Applied Mathematics and PDE Seminar

Applied Mathematics Seminar (Calendar)

UCSB Research Calendar of Events

Graduate Student Hypatian Seminar

Mathematics Department Seminars

Kavli Institute for Theoretical Physics (KITP)

Institute for Theoretical Physics (ITP)

Materials Research Laboratory (MRL)

California Nanosystems Institute (CNSI)

Center for Control, Dynamical Systems, and Computation (CCDC)

Molecular Cellular and Developmental Biology (MCDB)

Institute for Polymers and Organic Solids (IPOS)

Center for Interdisciplinary Research in Fluids Physics (CIRF)

Neuroscience Institute Seminars

Chemistry Department Seminars

Chemical Engineering Seminars

Electrical Engineering Seminars

Materials Department Seminars

*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.

References:

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.

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