Joint Fudan - RICAM Seminar on Inverse Problems
This talk is about the heuristic (or data-driven) choice of the regularization
parameter in the regularization theory of ill-posed problems. Here, heuristic
means that the parameter is chosen independent of the knowledge of the noise level
(or any other supplementary information).
Recently, a convergence theory for several heuristic parameter choice methods (for linear regularization) has been developed on basis of the so-called noise-restricted convergence analysis. Withing this framework, one can circumvent the restrictions of the so-called Bakushinskii veto.
We outline the corresponding theory and present theoretical results for the most important examples of heuristic parameter choice rules.
We furthermore discuss some recent results in this direction for convex, nonlinear Tikhonov regularization, together with some open research questions.
Joint Fudan - RICAM Seminar
Wed, Nov 25, 2020 1:00 PM - 2:30 PM (CET)
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Fast, high-accuracy algorithms for acoustic and electromagnetic scattering from axisymmetric objects are of great importance when modeling physical phenomena in optics, materials science (e.g. meta-materials), and many other fields of applied science. In this talk, we develop an FFT-accelerated separation of variables solver that can be used to efficiently invert integral equation formulations of Maxwell's equations for scattering from axisymmetric bodies. The solver is also extended to geometries with non-smooth generating curves and the scattering from large cavities. We then discuss the applications of the solver in the inverse scattering problems. In particular, based on the multi-frequency data and recursive linearization method, we are able to accurately recover the location and shape of the unknown scatterer.
Jun Lai. He received the B.S. degree in mathematics from Nanjing University, China, in 2008, and the Ph.D. degree in applied mathematics from Michigan State University, USA, in 2013. Then he was a post-doctoral fellow in Courant Institute of Mathematical Sciences in New York University and later became a Courant Instructor. In 2016, he joined Zhejiang University and currently he is an assistant professor in the department of mathematical sciences. His main research interests are computational electromagnetics and inverse problems, including scattering and inverse scattering, boundary integral equations and the fast multipole method.
The total variation (of the gradient) is widely applied in the regularization of inverse problems. It is most useful when the true data is expected to be nearly piecewise constant, for example in the recovery of relatively simple images consisting of well-defined objects with limited texture, or identification of physical parameters which are expected to contain inclusions or discontinuities. A basic question for any regularization method is consistency in the low noise regime. For total variation regularization, basic compactness considerations yield convergence in $L^p$ norms, while adding a source condition involving the subgradient at the least-energy exact solution allows for convergence rates in Bregman distance. However, these distances do not provide much information in the setting of nearly piecewise constant functions that motivates the use of the total variation in the first place.
A different, perhaps more adequate choice is convergence of the boundaries of level sets with respect to Hausdorff distance, which can be loosely interpreted as uniform convergence of the objects to be recovered. Such a result requires an adequate choice of (possibly Banach) spaces for the measurements, dual stability estimates to account for the noise, and uniform density estimates for quasi-minimizers of the perimeter. We present some recent results obtaining this type of convergence for regularization of linear inverse problems under the same type of source condition, and for denoising of simple data without source condition.
A Riemannian gradient descent algorithm and a truncated variant will be presented for solving systems of phaseless equations $|Ax|^2 = y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. In addition, we will present a loss function without spurious local minima when the sampling complexity is optimal.
Ke Wei is currently a tenure track professor at School of Data Science, Fudan University. He obtained his DPhil degree in the University of Oxford, followed by three years postdoctoral research in the Hong Kong University of Science and Technology and the University of California at Davis. His reasearch interests are mainly in signal and image processing, mathematical data science and nonconvex optimization.
Two-Point Gradient (TPG) methods are a class of gradient-based iterative regularization methods for solving nonlinear ill-posed problems, and are inspired by Landweber iteration and Nesterov's acceleration scheme. Simple to implement and numerically efficient in practice, these methods have the potential to become useful alternatives to second order iterative schemes. In this talk, we present our initial convergence analysis and numerical experience with TPG methods, and provide a short overview over further works by other researchers which it inspired.
The novel corona virus pneumonia (COVID-19) is a major event in the world. Whether we can establish the mathematical models to describe the characteristics of epidemic spread and evaluate the effectiveness of the control measures we have taken is a question of concern. From January 26, 2020, our team began to conduct research on the modeling of new crown epidemic. A kind of linear nonlocal dynamical system model with time delay is proposed to describe the development of covid-19 epidemic. Based on the public data published by the government, the information of transmission rate, isolation rate and other information, which may not be directly observed in the process of epidemic development is obtained through inversion method, and on the basis of that, a "reasonable" prediction of the development of the epidemic is made. To provide some reasonable data support for government decision-making and various needs of the public.
In this talk, we report some recent results on inverse problems associated with randomness. The first part focuses on the continuous asymptotical regularization on the statistical inverse problems in presence of white noise, where infinite-dimensional stochastic integration shall be treated carefully. The second part considers the convergence analysis on the random projection of discrete inverse problems and we briefly explain how to handle the randomness there.
In this talk I will speak about some recent results on the study of linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We show that regularisation by projection and variational regularisation can be formulated by using the training data only and without making use of the forward operator. I will provide some information regarding convergence and stability of the regularised solutions. Moreover, we show, analytically and numerically, that regularisation by projection is indeed capable of learning linear operators, such as the Radon transform. This is a joint work with Yury Korolev (University of Cambridge) and Otmar Scherzer (University of Vienna and RICAM).