Special Keynote Address:Peter Markowich, King Abdullah University of Science and Technology, Saudi Arabia
Calderon Prize Lecture:Peijun Li, Purdue University, USA
Plenary speakers:Takashi Kako, University of Electro-Communications, Tokyo, Japan
Katya Krupchyk, University of California - Irvine, USA
Gitta Kutyniok, TU Berlin, Germany
Armin Lechleiter, University of Bremen, Germany
Hongyu Liu, Hong Kong Baptist University
Eero Saksman, University of Helsinki, Finland
Thomas Schuster, University of Saarland, Germany
Xiaoqun Zhang, Shanghai Jiao Tong University, China
Three Forward PDE Problems with Urgent Need of Data AssimilationPeter Markowich, King Abdullah University of Science and Technology, Saudi Arabia
Abstract: I discuss three very different forward PDE problems which need data assimilation/inverse approaches to make them potentially useful in practical applications. The first problem is a reaction-diffusion system for biological transportation network formation and adaptation, the second is a highly nonstandard parabolic free boundary problem describing price formation in economic markets and the third problem is the incompressible Navier-Stokes-Forchheimer-Brinkmann system for flow in porous media.
Near-Field Imaging of Rough SurfacesPeijun Li, Purdue University, USA
Abstract: In this talk, our recent progress on a class of inverse surface scattering problems will be discussed. I will present new approaches to achieve subwavelength resolution for these inverse problems. Based on transformed field expansions, the methods convert the problems with complex scattering surfaces into successive sequences of two-point boundary value problems, where explicit reconstruction formulas are made possible. The methods require only a single incident field and are realized by using the fast Fourier transform. The convergence and error estimates of the solutions for the model equations will be addressed. I will also highlight some ongoing projects in rough and random surface imaging.
Spectral Estimates and Inverse Boundary Problems for Elliptic OperatorsKatya Krupchyk, University of California - Irvine, USA
Abstract: We shall discuss some recent progress concerning Lebesgue-space estimates for eigenfunctions and resolvents of elliptic partial differential operators. Applications to inverse boundary problems for elliptic operators with coefficients of low regularity as well as to spectral theory for periodic Schrödinger operators will be presented. This talk is based on joint works with Gunther Uhlmann.
Anisotropic Structures and RegularizationGitta Kutyniok, TU Berlin, Germany
Abstract: Many important problem classes are governed by anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic features for regularization of inverse problems is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations.
In this talk, we will provide an introduction to the anisotropic representation system of shearlets, in particular, compactly supported shearlets, and present the main theoretical results. We will then analyze the effectiveness of using shearlets for regularization of exemplary inverse problems such as feature extraction and recovery of missing data both theoretically and numerically.
Resonance and shape design/identification problemTakashi Kako, University of Electro-Communications, Chofu-Tokyo, Japan
Abstract: In several problems related to wave propagation, resonance phenomena are very important to characterize and to study the problems. In this lecture, we treat among others the vocal tract shape design/identification problem in voice generation and the structure-structure interaction problem through soil foundation via seismic elastic wave. The mathematical formulation of these phenomena is based on scattering theory and generalized eigen-functions and resonant poles related to frequency response function play essential roles in the investigation of the problems. Some numerical methods are considered together with the optimization procedures to solve design problems.
Inside-Outside Duality in Time-Harmonic Wave ScatteringArmin Lechleiter, University of Bremen, Germany
Regularized partial and full cloaks of acoustic and electromagnetic wavesHongyu Liu, Hong Kong Baptist University
Abstract: This talk concerns the invisibility cloaking via the transformation-optics approach for acoustic and electromagnetic waves. Ideal cloaks make use of singular metamaterials, which poses server difficulties for both theoretical analysis and practical realization. Regularization is naturally introduced to avoid the singular structure, and instead of the ideal cloak, one considers the approximate cloak. The speaker will talk about several general regularized cloaking schemes, which can produce customized cloaking effects with full or limited apertures of detection and observation.
On adaptive Markov Chain Monte Carlo MethodsEero Saksman, University of Helsinki, Finland
MCMC (Markov Chain Monte Carlo) methods are increasingly used in many areas of science as a useful tool for simulation. Adaptive MCMC algorithms try to adapt the parameters of the algorithm to enhance efficiency. They do this on fly, i.e automatically while running the algorithm. The talk gives a review of our theoretical understanding of such algorithms, and further discusses some open problems.
Vector tomography in cone beam and inhomogeneous geometriesThomas Schuster, University of Saarland, Germany
Vector tomography is the inverse problem of computing a 2D or 3D vector field given integrals of the searched field over geodesic curves. The most common setting is the Doppler transform which are line integrals of the vector field and thus is the analogue to the X-ray transform for standard computerized tomography. Vector tomography has a wide variety of applications in such different fields as medicine, industry, oceanography, plasmaphysics, polarization tomography or electron microscopy. In the talk we introduce to this challenging research field and focus on the cone beam geometry for 3D vector fields as well the case of an inhomogeneous medium with a variable refractive index. We present an inversion formula for the cone beam case which has been recently developed in a joint work with Alexander Katsevich and a numerical solution approach to the very demanding and nonlinear problem of reconstructing the refractive index from time-of-flight measurements. Our results are illustrated by some numerical experiments.
Computational methods for sparsity promoting in Inverse problemsXiaoqun Zhang, Shanghai Jiao Tong University, China
Sparse promoting regularization and related computational methods has been one of the dominant topics in inverse problems over the last years. In this talk, I will present several popular models and numerical schemes for promoting sparsity in the context of signal and image processing, from both convex and nonconvex prospective. In particular, operator and variable splitting techniques will be present to design efficient algorithms for solving L1 based convex regularizations models. Numerical schemes and theoretical analysis for nonconvex sparsity promoting models, such as L0 regularization, low rank matrix factorization and regularized nonlinear least square will be discussed and illustrated through various imaging applications.