Applied/ACMS/absF11

Sigurd Angenent, UW-Madison

Deterministic and random models for polarization in yeast cells

I'll present one of the existing models for "polarization in yeast cells." The heuristic description of the model allows at least two mathematical formulations, one using pdes (a reaction diffusion equation) and one using stochastic particle processes, which give different predictions for what will happen. The model is simple enough to understand and explain why this is so.

John Finn, Los Alamos

Symplectic integrators with adaptive time steps

TBA

Jay Bardhan, Rush Univ

Understanding Protein Electrostatics using Boundary-Integral Equations

The electrostatic interactions between biological molecules play important roles determining their structure and function, but are challenging to model because they depend on the collective response of thousands of surrounding water molecules. Continuum electrostatic theory -- e.g., the Poisson equation -- offers a successful and simple theory for biomolecule science and engineering, and boundary-integral equation formulations of the problem offer several theoretical and computational advantages. In this talk, I will highlight some recent modeling advances derived from the boundary-integral perspective, which have important applications in biophysics and whose mathematical foundations may be useful in other domains as well. First, one may derive a fast electrostatic model that resembles Generalized Born theory, but is based on a rigorous operator approximation for rapid, accurate estimation of a Green's function. In addition, we have been exploring a boundary-integral approach to nonlocal continuum theory as a means to model the influence of water structure, an important piece of molecular physics left out of the standard continuum theory.

Omar Morandi, TU Graz

Modeling quantum transport with the phase-space formalism

Quantum modeling is becoming a crucial aspect in nanoelectronics research in perspective of analog and digital applications. Devices like interband tunneling diodes or graphene sheets are examples of solid state structures that are receiving a great importance in the modern nanotechnology for high-speed and miniaturized systems. Differing from the usual transport where the electronic current flows within a single band, the remarkable feature of such solid state structures is the possibility to achieve a sharp coupling among states belonging to different bands. As a consequence, the single band transport or the classical phase-space description of the charge motion based on the Boltzmann equation are not longer accurate. Moreover, in a crystal where the effective Hamiltonian is expressed by a partially diagonalized basis (e. g. in graphene or in semiconductors), the usual definitions of the macroscopic quantities, as for example the mean velocity or the particle density, no longer apply. The theory of Berry phases offers an elegant explanation of this effect in terms of the intrinsic curvature of the perturbed band. Different approaches have been proposed to achieve a full quantum description of electron transport where the interaction among the different bands can be included. Among them, the phase-space formulation of quantum mechanics based on the concept of “Wigner-Weyl quantization”, offers a framework in which the quantum phenomena can be described with a classical language and the question of the quantum-classical correspondence can be directly investigated. In this contribution, an extension of the original Wigner-Weyl theory based on a suitable projection procedure, is presented. The applications of this formalism span among different subjects: the multi-band transport and applications to nano-devices, the infinite- order 􏰀-approximations of the motion and the characterization of a system in terms of Berry phases or, more generally, the representation of a quantum system as a Riemann manifold with a suitable connection. Furthermore, some asymptotic procedures devised for the approx- imation of the quantum Wigner-Weyl solution have shown a very attractive connection with the Dyson-Feynmann theory of the particle interaction, which allows us to describe quantum transition by means of an effective Boltzmann process.

Guowei Wei, Michigan State

Variational multiscale models for biomolecular systems

A major feature of biological science in the 21st Century will be its transition from a phenomenological and descriptive discipline to a quantitative and predictive one. Revolutionary opportunities have emerged for mathematically driven advances in biological research. However, the emergence of complexity in self-organizing biological systems poses fabulous challenges to their quantitative description because of the excessively high dimensionality. A crucial question is how to reduce the number of degrees of freedom, while returning the fundamental physics in complex biological systems. This talk focuses on a new variational multiscale paradigm for biomolecular systems. Under the physiological condition, most biological processes, such as protein folding, ion channel transport and signal transduction, occur in water, which consists of 65-90 percent of human cell mass. Therefore, it is desirable to describe membrane protein by discrete atomic and/or quantum mechanical variables; while treating the aqueous environment as a dielectric or hydrodynamic continuum. I will discuss the use of differential geometry theory of surfaces for coupling microscopic and macroscopic scales on an equal footing. Based on the variational principle, we derive the coupled Poisson- Boltzmann, Nernst-Planck (or Kohn-Sham), Laplace-Beltrami and Navier-Stokes equations for the structure, dynamics and transport of ion-channel systems. As a consistency check, our models reproduce appropriate solvation models at equilibrium. Moreover, our model predictions are intensively validated by experimental measurements. Mathematical challenges include the well-posedness and numerical analysis of coupled partial differential equations (PDEs) under physical and biological constraints, lack of maximum-minimum principle, effectiveness of the multiscale approximation, and the modeling of more complex biomolecular phenomena. References  Guo-Wei Wei, Differential geometry based multiscale models, Bulletin of Mathematical Biology, 72, 1562-1622, (2010). http://www.springerlink.com/content/8303641145x84470/fulltext.pdf  Zhan Chen, Nathan Baker and Guo-Wei Wei, Differential geometry based solvation model I: Eulerian formulation, Journal of Computational Physics, 229, 8231-8258 (2010). http://math.msu.edu/~wei/paper/p141.pdf  Qiong Zheng and Guo-Wei Wei, Poisson-Boltzmann-Nernst-Planck model. Journal of Chemical Physics, 134 (19), 194101, (2011). http://jcp.aip.org/resource/1/jcpsa6/v134/i19/p194101_s1

George Hagedorn, Virginia Tech

Time Dependent Semiclassical Quantum Dynamics: Analysis and Numerical Algorithms

We begin with some elementary comments about time-dependent quantum mechanics and the role of Planck's constant. We then describe several mathematical results about approximate solutions to the Schr\"odinger equation for small values of the Planck constant. Finally, we discuss numerical difficulties of semiclassical quantum dynamics and algorithms that have recently been developed, including some work in progress.

Frederic Coquel, Ecole Polytechnique Paris

Coupling of Hyperbolic PDEs: thin versus thick coupling interfaces

The talk will give an overview of some results obtained with several co-workers on the mathematical coupling of nonlinear hyperbolic PDEs. The (well-separated) multi-scale phenomena taking place in various technological setups indeed requires to address Cauchy problems built from a hierarchy of hyperbolic models with relaxation that are formulated on a partition of the physical domain into subregions. At the interface of two subregions, discontinuities in the modeling arise and transient exchange conditions, the so-called coupling conditions, have to be prescribed. I will present a mathematical formalism which models the coupling interfaces in terms of standing waves for an augmented PDE model which is set over the whole physical domain. The augmented equations which may be seen as a first order system with discontinuous coefficients can in turn support various regularization mechanisms. We first adopt the viscous regularization {\it \`a la} Dafermos and prove existence of self-similar weak solutions for the coupling of two hyperbolic systems in a single space dimension under fairly general conditions. However, failure of uniqueness is observed in the limit of a vanishing viscosity, as a consequence of a resonance phenomena. To recover uniqueness, we will promote another regularization mechanism based on thickened coupling interfaces. The proposed framework naturally allows for the definition of multi-dimensional and multi-component couplings with possible covering. Numerical illustrations will be given all along the lecture.

Qiang Deng, UW-Madison

Tropical cyclogenesis in a 3D Boussinesq model with simple cloud physics

TBA

Ray Pierrehumbert, U of Chicago

Bifurcation and climate sensitivity

The concept of climate sensitivity lays at the heart of assessment of the magnitude of the imprint of human activities on the Earth's climate. Most commonly, the "climate" is represented by a simple projection such as a global mean temperature, and we wish to know how this changes in response to changes in a single control parameter -- usually atmospheric CO2 concentration. This problem is an instance of a broad class of related problems in parameter dependence of dynamical systems. I will discuss the shortcomings of the traditional linear approach to this problem, particularly in light of the spurious "runaway" states produced when feedback becomes large. The extension to include nonlinear effects relates in a straightforward way to bifurcation theory. I will discuss explicit examples arising from ice-albedo, water vapor, and cloud feedbacks. Finally, drawing on the logistic map as an example, I will discuss the problem of defining climate sensitivity for problems exhibiting structural instability.

Bokai Yan, UW-Madison

Asymptotic-preserving schemes for kinetic-fluid coupling model

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker-Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker-Planck solver or an incompressible Navier-Stokes solver, only the conjugate-gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the schemes, with several interesting applications.

Henri Berestycki, EHESS

The Explosion Problem with a Flow

The classical explosion problem or Gelfand problem is a semi-linear elliptic equation with exponential non-linearity and a parameter. My talk is about aspects of this problem when one further takes into account the effect of transport by an incompressible flow. I will report here on joint works with X. Cabre, A. Kiselev, A. Novikov and L. Ryzhik

David Anderson, UW-Madison

Computational methods for stochastic models in biology

I will focus on computational methods for continuous time Markov chains, which includes the large class of stochastically modeled biochemical reaction networks and population processes. I will show how different computational methods can be understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.

Carsten Conradi, MPI-Magdeburg

Multistationarity and switching in mass action networks

Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions): switching is, for example, the basis of intracellular processes like cellular signal transduction, information processing and cell cycle control. Due to predominant parameter uncertainty numerical methods are generally difficult to apply to realistic models originating in Systems Biology. Hence analytical tools that allow the direct computation of states and parameters where switching occurs are desirable. Here conditions for mass action networks are presented that take the form of linear inequality systems.

Jianfeng Lu, Courant Institute

Multiscale analysis of solid materials: From electronic structure models to continuum theories

Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will focus on macroscopic limit and microstructure pattern formation of electronic structure models.

Anne Shiu, U of Chicago

Chemical reaction systems with toric steady states

Chemical reaction networks taken with mass-action kinetics are dynamical systems governed by polynomial differential equations that arise in chemical engineering and systems biology. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. This talk focuses on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to admit toric steady states. Furthermore, we analyze the capacity of such a system to exhibit multiple steady states. An important application concerns the biochemical reaction networks networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism. No prior knowledge of chemical reaction network theory or binomial ideals will be assumed. This is joint work with Carsten Conradi, Mercedes Pérez Millán, and Alicia Dickenstein.

Willard Miller, Minnesota

Compressive sampling, or how to get something from almost nothing (probably)

Is it possible to fully reconstruct a signal if we only have very few samples of the signal ? Mathematically, this is the problem of solving m equations for n unknowns where m<< n. The surprising answer is yes, if the signal is nearly sparse. With the help of compressive sampling, a signal processing technique that employs random matrix theory, sparse signals can be recovered with great accuracy even with a small number of samples. I will give a simple explanation of compressive sampling and use live Matlab simulations to show that it actually works.

Peter Thomas, Case Western

Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit

Rhythmic behaviors in neural systems often combine features of limit cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small perturbations satisfying a net inflow condition will generically give rise to a stable limit cycle (Reyn, 1980; Guckenheimer and Holmes, 1983). Here we consider the asymptotic behavior of the infinitesimal phase response curve (iPRC) for examples of two systems satisfying Reyn's inflow criterion, (i) a smooth system with a chain of four hyperbolic saddle points and (ii) a piecewise linear system corresponding to local linearization of the smooth system about its saddle points. For system (ii), we obtain exact expressions for the limit cycle and the iPRC as a function of a parameter $\mu>0$ representing the distance from a heteroclinic bifurcation point. In the $\mu\to 0$ limit, we find that perturbations parallel to the unstable eigenvector direction in a piecewise linear region lead to divergent phase response, as previously observed (Brown, Moehlis and Holmes, 2004). In contrast to previous work, we find that perturbations parallel to the stable eigenvector direction can lead to either divergent or convergent phase response, depending on the phase at which the perturbation occurs. In the smooth system (i), we show numerical evidence of qualitatively similar phase specific sensitivity to perturbation. Having the exact expression for the iPRC for the piecewise linear system allows us to investigate its stability under diffusive coupling. In addition, we qualitatively compare iPRCs obtained for systems (i) and (ii) to iPRCs for the Morris-Lecar equations near a bifurcation from limit cycles to a saddle-homoclinic orbit. Joint work with K. Shaw, Y. Park, and H. Chiel.

Ian Tice, Universit Paris-Est Crteil

Some stability problems in interfacial fluid mechanics

Interfacial problems in fluid mechanics are ubiquitous in nature, appearing at a huge range of scales and in a multitude of physical configurations. As such, the stability of these problems is of significant interest. In this talk I will present recent results on the nonlinear stability / instability of three distinct problems: the viscous surface wave problem, the viscous surface-internal wave problem with surface tension, and the viscous gaseous star problem.

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Archived semesters

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