Joint Fudan - RICAM Seminar on Inverse Problems

Wednesday, June 23, 2021, 2:00 p.m. (Austrian time) / 8:00 p.m. (Shanghai time)

Stochastic Primal dual splitting methods for convex and nonconvex optimization in imaging and data sciences

Xiaoqun Zhang (Shanghai Jiaotong University)

Primal dual splitting algorithms are largely adopted for composited optimization problems arising in imaging. In this talk, I will present stochastic extensions of some popular composite optimization algorithms in both convex and nonconvex settings and their applications to image restoration. The first class of algorithms is designed for convex linearly composite problems by combining stochastic gradient with the so-called primal dual fixed-point method (PDFP). As a natural generalization of proximal stochastic gradient types methods, we proposed stochastic PDFP (SPDFP) and its variance reduced version SVRG-PDFP which do not require subproblem solving. The convergence and convergence rates are established for the proposed algorithms based on some standard assumptions. The numerical examples on graphic Lasso, graphics logistic regressions and image reconstruction are provided to demonstrate the effectiveness of the proposed algorithm. In particular, we observe that for large scale image reconstruction problems, SVRG-PDFP exhibits some advantages in terms of accuracy and computation speed, especially in the case of relatively limited high-performance computing resource. The second class of algorithms is based on Alternating direction method of multipliers (ADMM) for nonconvex composite problems. In particular, we study the ADMM method combined with a class of variance reduction gradient estimators and established the global convergence of the sequence and convergence rate under the assumption of Kurdyka-Lojasiewicz (KL) function. Moreover, we also show that the popular SAGA and SARAH gradient estimators satisfy the variance reduction property. Finally, the efficiency of the algorithms is verified through statistical learning examples and L0 based sparse regularization for 3D image reconstruction.

Fourier reconstruction for diffraction tomography of an object rotated into arbitrary orientations

Eric Setterqvist (RICAM)

In this talk we report on recent work concerning tomographic imaging of objects undergoing irregular motion, the application in mind being imaging of microscopic biological particles that are moved using optical tweezers. As the wavelength of the imaging beam typically is comparable to the object size in such experiments, the wave nature of light should be taken into account meaning that classical projection tomography has limited applicability. Instead diffraction tomography within the Born approximation will be considered. Working in frequency space and assuming plane wave illumination, we introduce a Fourier diffraction theorem relating measurements of the scattered waves to the refractive index distribution of the object. Based on the Fourier diffraction theorem, reconstruction formulae covering a large class of rotations will be presented.

Joint Fudan - RICAM Seminar

Wed, June 23, 2021 2:00 PM - 3:30 PM (CET)

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Wednesday, June 2, 2021

Composite scattering and inverse scattering for the rough surfaces and obstacles

Lei Zhang (School of Mathematical Sciences, Heilongjiang University)

This talk will discuss some recent progress on the mathematical analysis and numerical calculation of the composite scattering and inverse scattering for the rough surfaces and obstacles. Based on the boundary integral equation and variation method, we prove the composite scattering problem's well-posedness. For the inverse problem, we show that any two obstacles and unbounded rough surfaces are identical if they generate the same data. Furthermore, we studied the numerical methods for the direct and inverse problems. We show some results for the inverse problems by RTM. The impact of different parameters on the inversion algorithm is also discussed. Finally, we will introduce our ongoing research work. It is a joint work with Gang Bao (Zhejiang University), Peijun Li（Purdue University, Huayan Liu(Zhejiang University), and Jue Wang (Harbin Engineering University).

A two-step method for the numerical solution of lateral Cauchy problems

Leonidas Mindrinos (Faculty of Mathematics, University of Vienna)

In this work we propose a two-step method for the numerical solution of parabolic and hyperbolic Cauchy problems in two dimensions. The method can be applied in both direct and inverse problems. It is a combination of a semi-discretization with respect to the time variable together with a boundary integral equation method for the spatial variables. The time discretization results to a sequence of elliptic stationary problems. We describe the derived coefficients using a single-layer ansatz for some unknown boundary density functions. We solve the discretized problem on the boundary with the collocation method applying quadrature rules for handling the singularities. Numerical results are presented for the wave, Navier and heat equations. This is a joint work with R. Chapko (Ivan Franko University of Lviv, Ukraine) and B. T. Johansson (Linköping University, Sweden).

Wednesday, May 19, 2021

Fourier based wavefront sensing and its application in Adaptive Optics

Victoria Hutterer (Johannes Kepler University Linz)

Wavefront sensors encode phase information of an incoming wavefront into intensity patterns measured by a camera. We consider Fourier based wavefront sensors which use Fourier filtering with an optical element located in the focal plane. These sensors are used in astronomical adaptive optics to correct for atmospheric turbulence which degrades the quality of observations from ground-based telescopes. They can also be utilised in retinal imaging for medical diagnostics, especially early detection of anomalies and diseases. In ophthalmic AO, the distortions of the laser beam are not caused by turbulence of the air, but mainly by the patient’s eye itself.

In this talk we investigate underlying mathematical models of Fourier based wavefront sensors which are,
e.g., variations of the Hilbert transform. We additionally present wavefront reconstruction algorithms
based on a thorough analysis of the nonlinear models like a singular value type reconstructor or iterative
methods.

As an example of a Fourier based wavefront sensor we particularly study the pyramid wavefront sensor
which is the baseline for future instruments of extremely large telescopes.

Numerical Algorithms for Inverse Spectral Problems Based on Trace Formulas

Xiang Xu (Zhejiang University)

In this talk, we will discuss some recent progress on numerical algorithms for inverse spectral problems for the Sturm-Liouville, Euler-Bernoulli and damped wave operator. Instead of inverting the map from spectral data to unknown coefficients directly, we propose a novel method to reconstruct the coefficients based on inverting a sequence of trace formulas which bridge the spectral and geometry information clearly in terms of a series of nonlinear Fredholm integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed. This is a joint work with Gang Bao (Zhejiang U) and Jian Zhai (HKUST).

Wednesday, April 21, 2021

Boundedness and unboundedness in total variation regularization

Gwenael Mercier (University of Vienna)

We consider whether minimizers for total variation regularization of linear inverse problems belong to $L^\infty$ even if the measured data does not. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for small enough noise under a source condition and adequate a priori parameter choices. To show that such a result cannot be expected for every fidelity term and dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving the equivalence of weighted one-dimensional denoising with a generalized taut string problem. This is a joint work with K. Bredies (Graz) and J.A. Iglesias (RICAM, Linz).

Inverse Problems of the Yang-Mills Equation

Xi Chen (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge)

The Yang-Mills equation addresses the fields of the electroweak and strong interactions of gauge bosons. We shall discuss the uniqueness of the recovery of the fields of gauge bosons by doing local measurements. In terms of differential geometry, the fields of gauge bosons are modelled by the connections on a principal G-bundle, the parallel transport of the Yang-Mills connection is viewed as a broken light ray transform, the detection of gauge bosons amounts to the reconstruction of the connections via the light ray transform. This is a joint work with M. Lassas (Helsinki), L. Oksanen (Helsinki) and G. Paternain (Cambridge).

Wednesday, April 7, 2021

Two-point gradient methods for nonlinear inverse problems based on Bregman projections

Min Zhong (School of Mathematical Sciences, Southeast University)

We propose and analyze a projected two-point gradient method for solving nonlinear inverse problems. The approach is based on the Bregman projection onto stripes the width of which is controlled by both the noise level and the structure of the operator, and the two-point gradient method is efficient for acceleration. The method allows to use L1−liked penalty terms, which is significant in sparsity reconstructions. We present a proof for the regularizing properties of the method, some parameter identification examples are presented to illustrate the effectiveness of the proposed method. It is a joint work with Wei Wang (Jiaxing University).

Localization of individual fixed dipole emitters in Cryo Superresolution Microscopy

Fabian Hinterer (Johannes Kepler University Linz)

Single molecule localization microscopy has the potential to resolve structural details of biological samples at the nanometer length scale. However, to fully exploit the resolution it is crucial to account for the anisotropic emission characteristics of fluorescence dipole emitters. In case of slight residual defocus, localization estimates may well be biased by tens of nanometers. We show that astigmatic imaging in combination with information about the dipole orientation allows to extract the position of the dipole emitters without localization bias and down to a precision of 1nm, thereby reaching the corresponding Cramér Rao bound. The approach is showcased with simulated data forvarious dipole orientations, and parameter settings realistic for real life experiments.

Wednesday, January 20, 2021

Aggregation by the Linear Functional Strategy in Regularized Domain Adaptation

Sergei Pereverzyev (RICAM)

In this talk we are going to discuss the problem of hyperparameters tuning in the context of learning from different domains known also as domain adaptation. The domain adaptation scenario arises when one studies two input-output relationships governed by probabilistic laws with respect to different probability measures, and uses the data drawn from one of them to minimize the expected prediction risk over the other measure.

The problem of domain adaptation has been tackled by many approaches, and most domain adaptation algorithms depend on the so-called hyperparameters that change the performance of the algorithm and need to be tuned. Usually, algorithm performance variation can be attributed to just a few hyperparameters, such as a regularization parameter in kernel ridge regression, or batch size and number of iterations in stochastic gradient descent training. In spite of its importance, the question of selecting these parameters has not been much studied in the context of domain adaptation. In this talk, we are going to shed light on this issue. In particular, we discuss how a regularization of numerical differentiation problem of estimating the Radon-Nikodym derivative of two measures from their samples can be employed in hyperparameters tuning.

Theoretical results will be illustrated by application to stenosis detection in different types of arteries.

The presentation is based on the research performed within FFG COMET project S3AI in cooperation with Nguyen Duc Hoan (RICAM), Bernhard Moser (SCCH), Sergiy Pereverzyev Jr. (Medical University of Innsbruck), Werner Zellinger (SCCH).

Wednesday, December 9, 2020

An inverse boundary value problem for a nonlinear elastic wave equation

Jian Zhai (IAS, Hong Kong University of Science and Technology)

We consider an inverse boundary value problem for a nonlinear model of elastic waves. We show that all the material parameters appearing in the equation can be uniquely determined from boundary measurements under certain geometric conditions. The proof is based on the construction of Gaussian beam solutions.

Dr. Jian Zhai is now a postdoc fellow at the Institute for Advanced Study, the Hong Kong University of Science and Technology. He received his PhD in Computational and Applied Mathematics from Rice University in 2018. His research interests include Inverse Problems, Partial Differential Equations, Microlocal Analysis and Scientific Computing.

Wednesday, November 25, 2020

How to choose alpha without delta

Stefan Kindermann (Johannes Kepler University Linz)

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.

Wednesday, November 11, 2020

A fast solver for scattering problems from axis-symmetric bodies and its applications in inverse problems

Jun Lai (School of Mathematical Sciences, Zhejiang University)

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.

Wednesday, October 28, 2020

Geometric convergence in total variation regularization

Jose Iglesias (RICAM)

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.

Wednesday, October 14, 2020

Solving systems of quadratic equations: Algorithms and global geometry

Ke Wei (School of Data Science, Fudan University)

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.

Wednesday, July 8, 2020

Two-Point Gradient Methods for Nonlinear Ill-Posed Problems

Simon Hubmer (RICAM)

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.

Wednesday, June 24, 2020

A linear time delay model for outbreak of COVID-19 and parameter identification

Jin Cheng (Fudan University)

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.

Wednesday, June 10, 2020

When randomness meets inverse problems

Shuai Lu (Fudan University)

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.

Data driven regularization by projection

Andrea Aspri (RICAM)

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).