|Dr. Giovanni S. Alberti
Department of Mathematics, University of Genoa, Italy
Working Areas: Inverse problems for partial differential equations, applied harmonic analysis, mathematical imaging, hybrid imaging inverse problems, Maxwell's equations, compressed sensing.
Title of the talk: Mathematical analysis of ultrafast ultrasound imaging
Abstract: Conventional ultrasound imaging is performed with focused ultrasonic waves. This yields relatively good spatial resolution, but clearly limits the acquisition time, since the entire specimen has to be scanned. Over the last decade, ultrafast imaging in biomedical ultrasound has been developed. Plane waves are used instead of focused waves, thereby limiting the resolution but increasing the frame rate considerably, up to 20,000 frames per second. The advantages given by the very high frame rate are many, and the applications of this new modality range from blood flow imaging, deep superresolution vascular imaging and functional imaging of the brain to ultrasound elastography.
In this talk, I will discuss a few mathematical aspects related to the analysis of ultrafast ultrasound imaging. In particular, I will describe how to successfully perform blood flow imaging by using an SVD decomposition of the data and present the route towards super-resolution.
|Prof. Dr. Laurent Bourgeois
Laboratoire POEMS, Paris, France
Working Areas: Method of quasi-reversibility to solve ill-posed Cauchy problems and application to inverse problems, inverse problems related to mechanics of materials, inverse scattering in waveguides with sampling methods, inverse scattering for generalized impedance boundary conditions, uniqueness and stability of ill-posed problems, in particular with Carleman estimates.
Title of the talk: On mixed formulations of quasi-reversibility and their application to inverse obstacle problems
Abstract: The method of quasi-reversibility, introduced 50 years ago by Lattès and Lions, is a simple and non iterative regularization technique for solving linear ill-posed PDE problems. Since such regularized problems are directly in the form of a weak formulation, they may be directly discretized with the help of a Finite Element Method. Recently, we have introduced a refinement of the method which consists in some mixed formulations of quasi-reversibility. Contrary to the original formulations, which require high-order finite elements, they enable us to use some classical ones.
In the first part of this talk, after presenting a general abstract framework for these mixed formulations, we will show how they are related to the classical Tikhonov regularization and why they are applicable to a large amount of ill-posed PDE problems. In the second part of the talk, we will show that these mixed formulations of quasi-reversibility can be combined with a simple level set method in order to solve some geometric inverse problems. More precisely, this "exterior approach" can be used to find some obstacles within a reference background medium from a single pair of Cauchy data (lateral Cauchy data for time evolution problems) on a subpart of the boundary. In particular, we will show some numerical experiments for the inverse obstacle problems in the case of the heat equation or the wave equation, either in the frequency or the time domain.
|Prof. Dr. Thorsten Hohage
University of Göttingen, Germany
Working Areas:Inverse problems in partial differential equations; regularization theory; statistical inverse problems; efficient algorithms; applications in helioseismology, phase retrieval problems in optics, and Magnetic Resonance Imaging
Title of the talk: Stability estimates and variational source conditions
Abstract: The total reconstruction error of regularization methods for inverse problems can usually be bounded in terms of an approximation error and a propagated data error. To bound the approximation error, typically source conditions are imposed, but unfortunately these conditions can often not be interpreted for many interesting inverse problems in partial differential equations.
In my talk I will report on some recent progress on using techniques from conditional stability estimates to verify source conditions for nonlinear inverse problems in the form of variational inequalities. For linear inverse problem this approach yields not only sufficient, but even necessary conditions for rates of convergence, which can often be characterized in terms of Besov spaces. Moreover, I will discuss sufficient conditions for rates of convergence for inverse scattering problems and parameter identification problem in linear and semilinear differential equations with distributed measurements.
|Prof. Dr. Thomas Schuster
University of Saarland, Germany
Working Areas: Inverse Problems (Vector- and tensor tomography, Parameter identification for elastic wave equation, Damage detection in fiber composites, Regularization techniques in Banach spaces), Numerical Analysis, Modeling in Magnetic Particle Imaging, Numerical methods for partial differential equations.
Title of the talk: Different views onto solving the nonlinear problem of terahertz tomography
Abstract: The inverse problem of terahertz tomography consists of reconstructing the complex refractive index of a specimen from reflection and transmission of electromagnetic waves that have a frequency of 0.1-100 THz. Terahertz waves are very sensitive with respect to humidity and appropriate for non-destructive testing of ceramics and plastics. The mathematical model is a nonlinear parameter identification problem for the Helmholtz equation. Physically terahertz waves are described as Gaussian beams which solve a paraxial approximation of the Helmholtz equation.
In the talk we outline two different solution approaches to this inverse problem. One tackles the parameter identification problem for the Helmholtz equation by iterative solvers as Landweber's method and sequential subspace optimization techniques. The other one relies on a modified algebraic reconstruction technique and takes reflection losses and refraction by Fresnel's equation and Snell's law into account. We demonstrate pros and cons for both approaches and show several numerical experiments.
|Prof. Dr. Masahiro Yamamoto
Graduate School of Mathematical Sciences, The University of Tokyo, Japan
Working Areas:Inverse problems for partial differential equations, fractional partial differential equations, industrial mathematics
Title of the talk: Mathematical analysis of inverse problems for coupling systems in fluid, viscoelastic dynamics
Abstract: We consider various coupling systems including the linear viscoelasticity, the Kelvin-Voigt model, compressible viscous fluid equations. The principal parts are not only coupled but also these systems are characterized by hyperbolic-integral isotropic Lame equations and hyperbolic-parabolic coupling, etc. By these difficulties, there are few results concerning inverse problems although these systems are important physically.
We establish the Lipschitz stability for inverse problems of determining spatially varying coefficients and/or factors in external force terms by a finite number of (often a single) maeasurments on suitable lateral subboundary, and our argument is a modificati0n of the Bukhgeim-Klibanov method which is based on the Carleman estimate. Moreover we can present also energy estimates called observability inequalities yielding the exact controllability.
For such inverse problems for hyperbolic-parabolic systems, only Hölder stability is known, while for viscoelasticity systems, most of the existing results require measurements on the whole lateral boundary. Naturally we should expect that such existing results can be improved to obtain the Lipschitz stability by data on suitable lateral subboundary.
In my talk, for proving such Lipschitz stability, we establish novel Carleman estimates for the viscoelasticity and new energy estimates for hyperbolic-parabolic systems.