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PDE/Applied Math Seminar - Abstracts Fall 2003

(Feb 5) John Greer Fourth order equations for image processing

A number of fourth order diffusion equations have recently been introduced for image smoothing and denoising. Although discrete implementations of these methods produce impressive results, very little is known about the mathematical properties of the equations themselves. I will discuss some of the first results regarding a few of these nonlinear diffusions. In particular, I will describe the use of energy methods to prove that a class of $H^1$ diffusions for image processing is well posed. I will discuss similar methods for showing that the `Low Curvature Image Simplifier' (LCIS) equation of Tumblin and Turk (SIGGRAPH, August, 1999) has smooth solutions locally in time in $R^2$ and globally in time in $R.$ I will demonstrate implementations of a new finite difference discretization of the LCIS equation that ensures the discrete Laplacian of the image intensity remains bounded. I will also discuss new model advection-diffusion equations motivated by image inpainting and will show how topology and dynamical system theory were used to prove analytical results for these new equations.

(Feb 12) Fumioki Asakura Schaeffer-Shearer's Case I and II -- existence of viscous profiles

We discuss the existence of viscous profiles for a $2 \times 2$ system of hyperbolic conservation laws: $U_t + F(U)_x =0, (x,t) \in {\bf R}\times {\bf R}_+.$ We say that the point $U^{\ast}$ is an umbilic point if the Jacobian matrix is expressed as $F'(U^{\ast}) = \lambda I_2.$ Schaeffer and Shearer obtained a normal form of quadratic flux, containing two parameteres, that is an approximation of $F(U)$ at an umbilic point modulo higher order terms (CPAM, 1987). In this talk, we will confine ourselves to these fluxes in Case I and II of their classification. The viscous profile is the solution to the system of ODE's: $ \frac{dU}{d\xi} = -s(U - U_-) + F(U) - F(U_-)$ satisfying $\lim_{\xi \to \pm\infty} = U_{\pm}. $ Generically, the above vector field has four critical points. Topological consideration implies that they are a single node and three saddles in Case I, and two nodes and two saddles in Case II. In the terminology of hyperbolic conservation laws, by considering $1$-shock wave with shock speed $s,$ there are three compressive 1-shocks in Case I and two compressive 1-shocks and an overcompressive 1-shock in Case II. We can prove that all of these shock waves have viscous profiles and hence {\it admissible\/.} The key point is that the above system turns out to be a gradient system: $\frac{dU}{d\xi} =\frac{1}{2} \nabla \phi(U)$ and the study of the integral curves consists in studying the level curves of $\phi(U).$ A generalization of Morse's fundamental lemma plays a crucial role.

(Feb 19) Shouhong Wang Bifurcation and Stability of Rayleigh-Benard Convection

In this talk, I shall present my recent work on bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Bénard convection. This problem goes back to the Bénard's experiments in 1900, and the pioneering work of Rayleigh in 1916. The best nonlinear theory for this problem was done by Rabinowitz and Yudovich in the 60's. However, most, if not all, known results on the bifurcation and stability analysis for this problem are restricted to certain subspaces of the entire phase space obtained by imposing certain symmetry when the first engenvalue of the linear problem is simple.

I shall present a nonlinear theory for this problem, which is established using a new notion of bifurcation and its corresponding theorem. This theory includes 1) the existence of bifurcation from the trivial solution when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_c$ for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue $R_c$ for the linear problem, 2) aymptotical stability of the bifurcated solutions, and 3) the roll structure and its stability in the physical space.

This is joint work with Tian Ma.

(Feb 26) Thomas Wanner Stochastic Cahn-Hilliard Dynamics

The Cahn-Hilliard equation is one of the fundamental models for phase separation dynamics in metal alloys. On a qualitative level, it can successfully describe phenomena such as spinodal decomposition and nucleation. Yet, as deterministic partial differential equation it does not account for thermal fluctuations or similar random effects. In this talk I will describe some dynamical aspects of a stochastic version of the model due to Cook. These include recent results on spinodal decomposition and nucleation. In addition, differences between the deterministic and stochastic dynamics are discussed. In particular, I will address the temporal evolution of microstructures generated through spinodal decomposition using methods of computational homology.

(March 4) Todd Troyer The dynamics of spiking: First passage time problems and simple model neurons

Nerve cells communicate by producing sudden pulses of voltage known as "spikes." The mechanism for spike production is often modeled as a stochastic threshold-crossing problem. Although a number of special cases have been solved, solutions usually concern the case of stationary inputs. I will outline a very basic argument that may give some insight into the dynamic response of these models to changing inputs, and will show simulation results from some simple approximations applied to extreme cases. Along the way, I will point to open problems and ways in which one could incorporate greater biological realism.

(March 11) Hyung Ju Hwang Hyperbolic chemotaxis model

We will consider a one dimensional hyperbolic system for chemosensitive movement, especially for chemotactic behavior. The model consists of two hyperbolic differential equations for the chemotactic species and is coupled with either a parabolic or an elliptic equation for the dynamics of the external chemical signal. The speed of the chemotactic species is allowed to depend on the external signal and the turning rates may depend on the signal and its gradients in space and time, as observed in experiments. Global classical solutions for regular initial data and a parabolic limit from hyperbolic model to diffusive chemotaxis model will be discussed.

(March 18) Eitan Tadmor A multiscale image representation using hierarchical $(BV,L^2)$ decompositions

We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation of general images. The starting point is a variational decomposition of an image, $f = u_0+v_0$, where $[u_0,v_0]$ is the minimizer of a $J$-functional, $J(f,c_0; X,Y)=\inf_{u+v=f} {\|u\|_X + c_0 \|v\|_Y^p}$. Such minimizers are standard tools for denoising, deblurring, compression, ... of images, e.g., [Mumford-Shah] and [Rudin-Osher-Fatemi]. Here, $u_0$ should capture `essential features' of $f$, to be separated from the spurious components in $v_0$, and $c_0$ is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step $[u_{j+1},v_{j+1}] = {\rm arginf}(v_j,c_0*2^j)$, leading to the hierarchical decomposition, $f = \sum_{j=0}^k u_j + v_k$. We focus our attention on the particular case of $(X,Y)=(BV,L^2)$ decomposition. The resulting hierarchical decomposition, $f \sim \sum_j u_j$, is essentially nonlinear. The questions of convergence, energy decomposition, localization and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler-Lagrange equations. Numerical results illustrate applications to synthetic and real images (both grayscale and colored images).

(April 1) Rustem Choksi Microphase Separation of Diblock Copolymers

Diblock copolymer melts, dubbed ``designer materials'', have the remarkable ability for self-assembly into various ordered structures. These structures are key to the many properties that make diblock copolymers of great technological interest.

In this talk, I will discuss modeling and analytical issues for both diblock copolymer melts and copolymer-homopolymer blends.

Modeling issues pertain to deriving averaged density functional theories from the microscopic behavior of the polymer chains. Analytical work addresses these density functional theories with ansatz-independent results for energy minimizing configurations.

(April 8) Björn Sandstede Absolute instability of standing pulses

We analyse instabilities of standing pulses in reaction-diffusion systems that are caused by a Turing or Hopf instability of the homogeneous background state. At a Turing instability, symmetric pulses emerge that are spatially asymptotic to the bifurcating Turing patterns. These pulses exist for any wavenumber inside the Eckhaus stability band. In contrast, an oscillatory instability of the background state leads to a unique modulated pulse that emits small-amplitude wave trains with a selected wavenumber. The existence of these patterns is investigated using blow-up techniques, while the stability considerations involve Evans-function computations for operators with algebraically decaying coefficients.

(April 15) Alexandre Freire The normalized mean curvature flow in Riemannian manifolds

The evolution under normalized mean curvature is the result of a complicated interaction between the geometry of the evolving surface and that of the ambient space. We identify a class of initial conditions for which we can prove global existence and describe asymptotic behavior and the law governing large-scale motion of the hypersurface inside the manifold.

This is joint work with N. Alikakos, UNT.

(April 22) Suncica Canic Effective, closed equations describing the flow of blood in compliant vessels

Due to a tremendous complexity of the human cardiovascular system it remains unfeasible to numerically simulate larger sections of the circulatory system using the full three-dimensional equations for blood flow coupled with the equations describing the vessel wall behavior. This is why simplified, effective equations capturing the most important features of blood flow in compliant vessel are called for. In this vein, a variety of one-dimensional models have been used to study the flow of blood in axially symmetric sections of the vascular system. They are typically quasilinear hyperbolic systems of partial differential equations obtained from the incompressible viscous Navier-Stokes equations using asymptotic and averaging techniques. In all of these models the typical question of closure related to averaging has been resolved by assuming an {\sl ad hoc} closure in the form of the prescribed (estimated, experimentally calculated) horizontal component of the velocity.

To avoid prescribing an ad hoc closure, in a work with Mikelic (University of Lyon 1, France) and now with Tambaca (University of Zagreb, Croatia) we have obtained a system of effective, simplified equations that include the closure. Using asymptotic techniques, coupled with the ideas from homogenization of flows through porous media, we obtained a Biot-type system of equations approximating the full three-dimensional fluid-structure interaction problem to the $\epsilon^2$-order, where $\epsilon$ is the aspect ratio (radius/length of vessel). The resulting system includes memory terms typically arising in homogenized equations describing wave-like phenomena. In our case the memory term describe the waves in the elastic structure (vessel wall) induced by the pressure waves of the fluid (blood).

Our system is "almost 1-dimensional". More precisely, the equations are two-dimensional (radial and axial coordinates) but they can be decoupled into a set of one-dimensional problems. As a result of the two-dimensional nature of the equations, our simplified equations capture the two-dimensional phenomena such as secondary flows ignored in the purely 1-D model.

Movies showing numerical simulations in various geometries will be shown. A background coming from various medical applications will be mentioned. Experimental laboratory at the Texas Heart Institute where a comparison with the experiment will be performed, will be shown.

(April 22) Shankar Venkataramani Multiple scale behaviors in thin elastic sheets

Everyday experience tells us that thin elastic sheets crumple when they are ``strongly'' forced. Crumpling results in very complex morphologies. At the same time it is a very robust phenomenon. It occurs in a range of situations from paper crushed in one's hands, to the ``crumple zones'' of automobiles in collisions. I will present some recent results on the crumpling phenomenon. In particular, I will show rigorous scaling laws for the energy and structure of a single ridge.

The ridge energy is ``nonlocal'' in sharp contrast to defect energies in many other condensed matter systems. Also, the scaling of the ridge energy is different from the scaling for other structures in thin elastic sheets, for example the energy of thin-film blisters. I will discuss the implications of our results to the problem of determining the appropriate reduced problem ($Gamma$-limit) for thin elastic sheets as their thickness goes to zero.

Time permitting I will also talk about recent work on the elasticity of negatively curved sheets -- the shape of leaves.

Georg Dolzmann

Last modified: Mon Apr 19 11:43:37 EDT 2004