Applied/ACMS/absS11

Cynthia Vinzant, UC Berkeley

The central curve in linear programming

The central curve of a linear program is an algebraic curve specified by the associated hyperplane arrangement and cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. Here we will discuss the algebraic properties of this curve and its beautiful global geometry. In the process, we'll need to study the corresponding matroid of the hyperplane arrangement. This will let us give a refined bound on the total curvature of the central curve, a quantity relevant for interior point methods. This is joint work with Jesus De Loera and Bernd Sturmfels appearing in arXiv:1012.3978.

József Farkas, University of Stirling, Scotland

Analysis of a size-structured cannibalism model with infinite dimensional environmental feedback

First I will give a brief introduction to structured population dynamics. Then I will consider a size-structured cannibalism model with the model ingredients depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system. We show how the point spectrum of the linearised semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.

Tatiana Márquez-Lago, ETH-Zurich

Stochastic models in systems and synthetic biology

Cells prevail as efficient decision makers, despite the intrinsic uncertainty in the occurrence of chemical events, and being embedded within fluctuating environments. The underlying mechanisms of this ability remain widely unknown, but they are critical for the correct understanding of biological systems output and predictability. Some advances have been achieved by considering biological processes as modular units, but the conclusions in many studies vary alongside experimental conditions, or easily break down once the system is no longer isolated. Moreover, sets of seemingly simple biochemical reactions can generate a wide range of highly non-linear complex behaviours, even in the absence of crosstalk. To illustrate some of these challenges, encountered in everyday biological/pharmaceutical research, I will present three short stories showing how iterations between mathematicians, computer scientists and biologists can generate successful ideas, testable in the laboratory. The first story revolves around a tunable synthetic mammalian oscillator, from the individual cell perspective and population behavior. The long term importance of this work lies in discerning whether it is possible to influence the underlying genetic clockwork to tune the expression of key genes. Answering this question may prove to be central in the design of future gene therapies, particularly those requiring a periodic input. In the second story I will show how closures on master equations describing negative self-regulation may yield diametrically opposed noise effects to those expected by exact solutions, discovering how any noise profile (and correlations between mRNA transcription and protein synthesis) can be created by the consideration of specific kinetic rates and network topologies. Lastly, I will illustrate in a third story how chemical adaptation can many times be considered a purely emergent property of a collective system (even in simple linear settings), how a simple linear adaptation scheme displays fold-change detection properties, and how rupture of biological ergodicity prevails in scenarios where transitions between protein states are mediated by other molecular species in the system.

Ari Stern, UC San Diego

Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces

In recent years, the success of "mixed" finite element methods has been shown to have surprising connections with differential geometry and algebraic topology---particularly with the calculus of exterior differential forms, de Rham cohomology, and Hodge theory. In this talk, I will discuss how the notion of "Hilbert complex," rather than "Hilbert space," provides the appropriate functional-analytic setting for the numerical analysis of these methods. Furthermore, I will present some recent results that analyze "variational crimes" (a la Strang) on Hilbert complexes, allowing the numerical analysis to be extended from polyhedral regions in Euclidean space to problems on arbitrary Riemannian manifolds. As a direct consequence, our analysis also generalizes several key results on "surface finite element methods" for the approximation of elliptic PDEs on hypersurfaces (e.g., membranes or level sets undergoing geometric evolution).

Jian-Guo Liu, Duke University

Dynamics of orientational alignment and phase transition

Phase transition of directional field appears in some physical and biological systems such as ferromagnetism near Currie temperature, flocking dynamics near critical mass of self propelled particles. Dynamics of orientational alignment associated with the phase transition can be effectively described by a mean field kinetic equation. The natural free energy of the kinetic equation is non-convex with a minimum level set consisting of a sphere at super-critical case, a typic spontaneous symmetry breaking behavior in physics. In this talk, I will present some analytical results on this dynamics equation of orientational alignment and exponential convergence rate to the equilibria for both supper and sub critical cases, as well at algebraic convergence rate at the critical case. A new entropy and spontaneous symmetry breaking analysis played an important role in our analysis.

Tim Reluga, Penn State University

Accounting for individual and community interests in the public-health management of infectious diseases

In his history of the Peloponnesian war, Thucydides provides one of the earliest accounts of the devastation that infectious diseases can cause cities and communities. Despite 2000 years of advancement, infectious diseases continue to plague nations around the world. While vaccines and modern medicine have greatly reduced disease burdens in many parts of the world, pressures from growing human populations and microbial evolution are eroding our advances. Today, management problems are as much social as biological. In this talk, I'll describe some contemporary challenges we face in managing infectious disease, and how mathematical methods can help us understand these challenges. Using dynamical systems, Markov processes, and game theory, we can formulate and solve a rich variety of problems with practical applications related to vaccines, disease prevention and treatment, and public health in general. These methods are suitable for use throughout the field of ecological-economics.

Yuri Lvov, Rensselaer Polytechnic Institute

Internal waves in the ocean - observations, theory and DNS

Spectral energy density of internal waves in the ocean exhibit a surprising degree of universality - it is given by the Garrett and Munk Spectrum of internal waves, discovered over 30 years ago. I will explain that situation is much more interesting, and will describe recent theoretical advances in understanding internal waves. I will demonstrate that when using traditional wave turbulence theory one runs to internal logical contradictions: the results of the theory (strong nonlinearity) contradict the underlying assumptions (weak nonlinearity) used to build the theory. I will demonstrate possible directions out of the puzzle and will elaborate on open questions and challenges.

Alex Kiselev, UW-Madison (Mathematics)

Biomixing by chemotaxis and enhancement of biological reactions

Many processes in biology involve both reactions and chemotaxis. However, to the best of our knowledge, the question of interaction between chemotaxis and reactions has not yet been addressed either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction (fertilization). The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemotactic attraction is sufficiently strong. Moreover, an interesting feature of the estimates in chemotactic case is that rates and timescales of the reaction (fertilization) process do not depend on the reaction amplitude coefficient.

Gerardo Hernández-Dueñas, University of Michigan

Shallow water flows in channels

The talk will discuss shallow water flows through channels of arbitrary geometry. They form a set of nonlinear hyperbolic conservation laws with geometric source terms. A Roe-type upwind scheme will be presented for geometries where the cross sections consist of vertical walls of variable width, followed by trapezoidal, piecewise trapezoidal and general cross-sectional areas. Considerations of conservation, near steady-state accuracy and positivity near dry states will be discussed, and numerical results will be shown for a variety of unsteady and near steady flows.

Anne Gelb, Arizona State University

Reconstruction of piecewise smooth functions from non-uniform Fourier data

We discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI). We summarize two standard techniques, convolutional gridding and uniform resampling, and address the issue of non-uniform sampling density and its effect on reconstruction quality. We compare these classical reconstruction approaches with alternative methods such as spectral re-projection and methods incorporating jump information.

Evangelos Coutsias, University of New Mexico

Protein loop modeling with inverse kinematics

Protein loops are the sections of the polypeptide chain connecting regions of secondary structure such as helices and beta strands. They may contain functional residues or have purely structural roles and often they can be the sites of evolutionary changes. In contrast to the relatively rigid helices and strands, loops can be flexible, allowing a protein to rapidly respond to changes and bind to ligands. Structure determination of flexible loops with given endpoints is a challenging problem, commonly referred as the Loop Closure problem. Loop closure has been studied by computational methods since the pioneering work of Go and Scheraga in the '70s. Our Triaxial Loop Closure (TLC) method provides a simple and robust algebraic formulation of the loop closure problem for loops of arbitrary length and geometry. We present results of several recent studies showing that TLC samples loop conformations more efficiently than other currently available methods: TLC sampling augmented with a simulated annealing protocol using the Rosetta scoring potential was able to predict the native structures of several standard loop test sets with up to 12 residue loops with sub-­‐Angstrom mean accuracy; TLC with a Jacobian guided Fragment Assembly scheme was shown to outperform other methods in generating near native ensembles; and finally, TLC based local moves were incorporated in a new Monte Carlo scheme that hierarchically samples backbone and sidechains, making it possible to make large moves that cross energy barriers. The latter method, applied to the flexible loop in triosephosphate isomerase that caps the active site, was able to generate loop ensembles agreeing well with key observations from previous structural studies. Further applications of kinematic geometry to protein modeling will be discussed as time permits.

Michael Holst, UC San Diego

Some far-from-CMC existence results for the Einstein constraint equations

In this lecture, we consider the non-dynamical subset of the Einstein equations known as the Einstein constraints. This coupled nonlinear elliptic system must be solved numerically to produce initial data for gravitational wave simulations, and to enforce the constraints during dynamical simulations, as needed for LIGO and other gravitational wave modeling efforts. The Einstein constraint equations have been studied intensively for half a century; our focus in this lecture is on a thirty-five-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature. Until 2009, all known existence results involved assuming either constant (CMC) or nearly-constant (near-CMC) mean extrinsic curvature. After giving a survey of known CMC and near-CMC results through 2009, we outline a new theoretical framework for analyzing existing of solutions that is fundamentally free of both CMC and near-CMC conditions, resting on the construction of "global barriers" for the Hamiltonian constraint. We then present such a barrier construction for case of closed manifolds with positive Yamabe metrics, giving the first known existence results for arbitrarily prescribed mean extrinsic curvature. Our results are developed in the setting of a ``weak background metric, which requires building up a set of preliminary results for for elliptic operators on manifolds with weak metrics. However, this allows us to recover the recent ``rough CMC existence results of Choquet-Bruhat (2004) and of Maxwell (2004-2006) as two distinct limiting cases of our non-CMC results. Our non-CMC results also extend to other cases such as compact manifolds with boundary. This is joint work with Gabriel Nagy and Gantumur Tsogtgerel [1-2]. [1] M. Holst, G. Nagy, and G. Tsogtgerel, "Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics", Physical Review Letters, Vol. 100 (2008), No. 16, pp. 161101.1-161101.4. http://arxiv.org/abs/0802.1031 [2] M. Holst, G. Nagy, and G. Tsogtgerel, "Rough Solutions of the Einstein Constraints on closed manifolds without near-CMC conditions", Comm. Math. Phys., Vol. 288 (June 2009), No. 2, pp. 547-613. http://arXiv:gr-qc/0712.0798

Smadar Karni, University of Michigan

Numerical approximation of shock waves in non-conservative hyperbolic systems

Non-conservative hyperbolic systems arise in a wide range of applications, which makes their theoretical study and numerical approximation very important. The mathematical theory of weak solutions has been generalized to the nonconservative setup using vanishing viscosity solutions and viscous paths. While advances have been made on the theoretical front, those advances have been slow to translate into successful numerical methods. The underlying difficulty is that shock relations depend not only on the immediate states ahead/behind the shock, but also on the viscous path that connects them. In this talk, we shed light on some of the difficulties involved by considering an illuminating example from gas dynamics. This is joint work with Remi Abgrall, University of Bordeaux, France.

Tim Barth, NASA Ames

Energy stable space-time finite element approximation of the 2-fluid Euler-Maxwell plasma equations

Energy stable variants of the space-time discontinuous Galerkin (DG) finite element method are developed that approximate the ideal two-fluid Euler-Maxwell plasma equations. Using standard symmetrization techniques, the two-fluid plasma equations are symmeterized via a convex entropy function and the introduction of entropy variables. Anaysis results for the DG formulation assuming general unstructured meshes in space and arbitrary order polynomial approximation include a cell entropy bound for the semi-discrete formulation, a global two-sided entropy bound and L_2 stability for the space-time formulation, a modification of the DG method with provable stability when entropy variables are not used. Numerical results of the 2-fluid system including GEM magnetic reconnection are presented verifying the analysis and assessing properties of the formulation. This is joint work with James Rossmanith, UW-Madison.

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Archived semesters

• Spring 2011
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