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 Assimilation

Peter 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 Surfaces

Peijun 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 Operators

Katya 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 Regularization

Gitta 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 problem

Takashi 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 Scattering

Armin Lechleiter, University of Bremen, Germany

Abstract: TBA.

Regularized partial and full cloaks of acoustic and electromagnetic waves

Hongyu 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 Methods

Eero 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 geometries

Thomas 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 problems

Xiaoqun 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.

Introducing special speakers

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  • Gitta Kutyniok from TU Berlin is an expert on "sparsity-promoting" reconstruction methods. Inverse problems are about recovering objects based on measurement data which is insufficient. The data needs to be complemented with extra information about the object, such as sparsity. Sparsity means representing the object using building blocks specifically chosen so that only very few of them are needed. Professor Kutyniok often uses "shearlets" for representing images. Shearlets are versatile building blocks adapting to image details of any scale and representing edges with a variety of orientations.

    In the attached picture she applies shear let reconstruction to an inverse scattering problem, resulting in a result much improved over a traditional method. In her plenary talk at the AIP2015 conference, Professor Kutyniok gives an introduction to the theory and computational use of the shearlet transform.

  • Peter Markowich from KAUST is an expert of partial differential equations which arise from systems depending on many variables and involving change. Due to the generality of mathematics, such models apply to wildly different areas of application.

    In his Special Keynote Address, Professor Markowich discusses biological transportation networks, price formation in economic markets and fluid flow in porous matter. The picture shows models for a large crowd of people in three groups exiting a building as fast as possible. Different models of human behaviour lead to different dynamics. This is a joint work with Martin Burger, Marco Di Francesco and Marie-Therese Wolfram.

  • Peijun Li from Purdue University studies direct and inverse scattering problems. One of the central contributions in his work is the design of imaging methods accepting realistic near-field measurements (as opposed to mathematically ideal far-field patterns). In the picture is shown reconstructions of a two-dimensional shape. Here the unknown shape is probed with acoustic waves send from different directions. Various datasets are considered with limited angles of view. Observe that the "dark side" of the shape is more difficult to recover. This work is joint between Peijun Li and Yuliang Wang.

    In his plenary talk at AIP, Peijun Li will describe his recent work on achieving sub-wavelength resolution for inverse surface scattering problems.

  • Hongyu Liu from Hong Kong Baptist University knows how to recover objects from remote measurements. Below is an example of sending elastic vibrations through an unknown body, and recovering inhomogeneities (red) inside. This 2013 result is a joint work between four authors: Guanghui Hu, Jingzhi Li, Hongyu Liu and Hongpeng Sun.

    At AIP, Professor Liu will explain how to hide objects from remote sensing. Such cloaking techniques are already used widely in fiction: think Harry Potter and his invisibility cloak.

  • Xiaoqun Zhang from Shanghai Jiao Tong University is an expert in inverse problems related to image processing. Here is an example of her work (this one done jointly with Tony Chan). On the left is the original "Barbara" image. Second image from left shows many missing pixels that should be filled back in using so-called "inpainting." Third image from left shows the result of a standard baseline technique, whereas the rightmost picture shows the excellent inpainting result using a nonlocal method developed by Zhang & Chan in 2010.

  • Recent work of Thomas Schuster from Saarland University, Germany, (joint with Arne Wöstehoff) paves the way to self-diagnosing airplanes. The idea is to equip the aircraft with vibration sources and sensors. Cracks and other defects can be detected by sending vibrations along the plane, and measuring the response at the sensors.

    Prof. Schuster's plenary talk at AIP will be about vector tomography, which allows new imaging techniques in the fields of medicine, industry, oceanography, plasma physics, polarization tomography and electron microscopy.

  • Katya Krupchyk from University of California at Irvine, USA. Professor Krupchyk is an expert on mathematical models of a range of indirect physical measurements. In one of her works, joint with Matti Lassas and Samuli Siltanen, she studied an extension of the imaging method called electrical impedance tomography.

    In this work, electrical voltage-to-current measurements are preformed on the boundary of a physical body. The resulting currents flowing inside the body produce heat. The surface of the body is covered with heat flow sensors (interlaced with electrodes used for electrical measurements), providing extra information. Now the electrical and thermal measurements can be combined to yield improved information about the internal structure of the body.

  • Takashi Kako from University of Electro-Communications, Chofu-Tokyo, Japan, is an expert on resonances, and he will talk about their role in the formation of vowels in human speech. The related inverse problem is quite tricky: given a recording of a vowel sound, recover the shape of the vocal tract and the excitation signal arising from the vocal folds flapping against each other.

    Pictured are simplified vocal tract models for the five Japanese vowels: /a/, /i/, /u/, /e/, /o/.

  • Eero Saksman, University of Helsinki: Adaptive Markov chain Monte Carlo (MCMC) methods (joint with Johanna Tamminen and Heikki Haario). In Bayesian inversion, one often needs to compute high dimensional integrals (posterior mean). Due to the "curse of dimensionality" it is not a good idea to use a quadrature method.

    Instead, MCMC shoots plenty of points in the space, distributed according to the posterior probability. The average of the points is close to the integral. Now if the posterior probability has a weird shape, regular MCMC may not visit all corners of positive probability. Adaptive MCMC monitors the chain and modifies the search strategy on the fly, guiding the process to all relevant areas.