# Seminars in 2014

### Towards a GPU/CPU FPM tool on shallow water equation

Dr. Panchatcharam Mariappan

Fraunhofer ITWM, Germany

Date: Friday, March 7, 2014

Abstract: In this talk, we demonstrate modern software that takes benefits of advanced graphics processing units (GPUs) to accelerate shallow flow simulations when com- pared to the CPU approach. The finite pointset method (FPM), a meshfree particle method is used to create a robust framework suitable for different types of flows. For a real time simulation, the FPM requires millions of particles to get the efficient solution, which consumes more time in CPU. GPU-computed models have been al- ready applied in computational fluid dynamics (CFD) successfully. In this paper, we present an efficient PGI Fortran (CUDA Fortran) implementation of linear solvers, particle management and neighbor searching methods on the FPM. Finally, some numerical tests are presented to show that the efficiency of the GPU dominates the CPU in many cases.

### Model-Resolution Based Deconvolution of Diffuse Optical and Photoacoustic Images

Prof. Phaneendra K. Yalavarthy

SERC, IISc, Bangalore

Date: Wednesday, March 19, 2014

Abstract: Biomedical optical imaging is a promising imaging modality that provides molecular (functional) level information of the soft biological tissues, with prime imaging applications including breast and brain tissue vasculature invivo This uses near infrared light (600 nm - 900 nm) as the probing media, giving an advantage of being non-ionizing imaging modality, with diffuse optical imaging and photoacoustic imaging being the strong contenders.

The image reconstruction (inverse) problem in these scenarios is highly ill-posed due to limited data availability and the estimation problem often requires regularization to bound the solution space. Usage of regularization, especially L2-norm based (Tikhonov type), in these scenarios, converts this inverse problem from exact to approximate, leading to blurry (blobby) images. To over come this, a model resolution matrix based framework was proposed, which can provide a model for the induced blur due to regularization. This model-resolution matrix framework was later utilized for deconvolution, where a basis pursuit deconvolution based on Split augmented Lagrangian shrinkage algorithm (SALSA) algorithm was used along with the Tikhonov regularization step making the image reconstruction into a two-step procedure. This new two-step approach was found to be robust with noise and was able to better delineate the structures which was evaluated using numerical and gelatin phantom experiments. For large-scale image reconstruction problems, like the one encountered in photoacoustic imaging, this deconvolution including the building of model-resolution matrix is achieved via the Lanczos bidagonalization (least squares QR) making this approach computationally efficient and deployable in real-time. The same is also evaluated using numerical experiments.

### Interfacial elastic waves and frictional slip instabilities

Prof. Ranjith Kunnath

SRM Research Institute

Date: Thursday-Friday, April 24-25, 2014

Abstract: Elasticity theory permits several interfacial wave solutions. In anti-plane elasticity, the Love wave occurs in bonded contact of an elastic layer on a dissimilar elastic half-space. In in-plane elasticity, the slip wave (also called the generalized Rayleigh wave) exists at a freely slipping interface between dissimilar elastic half-spaces, and the Stoneley wave occurs in bonded contact of dissimilar elastic half-spaces. It is shown that these interfacial elastic waves are often destabilized when frictional slipping occurs. Simple friction models such as the Coulomb law, as well as experimentally motivated rate- and state-dependent friction laws will be considered. With the Coulomb law, the stability problem for in-plane sliding of dissimilar elastic half-spaces is mathematically ill-posed for arbitrarily small friction due to destabilization of the slip wave at short wavelengths. Regularization procedures in such cases will be discussed. Long wavelength Love and Stoneley waves are also shown to be destabilized in slow sliding, indicating that the quasi-static limit is not attained in slow sliding when interfacial waves in bonded contact exist.

### Stochastic Navier-Stokes Equations in Unbounded Channel Domains

Manil Mohan

Senior Research Fellow (SRF), School of Mathematics, IISER, Thiruvananthapuram

Date: August 21, 2014

Abstract: In this talk, I will first describe the difficulty in using the standard L^2-energy methods for the solvability of the stochastic Navier-Stokes equations with non-zero, finite, random flux in unbounded channel domains with outlets having constant width. By considering an “admissible” channel domain, we will construct a unique basic vector field having the same flux as that of the original problem. A perturbed vector field with zero net flux is constructed using a suitable transformation involving the constructed basic vector field. Then a unique pathwise strong solution to this perturbed field will be proved by exploiting local monotonicity arguments. The perturbed pressure will be characterized using a generalization of the de Rham’s Theorem to processes.

### On the controllability of non-homogeneous viscous Navier-Stokes Equations

Prof. Sylvain Ervedoza

Institut de Mathématiques de Toulouse and CNRS, France

Date: September 4, 2014

Abstract: In this talk, I will report on some recent work done with M. Badra and S. Guerrero done on the local exact controllability trajectories for non-homogeneous Navier-Stokes equations. This question addresses the possibility to exactly reach a target trajectory solution of the system in some time $T>0$ when the errors in the initial data are small. In order to modify the state, we allow boundary controls, possibly from one part of the boundary only corresponding, roughly speaking, to the part in which the fluid enters the domain. The difficulty of our result mainly comes from the coupling of the transport equation satisfied by the density and of the Stokes equation satisfied by the fluid. This requires in particular to develop new Carleman estimates with weight functions following the flow of the target trajectory. Besides, our result will require a geometric condition on the control set and the time of controllability, namely that the flow transports all the particles outside the domain in time T. This condition is of course remanent from the transport equation satisfied by the density.

### A kinetic model for mixtures with application to plasma flow

Prof. Christian Klingenberg

Univ. of Wuerzburg

Date: September 16, 2014

Abstract: We consider a multi component gas mixture without chemical reactions. This mixture is modeled by a system of kinetic BGK equations featuring multiple interaction terms on the right hand side. It is motivated by physical consideration on the collision frequencies and the fluxes of momentum and energy. We prove consistency of our model: conservation properties, H-theorem and convergence to a global equilibrium in shape of a Maxwell distribution. By taking the special case of ions and electrons, we derive the macroscopic equations of ideal MHD from our model. We shall give an outlook on efficient numerical techniques for this model. This is joint work with Marlies Pirner and Gabriella Puppo.

### The Suliciu-Relaxation and its application to some balance laws

Markus Zenk

Univ. of Wuerzburg

Date: October 9, 2014

Abstract: This talk considers the approximation of some conservation laws and balance laws. To this end, first a class of approximate Riemann solvers is introduced, namely the relaxation solvers and specifically the Suliciu relaxation. Properties of this numerical technique such as positivity preserving and entropy stability will be shown. The second part considers the application of this technique to balance laws. Of prior importance are the equilibria of these systems. When dealing with simulations near a equilibrium of a system, spurious oscilations can occur if the source term is not properly taken into account. This leads to the notion of well-balanced schemes. It will be shown, how the relaxation framework is helpfull in this manner.