool and computers
have been fast enough to allow expansive exploration of complex models for only
a short time. When compared to the evolution of other technological advances
such as cars and medicine, it is clear that numerical modeling is early in its development.
Our evaluation of the present status of the field suggests that three challenges compromise the utility of mathematical models of environmental systems: (1) a dizzying array of model analysis methods and metrics make it difficult to choose how to proceed with evaluation of model adequacy, sensitivity, and uncertainty; (2) the high computational demands of many popular currently model analysis methods (requiring 1,000’s, 10,000s, or more model runs) make them difficult to apply to complex models; and (3) many models are plagued by unrealistic nonlinearities arising from the numerical model formulation and implementation. A strategy is proposed to address these challenges through a careful combination of model analysis and implementation methods. In this strategy, computationally frugal model analysis methods (often requiring a few dozen parallelizable model runs) play a major role, and computationally demanding methods are used for problems where (relatively) inexpensive diagnostics suggest the frugal methods are unreliable. We also argue in favor of detecting and, where possible, eliminating unrealistic model nonlinearities - this increases the realism of the model itself and generally reduces model execution time. Removal of unrealistic model nonlinearities also improves the performance of reduced and surrogate models. Reducing model execution times and using model analysis methods that require fewer model runs enables greater exploration of individual models. Of great importance is that it enables greater exploration of the more general question of what methods tend to produce more useful models. Results from recent field and theoretical hydrologic studies will be discussed. We suggest that the strategy proposed would make exploration of models easier and enable a more productive, exciting, and societally consequential future for simulations of complex environmental systems.
|
|
Oct 28
|
Weizhang Huang
|
Title: Anisotropic mesh quality measures and adaptation for polygonal meshes
Abstract. In this talk we will discuss a mesh quality measure for convex polygonal meshes
and propose a practical algorithm for the generation of anisotropic adaptive polygonal meshes.
We will present mathematical justifications for the algorithm and numerical results to demonstrate
it feasibility and robustness.
|
|
Nov 4
|
Milena Stanislavova
|
Title: Periodic traveling waves of the short pulse equation: existence and stability
Abstract: We consider various periodic traveling waves solutions of the Ostrovsky/Hunter-Saxton/short pulse equation and its KdV regularized version. For the regularized short pulse model with small Coriolis parameter we describe a family of periodic traveling waves which are perturbation of appropriate KdV solitary waves. We show that these waves are spectrally stable. For the short pulse model, we construct a family of traveling peakons with corner crests. We show that the peakons are spectrally stable as well.
|
|
Nov 11
|
Cuong Ngo
|
Title: Moving mesh methods for numerical solutions of porous medium equations.
Abstract:Porous medium equations (PME) are found in many applications of the
physical sciences. These equations are nonlinear, degenerate, and in
many situations have free boundaries, which altogether pose great
challenges for mathematical and numerical analyses. In contrast with the
mathematical development of PME, which began in the 1950s and has since
had much success, studies of numerical solutions did not appear until
the 1980s. Though major progresses have been made since then for the 1D
cases, only limited success has been observed for many of the 2D cases,
for until this day, no second-order method has been seen in the
literature. In this CAM seminar meeting, we will propose several moving
mesh approaches which improve the convergence order and/or accuracy of
the PME's numerical solutions.
|
|
Nov 18
|
Selim Sukhtaiev(University of Missouri)
|
Title: Counting spectrum via the Maslov index for one dimensional $\theta-$periodic Schr\"odinger operators.
Abstract: We study the spectrum of the Schr\"odinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $\theta-$periodic boundary conditions. For two such operators, corresponding to different values of $\theta$, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. This talk is based on joint work with C.K.R.T. Jones and Y. Latushkin.
|
|
Nov 25
|
Thanksgiving
|
|
|
Dec 2
|
Hongguo Xu
|
Title: Staircase Forms - From matrices to matrix polynomials
Abstract: We review the staircase forms for computing the Jordan form of matrices
and for computing the Kronecker form of matrix pencils. We will discuss recent
progress in finding staircase forms of matrix polynomials.
|
|
Dec 9
|
Xianping Li (UMKC)
|
Title: Anisotropic Mesh Adaptation for 3D Anisotropic Diffusion Problems
Abstract: Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The M-uniform mesh approach is used, where an anisotropic adaptive mesh is generated as a uniform one in the metric specified by a tensor. In addition to mesh adaptation, preservation of the maximum principle is also studied. Four different metric tensors are investigated: one is the identity matrix, one focuses on minimizing an error bound, another one on preservation of the maximum principle, while the fourth combines both. Numerical examples show that these metric tensors serve their purposes. Particularly, the fourth leads to meshes that improve the finite element solution's satisfaction of the maximum principle while concentrating elements in regions where the error is large. Application of the anisotropic mesh adaptation to fractured reservoir simulation in petroleum engineering is also investigated.
|
|
Mar 3 (4:00PM Colloquium)
|
Roland Schnaubelt (Karlsruhe Institute of Technology)
|
Title: Error analysis of the ADI Splitting method for the Maxwell equations
Abstract: The ADI scheme is a very efficient explicit-implicit time integration
method for the linear Maxwell equations in certain situations. It is
based on a splitting of the curl operators, which allows to decouple
implicit steps into one dimensional sub problems. We establish error
bounds for this scheme and show near preservation of the divergence
conditions. We also treat the inhomogeneous system with currents and
charges, but we do not consider the space discretization. Our results
are based on wellposedness and regularity results for the full and the
split systems.
|
|
Mar 9
|
Andrew Steyer
|
Title:A Lyapunov exponents based stability theory for Runge-Kutta methods
Abstract: In this talk we consider the stability of variable step-size Runge-Kutta methods approximating bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees the discrete Lyapunov exponents of the numerical solution of a time-dependent linear problem with an integral separation structure approximate the Lyapunov exponents of the exact solution with the same order of accuracy as the Runge-Kutta method that is used. This result is used to justify using a one-dimensional asymptotically contracting real-valued nonautonomous linear test problem to characterize the stability of a Runge-Kutta method approximating a time-dependent problem. The linear stability result is applied to show the stability of the numerical solution of stable nonlinear problems through an application of the discrete variation of constants formula. We conclude the talk by showing how these results extend to strictly stable linear multistep methods
and discussing applications in the computation of integral manifolds and step-size selection.
|
|
Mar 16
|
|
Spring break
|
|
Mar 23
|
Hongguo Xu
|
Title: Weighted Golub-Kahan-Lanczos algorithms
Abstract: Standard Golub-Kahan-Lanczos algorithms are Krylov subspace
methods based on a bidiagonal factorization of a general matrix. The
proposed weighted algorithms generalize these methods by replacing the
standard orthogonality with orthogonality with respect to weighted inner
products related to the given matrices. Formally, the weighted algorithms
are developed for solving the eigenvalue problem of a product of two Hermitian
positive definite matrices, but they can be applied to solve eigenvalue problems
arising from linear response problem. Also, the weighted algorithms have a
natural connection with conjugate gradient algorithm.
|
|
Mar 30
|
Erik Van Vleck
|
Title: Data assimilation and applications
Abstract:
|
|
Apr 6
|
Joshua Roundy (KU, CEAE)
|
Title: A Stochastic Model for Seasonal Prediction of Drought
Abstract: Extremes in the water cycle, such as drought, cause significant impacts on society that can be reduced through preparations made possible through seasonal prediction. Improvements in drought prediction require a better understanding of the mechanisms that dictate drought development. Feedbacks between the land and the atmosphere play a role in drought intensification, persistence and recovery. Recent work has developed a classification of land-atmosphere interactions that summarizes the complex interactions between the land and the atmosphere into coupling regimes. One of these regimes is dry coupling, which is connected with drought initialization, persistence and intensification through feedbacks between the land and the atmosphere. The temporal persistence of this coupling regime and its impact on the water cycle is summarized through the Coupling Drought Index (CDI). Recent work demonstrated that seasonal forecasts from a Global Climate Model exhibit a climatological bias in the CDI that impacts the prediction of precipitation and has ramifications on the ability to predict drought. These shortcomings in the seasonal prediction motivated the development of the Coupling Stochastic Model (CSM), which utilizes the persistent coupling states to make seasonal predictions through a Markov Chain model coupled to a probabilistic prediction of precipitation and temperature. The CSM model can be used as an independent forecast model or to correct the known issues with existing climate models and was shown to improve the prediction of precipitation during drought. To this point the statistical model has relied on the initial conditions and persistence probabilities from reanalysis, however the needed variables to calculate the CDI and the associated statistical model parameters are available through satellite remote sensing. This provides a means to incorporate a more observationally based land-atmosphere product into the CSM model that could improve the seasonal predictions. Furthermore, incorporating other statistical observations with in the CSM framework could also provide a method for improved seasonal prediction.
|
|
Apr 13
|
Kyle Claassen
|
Title: Nondegeneracy of Antiperiodic Bound States for Fractional Nonlinear Schrodinger Equations
Abstract: We establish that the linearization of the Fractional Nonlinear Schrodinger Equation (fNLS) has a nondegenerate T-antiperiodic kernel, a key ingredient for proving stability of solutions in the presence of symmetries.
|
|
Apr 20
|
Hermen Jan Hupkes (Leiden University)
|
Title: Understanding pollution with Wiener-Hopf lattice factorizations
Abstract: We study optimal control problems with time delays posed on lattices,
which can be used to weigh the costs and benefits of utilizing
polluting agents to enhance crop yields. The conditions
defining optimal strategies turn out to
be Hilbert-space valued functional differential equations of mixed type (MFDEs). We develop
tools such as exponential dichotomies and Wiener-Hopf factorizations
for such systems to determine whether optimal strategies
can retain their optimality under small variations in their initial conditions.
Complications are caused by the fact that the modelling state space
is only half of the natural mathematical state space.
|
|
Apr 27
|
Blake Barker (Brown Univeristy)
|
Title: Numerical proof of stability of viscous shock profiles
Abstract: We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number ≈ 1.86, below which profiles are known by energy estimates to be stable, to the upper stability boundary of ≈ 1669, above which profiles are expected to be stable by rigorous asymptotic analysis. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.
|
|
July 11
|
Chak Shing Lee (TAMU)
|
Title: Generalization of mixed multiscale finite element methods with applications
Abstract: Many science and engineering problems exhibit scale disparity and high contrast.
The small scale features cannot be omitted in the physical models because they
can affect the macroscopic behavior of the problems. However, resolving all the scales
in these problems can be prohibitively expensive. As a consequence, some types of
model reduction techniques are required to design efficient solution algorithms.
For practical purpose, we are interested in mixed finite element problems as they
produce solutions with certain conservative properties. Existing multiscale methods
for such problems include the mixed multiscale finite element methods. We will show
that for complicated problems, the mixed multiscale finite element methods may fail to
produce reliable approximations. This motivates the need of enrichment for coarse
spaces. Two enrichment approaches are proposed, one is based on generalized
multiscale finite element methods, while the other is based on spectral element-based
algebraic multigrid. Application of the algorithms in single- and two-phase flow
simulations are demonstrated.
|