Applied Mathematics

Seminars at UCSB

Applied Mathematics and PDE Seminar
Applied Mathematics Seminar (Calendar)

UCSB Research Calendar of Events
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Applied Mathematics and PDE Seminar

Cloaking by change of variables - The fix frequency case
Speaker: Niklas Wellander (Swedish Defence Research Agency, FOI)
Location and Time: January 31st, 2011 in South Hall 4519 from 1-2pm.
Host: Tom Sideris.

Abstract: We present the fundamentals for electromagnetic cloaking by means of change of variables. The method relies on the non-uniqueness of the inverse scattering problem. The scattering of electromagnetic energy is put into a variational form. The domain containing the cloak and the cloaked object is initially filled with the surrounding material (in general vacuum for the most interesting applications). The cloak is the effect of a singular transformation, which interpreted as an active transform defines the properties of the cloak explicitly. Greenleaf, Lassas and Uhlmann (2003) used a coordinate transform to define a surrounding heterogeneous medium for a cloak in the electrical impedance tomography problem. Kohn, Shen, Vogelius and Weinstein (2008) used a nonsingular transform to produce a near-cloak in a variational setting of electrical impedance tomography.

Well-posedness for the moving-boundary 3-D compressible Euler equations in physical vacuum
Friday, October 22, 2010; TBA
Steve Shkoller, University of California Davis
Host: Tom Sideris.

Abstract: We prove well-posedness for the 3-D compressible Euler equations with a moving surface of discontinuity comprised of the physical vacuum boundary, with an equation of state given by p(ϱ) = ϱ^γ for γ>1. The physical vacuum singularity requires the sound speed to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system. We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher-order Hardy-type inequality in conjunction with an approximation of the Euler equations consisting of a specific degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions. This is joint work with D. Coutand.

Dissipation-induced instabilities in Nature and Mathematics
Friday April 16, 2010; 12:00pm - 1:00pm; SH 6617
Rouslan Krechetnikov, Department of Mechanical Engineering, University of California Santa Barabra.
Host: Paul J. Atzberger.

Abstract: In this talk a joint work with Jerrold Marsden on a coherent theory of the counter-intuitive phenomena of dynamical destabilization under the action of dissipation is presented. While the existence of one class of dissipation-induced instabilities in finite-dimensional mechanical systems was known to Sir Thomson (Lord Kelvin), until recently it has not been realized that there is another major type of these phenomena hinted by one of theorems due to Russian mechanician Merkin; in fact, these two cases exhaust all the generic possibilities in finite dimensions. We put the main theoretical achievements in a general context of geometric mechanics, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature.

Next we develop a rigorous notion of dissipation-induced instability in the infinite-dimensional case, which inherent differences from classical finite degree of freedom mechanical systems make uncovering this concept more intricate. In building this concept of dissipation-induced instability we found Arnold's and Yudovich's nonlinear stability methods, for conservative and dissipative systems respectively, along with some new existence theory for solutions to be the essential ingredients. As a paradigm and the first infinite-dimensional example to be carefully analyzed, we use a two-layer quasi-geostrophic beta-plane model, which describes the fundamental baroclinic instability in atmospheric and ocean dynamics.

Solvability of Projected Equations in Banach Spaces
Friday April 2, 2010; 2:00pm - 3:00pm; SH 4607?
Monica Gabriela Cojocaru, Mathematics Department, University of Guelph; CANADA-US Fulbright Visiting Research Chair at University of California at Santa Barbara.
Host: (local).

Abstract: We will discuss the solvability of a class of nonlinear differential equations on Banach spaces that relate to variational inequalities and complementarity problems. The class is called projected differential equations and their characteristic is that their flow is only allowed to evolve within a subset of the underlying space. Such equations have been recently formulated in B-spaces, but their solvability has not yet been discussed. We offer a first insight into the question of existence of solutions for such equations and its implications for the study of applied problems related to such equations. They are a generalization of similar equations in Hilbert spaces, now widely used in applied equilibrium problems in networks, game theoretic and economic problems.

Influence of Cellular Substructure on Gene Expression and Regulation.
Thursday, March 4th-2010; 4:00pm - 5:00pm; South Hall 6617.
Samuel Isaacson, Boston University.
Host: Paul J. Atzberger.

Abstract: We will give an overview of our recent work investigating the influence of incorporating cellular substructure into stochastic reaction-diffusion models of gene regulation and expression. Extensions to the reaction-diffusion master equation that incorporate effects due to the chromatin fiber matrix are introduced. These new mathematical models are then used to study the role of nuclear substructure on the motion of individual proteins and mRNAs within nuclei. We show for certain distributions of binding sites that volume exclusion due to chromatin may reduce the time needed for a regulatory protein to locate a binding site.

A Hybrid Particle-Continuum Method for Hydrodynamics of Complex Fluids.
Friday Jan. 29-2010; 11:00am - 12:00pm; SH 6635
Aleksandar Donev, LBNL
Collaborators: John B. Bell, Alejandro L. Garcia, and Berni J. Alder.
Host: Paul J. Atzberger.

Abstract: We generalize a previously-developed hybrid particle-continuum method [J. B. Bell, A. Garcia and S. A. Williams, SIAM Multiscale Modeling and Simulation, 6:1256-1280, 2008] to dense fluids and two and three dimensional flows. The scheme couples an explicit fluctuating compressible Navier-Stokes solver with the Isotropic Direct Simulation Monte Carlo (DSMC) particle method [A. Donev and A. L. Garcia and B. J. Alder, J. Stat. Mech., 2009:P11008]. To achieve bidirectional dynamic coupling between the particle (microscale) and continuum (macroscale) regions, the continuum solver provides state-based boundary conditions to the particle domain, while the particle domain provides flux-based boundary conditions for the continuum solver, thus ensuring both state and flux continuity across the particle-continuum interface. Using the hybrid method we study the equilibrium diffusive motion of a large spherical bead suspended in a particle solvent and find that the hybrid method correctly reproduces the velocity autocorrelation function of the bead only if thermal fluctuations are included in the continuum solver. Finally, we apply the hybrid to the well-known adiabatic piston problem and find that the hybrid correctly reproduces the slow non-equilibrium relaxation of the piston toward thermodynamic equilibrium when fluctuations are included in the continuum solver. These examples clearly demonstrate the need to include fluctuations in continuum solvers employed in hybrid multiscale methods.

Global rough solutions to the critical generalized KdV equation.
Friday Jan. 8-2010; 2:00pm - 3:00pm; SH 4607
Luiz G. Farah (UCSB and ICEx/UFMG, Belo Horizonte, MG, Brazil).
Host: Carlos Garcia-Cervera.

Abstract: Following the $I$-method scheme, we prove that the initial value problem (IVP) for the critical generalized KdV equation $$u_t + u_{xxx} + (u5 )_x = 0$$ on the real line is globally well-posed in $H^s (R)$, $s > 3/5$, with the appropriate smallness assumption on the initial data.

A Direct Constrained Optimization Method for Solving the Kohn-Sham Equations.
Friday, Dec. 4, 2009 from 2:00pm - 3:00pm, South Hall 4607.
Juan C. Meza, Department Head and Senior Scientist, High Performance Computing Research, Lawrence Berkeley National Laboratory.
Host: Carlos Garcia-Cervera.

Abstract: Nanostructures have been proposed for many applications including solar cells for renewable energy, biomedical imaging, and other novel materials. To fully explore these ideas however, requires ab initio materials simulations. Today, these codes are usually based on Density Functional Theory (DFT) that are used for computing the ground state energy and the corresponding single particle wave functions associated with a many-electron atomistic system. At the heart of these codes, one typically finds a Self Consistent Field (SCF) iteration. In this talk, we propose an optimization procedure that minimizes the Kohn-Sham total energy directly. We point out the similarities between our new approach and SCF and show how the SCF iteration can fail when the minimizer of a particular surrogate produces an increase in the total energy. A trust region technique is introduced as a way to restrict the update of the wave functions within a small neighborhood of an approximate solution at which the gradient of the total energy agrees with that of the surrogate. Numerical examples demonstrate that the combination of these approaches is more efficient than SCF.

Parameterization of Turbulent Transport by Mesoscale Eddies.
Monday, Oct. 5, 2009 from 11:00am - 12:00pm, South Hall 6617.
Peter Kramer, Rensselaer Polytechnic Institute.
Host: Paul J. Atzberger.

Abstract: We employ homogenization theory to develop a systematic parameterization strategy for quantifying the transport effects of mesoscale coherent structures in the ocean which cannot be well resolved by large-scale weather and climate simulations. We work from the ground up with simple kinematic models and study in particular how the effective diffusivity depends on the governing parameters, such as Strouhal number and Peclet number, in a class of dynamical random vortex flows. We will also briefly describe some connections between the homogenized effective diffusivity and a recently introduced alternative mixing efficiency measure. This is joint work with Banu Baydil, Shane Keating, and Shafer Smith.

Solving Nonlinear Eigenvalue Problems in Electronic Structure Calculations.
Friday, May 15, 2009 4:00pm - 5:00pm, SH 4617.
Chao Yang, Lawrence Berkeley Laboratory.
Host: Carlos Garcia-Cervera.

Abstract: One of the fundamental problems in electronic structure calculations is to determine the electron density associated with the minimum total energy of molecules, solids or other types of nanoscale materials. The total energy minimization problem is often formulated as a nonlinear eigenvalue problem. It is also known as the Kohn-Sham problem. In this talk, I will discuss several numerical methods for solving this type of problem and examine their convergence properties.

Topological quantization of ensemble averages .
Friday, April 10, 2009 from 2 to 3 p.m., SH 4607.
Dr. Emil Prodan, Yeshiva University, New York, NY.
Host: Carlos Garcia-Cervera.

Abstract: Non-commutative geometry and calculus have been successfully used in the past to unlock the secretes of several important observations in condensed matter. In this talk I will discuss my own efforts to apply the non-commutative geometry and calculus to a new class of materials called topological insulators.

I will briefly review the non-commutative theory of the Integer Quantum Hall Effect, with emphasis on some relatively recent results concerning the edge physics. I will then discuss a result that underlines a general principle for the quantization of ensemble averages. I will use several examples to convey the implications of the result. These examples include quantization of conductance in metallic wires, quantization of edge conductance in 2D Chern insulators with random edges, robustness of the edge modes in 2D quantum spin-Hall systems against disorder. Notes on the 3D insulators will be also presented if time allows.

1. E. Prodan, Topological quantization of ensemble averages, J. Phys. A: Math. and Theor. 42, 065207 (2009)
2. E. Prodan, An edge index for the quantum spin-Hall effect, J. Phys. A: Math. and Theor. 42, 082001 (2009)
3. E. Prodan, The edge spectrum of Chern insulators with rough boundaries, arXiv:0809.2569v2 (2009)

Dynamics of a Stochastically Driven Neuronal Network Model.
Tuesday, February 17th, 2009; 4:00pm - 5:00pm, SH 4607.
Peter R. Kramer, Dept. Mathematical Sciences, Rensselaer Polytechnic Institute.
Host: Paul J. Atzberger.
Abstract: We study an all-to-all coupled network of identical excitatory integrate-and-fire (I\&F) neurons driven by an external spike train modeled as a Poisson process. Numerical simulations demonstrate that over a broad range of parameters, the network enters a synchronized state in which the neurons all fire together at regular intervals. We identify mechanisms leading to this synchronization for two regimes of the external driving current: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. In both cases, stochastic fluctuations play a central role in determining the oscillation period. We also develop a criterion for synchrony in the network through a probabilistic argument. This work is in collaboration with Katherine Newhall, Gregor Kovacic, David Cai, and Aaditya Rangan.

Title Here.
Tuesday, February 17th, 4:00pm - 5:00pm, SH 4607.
Speaker Name, Dept., Affiliation.

Abstract: Description of talk.

Seminar Scheduling (will be edited by speakers directly)