Math Colloquium

### Speaker: Brittany A. Erickson

Title: A Finite Difference Method for Plastic Response with an Application to the Earthquake Cycle

Location: V-157, VR-II.

Time: Monday, August 8 at 13:20.

### Abstract:

We are developing an efficient, computational framework for simulating multiple earthquake cycles with off-fault plastic response. Both rate-independent and viscoplasticity are considered, where stresses are constrained by a Drucker-Prager yield condition. The constitutive theory furnishes a nonlinear elliptic partial differential equation which must be solved through an iterative procedure. A frictional fault lies at an interface in the domain. The off-fault volume is discretized using finite differences satisfying a summation-by-parts rule and interseismic loading is accounted for at the remote boundaries through weak enforcement of boundary conditions. Time-stepping is done through an incremental solution process which makes use of an elastoplastic tangent stiffness tensor and the return-mapping algorithm to obtain stresses consistent with the constitutive theory. Solutions are verified by convergence tests along with comparison to a finite element solution. I will conclude with some application problems related to earthquake cycle modeling.

Math Colloquium

### Speaker: Sylvain Arguillère

Title: Constrained Shape Analysis Through Flows of Diffeomorphisms

Location: TG-227 (Tæknigarður, 2nd floor)

Time: Friday, June 24 at 13:20.

### Abstract:

The general purpose of shape analysis is to compare different shapes in a way that takes into account their geometric properties, such as smoothness, number of self-intersection points, convexity… One way to do this is to find a flow of diffeomorphisms that brings one (template) shape as close as possible to the other (target) shape while minimizing a certain energy. This is the so-called LDDMM method (Large Deformation Diffeomorphic Metric Matching).

Finding this minimizing flow requires solving an optimal control problem that can be seen as looking for (sub-)Riemannian geodesics on the infinite dimensional group of diffeomorphisms with respect to a right-invariant (sub-)Riemannian structure, creating a framework reminiscent of fluid mechanics, and opening the door to some new and exciting infinite dimensional geometries.

In this talk, I will introduce all these concepts, and give the geodesic equations for such structures. Then, I will extend this framework to the case where constraints are added to the shape, in order to better describe the objects they represent, and give some applications in computational anatomy.

Math Colloquium

### Speaker: Ana Carpio, Universidad Complutense de Madrid

Title: Well posedness of a kinetic model for angiogenesis

Location: VR-II, 158.

Time: Thursday, July 9, at 15:00-16:00.

### Abstract:

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling an integrodifferential kinetic equation of Fokker-Planck type with a diffusion equation for the tumor induced angiogenic factor. The chemotactic force field depends on the first velocity moment (flux of blood vessels) through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. Our theory combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity moments and compactness results for this type of kinetic and parabolic operators.

Math Colloquium

### Speaker: Luis L. Bonilla, Universidad Carlos III de Madrid

Title: Hybrid modeling of tumor induced angiogenesis

Location: VR-II, 158.

Time: Thursday, July 9, at 13:30-14:30.

### Abstract:

Angiogenesis is the formation of blood vessels induced by a deficit of oxygen in tissues. It is a basic process in wound repair, formation of vessel networks during growth. diseases of inflammatory character, cancer, etc. When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying elds. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic cell dynamics at their natural microscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. We set up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derive a mean eld approximation for their densities. This leads to a deterministic integro-partial differential system that describes the formation of the stochastic vessel network. We discuss the proper capillary injecting boundary conditions and the results of relevant numerical simulations.