International Workshop on Inverse Problems and Regularization Theory

Transcription

1 International Workshop on Inverse Problems and Regularization Theory Organized by Professor Jin Cheng (Shanghai) and Professor Bernd Hofmann (Chemnitz) Integrating the Chemnitz Symposium on Inverse Problems on Tour Fudan University, Shanghai, China September 26-29, 2013 Venue: Room 1801, East GuangHua Main Building, School of Mathematical Sciences, Fudan Univerisity, No. 220 Handan Road, Shanghai, China

2 Invited Speakers: Dinh Nho Hao (Hanoi, Vietnam) Markus Hegland (Canberra, Australia) Qinian Jin (Canberra, Australia) Philipp Kügler (Stuttgart, Germany, to be confirmed) Jijun Liu (Nanjing, China) Peter Maaß (Bremen, Germany, to be confirmed) Peter Mathé (Berlin, Germany) Sergei V Pereverzev (Linz, Austria) Robert Plato (Siegen, Germany) Ronny Ramlau (Linz, Austria) William Rundell (College Station, USA) Thomas Schuster (Saarbrücken, Germany) Samuli Siltanen (Helsinki, Finland) Jun Zou (Hong Kong, China, to be confirmed) Co-organized by: School of Mathematical Sciences, Fudan University, Shanghai, China Inverse Problem Group, TU Chemnitz, Germany Shanghai Key Lab of Comtemporary Applied Math, Shanghai, China Nonlinear Mathematical Modeling and Methods Laboratory, Ministry of Education, China Local Organizers: Jin Cheng Wenbin Chen Shuai Lu Sponsored by: 111 Project DFG-NSFC Joint Project Regularization of nonlinear ill-posed problems in Banach spaces and conditional stability Fudan University, Shanghai, China Shanghai Science and Technology Commission

3 Outlines Schedule...2 Detailed Program.5 Titles and Abstracts..9 Related Information...38 List of Participants.40 1

4 International Workshop on Inverse Problems and Regularization Theory Organized by Jin Cheng (Shanghai, China) and Bernd Hofmann (Chemnitz, Germany) September 26-29, 2013 Fudan University (Shanghai) Schedule September 26 (Thursday) Morning Session 09:00 AM 09:20 AM Opening ceremony 09:20 AM 10:10 AM William Rundell (TAMU, USA) 10:10 AM 10:40 AM Hanne Kekkonen (U of Helsinki, Finland) 10:40 AM-10:50 AM Tea Break 10:50 AM-11:40 AM Markus Hegland (ANU, Australia) 11:40 AM- 13:30PM Lunch Break Afternoon Session 13:30 PM 14:20 PM Ronny Ramlau (Linz U, Austria) 14:20 PM 15:10 PM Qinian Jin (ANU, Australia) 15:10 PM 15:30 PM Tea break 15:30 AM -16:00 AM Dana Uhlig (TU Chemnitz, Germany) 16:00 PM 16:30 PM Min Zhong (Fudan U, China) 16:30 PM 17:00 PM Yu Jiang (SHUFE, China) 2

5 September 27 (Friday) Morning Session 09:00 AM-09:50 AM Samuli Siltanen (U of Helsinki, Finland) 09:50 AM-10:20 AM Valeriya Naumova (RICAM, Austria) 10:20 AM 10:40 AM Tea break 10:40 AM 11:30 AM Peter Mathé (WIAS, Germany) 11:30 AM-13:30 PM Lunch Break Afternoon Session 13:30 PM 14:20 PM Dinh Nho Hào (Hanoi, Vietnam) 14:20 PM 15:10 PM Frank Werner (Göttingen U, Germany) 15:10 PM 15:30 PM Tea break 15:30 PM 16:00 PM Yikan Liu (U of Tokyo, Japan) 16:00 PM 16:30 PM Wei Wang (Jiaxing U, China) 16:30 PM 17:00 PM Kui Lin (Fudan U, China) 18:00 PM 21:00 PM Banquet 3

6 September 28th (Saturday) September 29 (Sunday) Morning Session 09:00 AM 09:50 AM Thomas Schuster (Saarland U, Germany) 09:50 AM-10:20 AM Xiang Xu (Zhejiang U, China) 10:20 AM -10:40 AM Tea break 10:40 AM 11:30 AM Jens Flemming (TU Chemnitz, Germany) 11:30 AM-13:30 PM Lunch Break 9:00AM-9:50AM Robert Plato (Siegen U, Germany) 09:50 AM 10:20 AM Kenichi Fujishiro (U of Tokyo, Japan) 10:20 AM -10:40 AM Tea break 10:40 AM 11:30 AM Jijun Liu (Southeast U, China) 11:30 AM 11:40 AM Closure 11:40 AM-13:30 PM Lunch Break Afternoon Session 13:30 PM 14:20 PM Sergei V Pereverzev (RICAM, Austria) 14:20 PM 15:10 PM Stephan W. Anzengruber (TU Chemnitz, Germany) 15:10 PM 15:30 PM Tea break 15:30 PM 16:00 PM Xiliang Lü (Wuhan U, China) 16:00 PM 16:30 PM Felix Lucka (Münster U, Germany) 16:30 PM 17:00 PM Zhiyuan Li (U of Tokyo, Japan) 17:00 PM 17:30 PM Shuai Lu (Fudan U, China) Departure 4

7 Detailed Program Thursday: September 26 th Morning session: Time Session/Activity Chairman 09:00-09:20 am Opening Ceremony 09:20-10:10 am 10:10-10:40 am 10:40-10:50 am Coffee break 10:50-11:40 am 11:40-13:30 pm William Rundell Some new results for inverse eigenvalue problems Hanne Kekkonen Tikhonov regularization in function spaces that are natural to white noise Markus Hegland Dilational interpolation inequalities, their theory and application to deconvolution problems Lunch Break at Fudan University, Danyuan Restaurant 3 rd floor Jin Cheng Afternoon session: Time Session/Activity Chairman Ronny Ramlau 13:30-14:20 pm TBA Bernd Qinian Jin Hofmann 14:20-15:10 pm Inexact Newton-Landweber iteration with non- smooth convex penalty terms 15:10-15:30 pm Coffee break 15:30-16:00 pm 16:00-16:30 pm 16:30-17:00 pm Dana Uhlig Copula density estimation as an ill-posed inverse problem Min Zhong Multiscale analysis for ill-posed problem with support vector approach Yu Jiang Recent new progress on the inverse problem for MRE Markus Hegland 5

8 Friday: September 27 th Morning session: Time Session/Activity Chairman 09:00-09:50 am Samuli Siltanen Regularization for electrical impedance tomography using nonlinear Fourier transform 09:50-10:20 am Valeriya Naumova Meta-Learning: towards flexibility and adaptivity in regularization William 10:20-10:40 am Coffee break Rundell 10:40-11:30 am Peter Mathé Projection schemes for signal detection in statistical inverse problems 11:30-13:30 pm Lunch Break at Fudan University, Danyuan Restaurant 3rd floor Afternoon session: Time Session/Activity Chairman 13:30-14:20 pm Dinh Nho Hào Determination of the ambient temperature and the heat transfer coefficient in transient heat conduction Ronny Frank Werner Ramlau 14:20-15:10 pm Convergence rates for inverse iroblems with impulsive noise 15:10-15:30 pm Coffee break 15:30-16:00 pm 16:00-16:30 pm 16:30-17:00 pm 18:00-21:00 pm Yikan Liu Direct and inverse problems for multiple hyperbolic equations and multi-term time-fractional diffusion equations Wei Wang Multi-parameter Tikhonov regularization with l sparsity constraint Kui Lin Bayesian geometric inverse problems arising in groundwater flow Conference dinner at Gd 365 Restaurant Samuli Siltanen 6

9 Saturday: September 28 th Morning session: Time Session/Activity Chairman 09:00-09:50 am Thomas Schuster TBA Xiang Xu 09:50-10:20 am Mathematical analysis in quantifying mechanical properties of nano materials Dinh Nho 10:20-10:40 am Coffee break Hào Jens Flemming 10:40-11:30 am Convergence rates for l -regularization without sparsity assumption 11:30-13:30 pm Lunch Break at Fudan University, Danyuan Restaurant 3rd floor Afternoon session: Time Session/Activity Chairman 13:30-14:20 pm Sergei V Pereverzev Discretized Tikhonov regularization for Robin boundaries localization Thomas Stephan W. Anzengruber Schuster 14:20-15:10 pm Convergence rates and numerical considerations for Tikhonov regularization of the Radon transform 15:10-15:30 pm Coffee break 15:30-16:00 pm 16:00-16:30 pm 16:30-17:00 pm 17:00-17:30 pm Xiliang Lü An active set algorithm for a class of nonconvex nonsmooth sparsity optimization Felix Lucka Computational and theoretical aspects of L1-type priors in Bayesian inverse problems Zhiyuan Li Initial-boundary value problems for diffusion equation with multiple time-fractional derivatives and applications to some inverse problems. Shuai Lu Parameter identification in non-isothermal nucleation and growth processes Robert Plato 7

10 Sunday: September 29 th Morning session: Time Session/Activity Chairman 09:00-09:50 am Robert Plato Quadrature methods for linear first kind Volterra integral equations with noisy data 09:50-10:20 am Kenichi Fujishiro Non-homogeneous boundary value problems for fractional diffusion equations and their approximate controllability Sergei V 10:20-10:40 am Coffee break Pereverzev 10:40-11:30 am Jijun Liu TBA 11:30-11:40 am Closure 11:40-13:30 pm Lunch Break at Fudan University, Danyuan Restaurant 3rd floor 8

11 Titles and abstracts Thursday: September 26 th William Rundell Department of Mathematics Texas A&M University, USA Some new results for inverse eigenvalue problems If we have a string with clamped endpoints then a measurement of the spectra is insufficient to determine an interior parameter such as the density. We can still solve the problem if we have certain apriori information on the density such as its value on half the interval or if it symmetric. In the case of general. we must give additional conditions such as the energy of the string in each vibrational mode or be able to change the boundary conditions, but we assume these such measurement are prohibited. We show how other spectral measurements can be made that compensate and so obtain both uniqueness and effective numerical reconstructions. We also look at the situation of star graphs which can be considered as connected intervals or edges with matching conditions at the node and unknown potentials along each edge; we provide conditions under which effective reconstruction of the potential coefficients can be obtained. The ill-conditioning of the problems together with regularization steps are discussed. 9

12 Hanne Kekkonen Department of Mathematics and Statistics University of Helsinki, Finland Tikhonov regularization in function spaces that are natural to white noise Let us consider an indirect noisy measurement M of a physical quantity U (1) M AU εδ, δ 0 where the realization u of U is a function on a domain of and ε is normalized Gaussian white noise. The inverse problem is to find U if we are given a realization m of the measurement M. Tikhonov regularization with chosen penalty function gives us an estimate (2) u arg min 1 1 u m Δ 1/2 2 2 Above we use 2 norm even though the realizations of ε are in 2 only with probability zero. On the other hand realizations of white noise are in with probability one when s /2. That is why we will show in this talk what happens when we use Tikhonov regularization for the noise that is a realization of the white noise having realizations in Sobolev space with negative smoothness index. We will also consider the question in which space does the estimate convergence to a correct solution when the noise variance goes to zero and what is the speed of the convergence. This is joint work with Matti Lassas and Samuli Siltanen (University of Helsinki) 10

13 Markus Hegland Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Australia Dilational interpolation inequalities, their theory and application to deconvolution problems I will talk about spectral sharpening which may be used in order to improve accuracy of peak location determination. In particular, some broadening mechanisms will be reviewed, Stokes correction considered and possibilities of radial basis functions mentioned. In the main part I will show some new error bounds for ideal sharpening or deblurring procedures. This is joint research with R.S. Anderssen. 11

14 Ronny Ramlau Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Austria and Industrial Mathematics Institute, Johannes Kepler University Austria TBA TBA 12

15 Qinian Jin Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Australia Inexact Newton-Landweber iteration with non-smooth convex penalty terms By making use of tools from convex analysis, we formulate an inexact Newton-Landweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme. The inner scheme provides increments by applying Landweber iteration with non-smooth uniformly convex penalty terms to local linearized equations. The outer iteration is then terminated by the discrepancy principle. The convergence of the method is shown under standard conditions on the nonlinearity. Finally, we report numerical simulations to test the performance of the method. 13

16 Dana Uhlig Fakultät für Mathematik TU Chemnitz, Germany Copula density estimation as an ill-posed inverse problem The reconstruction of the dependence structure of two or more Random variables ( d 2 ) is a big issue in finance and many other applications. Looking at samples of the random vector, neither the Common distribution nor the copula itself are observable. So the identification of the copula C or the copula density d C cu ( 1, ud ) u u 1 d can be treated as an inverse problem. In the statistical literature usually kernel estimators or penalized maximum likelihood estimators are considered for the non-parametric estimation of the copula density c from given samples of the random vector. Even though the copula C itself is unobservable we can treat the empirical copula as a noisy representation, since it is well known that the empirical copula converges for large samples to the copula. Hence we denote the empirical copula by C, where the noise level converges to zero if sample size goes to infinity. We propose solving the linear integral equation u1 u d Cu (, u) cs (, s) ds ds 1 d 1 d 1 d 0 0 for computing the copula density c. Due to the fact that solving the linear integral equation is an ill-posed inverse problem and that we have only noisy data C instead of C an appropriate regularization is needed. We present a Petrov-Galerkin projection for the numerical computation for solving the linear integral equation and discuss the assembling algorithm of the non-sparce matrices and vectors. Furthermore we analyze the stability of the discretized linear equation and discuss regularization methods. The presented talk is based on a joint work with Roman Unger(TU Chemnitz). 14

17 Min Zhong School of Mathematical Sciences Fudan University, China Multiscale analysis for ill-posed problem with support vector approach Based on the use of compactly supported radial basis functions, we extend in this paper the support vector approach (SVA) to a multiscale scheme for approximating the solution of a moderately ill-posed problem on bounded domains. In order to reduce the error induced by noisy data, regularization technique is performed by using the Vapnik's ϵ -intensive function to replace the standard l loss function. Convergence proof for noise-free data is then derived under an appropriate choice of the Vapnik's cut-off parameter and the regularization parameter. For the case of noisy data we show that a corresponding choice for the Vapnik's cut-off parameter gives the same order of error estimate as both the a posteriori strategy based on discrepancy principle and the noise-free a priori strategy. Numerical examples are constructed to verify the efficiency of the proposed SVA approach and the effectiveness of the parameter choices. 15

18 Yu Jiang Department of Applied Mathematics Shanghai University of Finance and Economics, China Recent new progress on the inverse problem for MRE We try to deal with an inverse problem for recovering the viscoelasticity of a living body from MRE (Magnetic Resonance Elastography) data. Based on a viscoelastic partial differential equation whose solution can approximately simulate MRE data, the inverse problem is transformed to a least square variational problem. This is to search for viscoelastic coefficients of this equation such that the solution to a boundary value problem of this equation fits approximately to MRE data with respect to the least square cost function. We will present some new regularization scheme to recover these unknown coefficients. This is a joint work with Prof. Gen Nakamura in Inha University, Korea. 16

19 Friday: September 27 th Samuli Siltanen Department of Mathematics and Statistics University of Helsinki, Finland Regularization for electrical impedance tomography using nonlinear Fourier transform The aim of electrical impedance tomography (EIT) is to reconstruct the inner structure of an unknown body from voltage-to-current measurements performed at the boundary of the body. EIT has applications in medical imaging, nondestructive testing, underground prospecting and process monitoring. The imaging task of EIT is nonlinear and an ill-posed inverse problem. A non-iterative EIT imaging algorithm is presented, based on the use of a nonlinear Fourier transform. Regularization of the method is provided by nonlinear low-pass filtering, where the cutoff frequency is explicitly determined from the noise amplitude in the measured data. Numerical examples are presented, suggesting that the method can be used for imaging the heart and lungs of a living patient. 17

20 Valeriya Naumova Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Austria Meta-Learning: towards flexibility and adaptivity in regularization (joint work with Sergei V. Pereverzyev, Sivananthan Sampath) Making accurate predictions is a crucial factor in many systems (such as in medical treatment and prevention, geomathematics, social dynamics, financial computations) for cost savings, efficiency, health, safety, and organizational purposes. At the same time, the situation mostly encountered in real-life applications is to have only at disposal incomplete or rough data, and extracting a predictive model from them is an impossible task unless one can rely on some a-priori knowledge of properties of the expected model. Taking our motivation from such an increased need in applications, in this talk we present a novel approach for performing reliable predictions from roughly measured data. The developed approach allows a dynamic adaptation of the unknown parameters in regularization schemes to each particular input. Such adaptivity and flexibility is achieved using the meta-learning concept [1,2] that employ the previous experience to construct selection rules which are intrinsically data-dependent. Finally, we confirm effectiveness and robustness of the proposed approach on the problem from diabetes therapy management with real-life data [3]. The material is patent pending, the patent application [4] has been filed jointly by Austrian Academy of Sciences and Novo Nordisk A/S(Denmark). References: [1]T.A.F.Gomes, R.B.C.Prudencio, C.Soares, A.L.D.Rossi, A.Carvalho, Combining metalearning and search techniques to select parameters for support vector machines, Neurocom puting 75 (2012),3-13. [2]V.Naumova, S.V.Pereverzyev, S.Sampath, A meta-learning approach to the regularized learning- Case study: Blood glucose prediction, Neural Networks 33 (2012), [3]V.Naumova, S.V.Pereverzyev, Blood Glucose predictors: an Overview on how recent developments help to unlock the problem of glucose regulation, Recent Patentson Computer Science 5 (2012),1-11. [4]S.Pereverzyev, S.Sivananthan, J.Randlv, S. McKennoch, Glucose predictor based on regularization networks with adaptively chosen kernels and regularization parameters, World Intellectual Property, WO2012/143505, 26October

21 Peter Mathé Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany Projection schemes for signal detection in statistical inverse problems We discuss how general regularization schemes, in particular projection schemes, can be used to design tests for signal detection in statistical inverse problems. We explain the statistical test framework. Then we show that such test can attain the minimax separation rates when the discretization level is chosen appropriately. It is also shown how to modify these tests in order to obtain (up to a loglog factor) a test which adapts to the unknown smoothness in the alternative. This talk is based on joint work with C. Marteau,Toulouse, General regularization schemes for signal detection in inverse problems,

22 Dinh Nho Hào Hanoi Institute of Mathematics Vietnam Determination of the ambient temperature and the heat transfer coefficient in transient heat conduction The restoration of the space- or time-dependent ambient temperature and of the space- or time-dependent heat transfer coefficient which links the boundary temperature to the heat flux through a third-kind Robin boundary condition in transient heat conduction is investigated. The reconstruction uses average surface temperature measurements. Least-squares penalized variational formulations are proposed and new formulae for the gradients are derived. Numerical results obtained using the conjugate gradient method combined with a boundary element direct solver are presented and discussed. 20

23 Frank Werner Institute for Numerical and Applied Mathematics University of Göttingen, Germany Convergence rates for inverse problems with impulsive noise We study inverse problems F f g where the data g is corrupted by so-called impulsive noise which is concentrated on a small part of the observation domain. Such noise occurs for example in digital image acquisition. To reconstruct f from noisy measurements we use Tikhonov regularization where it is well-known from numerical studies that L data fitting yields much better reconstructions than classical L data fitting. Nevertheless, so far rates of convergence are known only if the L -norm of the noise itself tends to zero, which does not fully explain the remarkable quality of the reconstructions obtained by L data fitting. We introduce a continuous model for impulsive noise depending on an impulsiveness parameter η 0 and prove convergence rates as η tends to zero. We therefore use a recently developed variational formulation of the noise level and derive expressions for it in terms of eta. It turns out that the rates of convergence (compared to the state of the art) clearly improve depending also on the smoothing properties of the forward operator F. Finally we present numerical results which are in good agreement with the prediction of our theory. 21

24 Yikan Liu Graduate School of Mathematical Sciences The University of Tokyo, Japan Direct and inverse problems for multiple hyperbolic equations and multi-term time-fra ctional diffusion equations Numerous industrious applications and treatments of environmental problems hav e witnessed the increasing importance of underlying mathematical models and have b een benefited from solving the corresponding inverse problems. In this talk, we discus s two kinds of evolution equations originated from respective backgrounds and model ings, namely, multiple hyperbolic equations describing structure generation kinetics a nd multi-term time-fractional diffusion equations describing anomalous diffusion phe nomena. First we investigate the multiple hyperbolic equations derived from Cahn's time c one model for nucleation and growth mechanisms. Cahn's model is an integral equatio n, preventing us from smooth arguments for both forward and inverse problems. To th is end, we derive a class of multiple hyperbolic systems from the original model. Than ks to this reduction, dramatically fast forward solver is developed in practical spatial d imensions. Next, on basis of the hyperbolic systems, we study two inverse source pro blems of determining the nucleation rate and one inverse coefficient problem of deter mining the growth rate to establish corresponding uniqueness and stability results. Next we study the multi-term time-fractional diffusion equations, which are natur al extensions to its single-term counterpart. By exploiting several properties of the mu ltinomial Mittag-Leffler functions which appear in the explicit solution, we establish t he Lipschitz stability of the forward problem with respect to the fractional orders of ti me derivatives and the diffusion coefficient. For their simultaneous reconstruction, we propose a minimization approach and show the existence of a minimizer by use of th e Lipschitz stability. 22

25 Wei Wang College of Mathematics Physics and Information Engineering Jiaxing University, China Multi-parameter Tikhonov regularization with l 0 sparsity constraint Using a sparsity promoting penalty term in the Tikhonov regularization scheme, it has been verified to be efficient in reconstructing solutions which own a sparse structure. These penalty terms usually take the form of l (quasi-)norm with p 0,2. At the same time, some examples have been shown that there is no regularizing property for the extreme penalty term with p 0. Here we examine the multi-parameter Tikhonov regularization with both l and l penalty terms for nonlinear ill-posed problems. We will show, in the sequel, that the proposed multi-parameter Tikhonov functional has a minimizer as well as the convergence result when the noise level goes to 0. Error estimates for the discrepancy principle and sequential discrepancy principle are verified under appropriate source conditions and nonlinearity assumptions. Finally, two numerical examples shed light on the appropriateness of the proposed method. 23

26 Kui Lin School of Mathematical Sciences Fudan University, China Bayesian geometric inverse problems arising in groundwater flow In this talk, we consider the inverse problem of determining the permeability of the subsurface from hydraulic head measurements, within the framework of a steady Darcy model of groundwater flow. We study geometrically defined permeability tensors, primarily with piecewise constant scalar form. We adopt a Bayesian framework showing existence and a class of well-posedness of the posterior distribution. We also introduce novel MCMC methods to explore the posterior and illustrate the methodology with some numerical experiments. This is joint work with Marco Iglesias and Andrew Stuart (Warwick). 24

27 Saturday: September 28 th Thomas Schuster Working group for Numerical Analysis and Applied Mathematics Saarland University, Germany TBA TBA 25

28 Xiang Xu Department of Mathematics Zhejiang University, China Mathematical analysis in quantifying mechanical properties of nano materials Nanotechnology deals with structures sized between 1 to 100 nanometer and involves investigating and designing materials or devices within that scale. Due to its small size, it is difficult to quantify the mechanical properties of nanomaterials which significantly rely on measurement techniques, conditions and environment. In this talk, we propose a novel model based on Euler-Bernoulli equation with a stochastic source term accounting for the effect of initial bending, surface roughness and white noise during the measurement. The forward problem is demonstrated to have a unique and explicit path-wise solution. Furthermore, the inverse problem consists of identifying the elastic modulus of nanobelt and reconstructing the random source structure, i.e., the mean and the variance. Based on the explicit formula of the direct problem, the inverse problem can be reduced into a first kind of Fredholm type integral equation. Two kinds of regularization techniques are introduced to obtain stable solutions. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods. 26

29 Jens Flemming Fakultät für Mathematik TU Chemnitz, Germany Convergence rates for -regularization without sparsity assumption Abstract: Sparsity promoting regularization techniques have been one of the most active fields of research in the inverse problems community for the last ten years. Especially -regularization, that is, Tikhonov regularization with -penalty, is well understood now. But all theoretic results obtained so far, in particular error estimates and convergence rates, rely on the so called sparsity assumption. This means that the unknown solution to the underlying ill-posed equation requires only finitely many coefficients with respect to a fixed chosen basis. In the talk we present convergence rates for -regularization which also hold for non-sparse solutions. Such rates can be achieved with the help of a smoothness assumption on the basis with respect to which the -norm is penalized. The results presented in the talk are joint work with Martin Burger (Münster, Germany) and Bernd Hofmann (Chemnitz, Germany). 27

30 Sergei V Pereverzev Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Austria sergei.pereverzyev@oeaw.ac.at Discretized Tikhonov regularization for Robin boundaries localization We deal with a boundary detection problem arising in nondestructive testing of materials. The problem consists in recovering an unknown portion of the boundary, where a Robin condition is satisfied, with the use of a Cauchy data pair collected on the accessible part of the boundary. We combine a linearization argument with a Tikhonov regularization approach for the local reconstruction of the unknown defect. Moreover, we discuss the regularization parameter choice by means of the so-called balancing principle and present some numerical tests that show the efficiency of the method. The presentation is based on a joint research with Hui Cao (Sun Yat-sen University) and Eva Sincich (University of Nova Gorica). 28

31 Stephan W. Anzengruber Fakultät für Mathematik TU Chemnitz, Germany Convergence rates and numerical considerations for Tikhonov regularization of the Radon transform Tikhonov regularization with sparsity promoting l p constraints (0 1) has attracted considerable attention over the last decade. Its regularizing and convergence properties are well understood, but results on convergence rates often depend on the sparsity of the unknown solution and on a structural assumption linking the underlying basis to the range of the adjoint of the operator [2]. In this talk, we first illustrate by means of the Radon transform how the latter assumption is related to the smoothness of the basis, and then present convergence rates results [1] that do not require sparsity of the unknown solution even in the non-convex case p 1. To compute the minimizers numerically we use superposition operators to formulate an equivalent nonlinear minimization problem with p 1 [3] and propose a dual TIGRA method to solve the nonlinear problem [4]. [1] S.W. Anzengruber, B. Hofmann, and R. Ramlau. On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization. (submitted) [2] M. Burger, J. Flemming and B. Hofmann. Convergence rates in l 1 -regularization if the sparsity assumption fails. Inverse problems, 29(2): pp, [3] C.A. Zarzer. On Tikhonov regularization with non-convex sparsity constraints. Inverse Problems, 25:02006, 13pp, [4] W. Wang, S.W. Anzengruber, R. Ramlau and B. Han. A global minimization algorithm for Tikhonov functionals with nonlinear operator and sparsity constraints (in preparation). 29

32 Xiliang Lü School of Mathematics and Statistics Wuhan University, China An active set algorithm for a class of nonconvex nonsmooth sparsity optimization In this talk we consider the problem of recovering a sparse vector from noisy measurement data. Nonconvex penalties, such as the quasi-l -norm penalty with 0 q 1 or the smoothly clipped absolute deviation penalty are often applied in signal and imaging processing, machine learning and statistics. These gives rise to diverse nonsmooth nonconvex sparsity optimization problems. A unified algorithm of primal-dual active set type for a class of nonconvex penalties is developed. First we establish the existence of a finite global minimizer for the class of optimization problems. Then we derive a novel necessary optimality condition for the global minimizer in terms of the thresholding operator associated with the nonconvex penalty. The solutions to the optimality system are necessarily coordinate-wise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined from the primal and dual variables. This relation naturally lends itself to an iterative algorithm of active set type which at each step involves updating the primal variable only on the active set and then updating the dual variable explicitly. Numerical experiments demonstrate the efficiency of the algorithm, and numerically a local superlinear convergence is observed. 30

33 Felix Lucka Institute for Computational and Applied Mathematics University of Münster, German Computational and theoretical aspects of L1-type priors in Bayesian inverse problems Using sparsity constraints in variational regularization techniques has become a key concept for solving of high-dimensional, ill-posed inverse problems. A popular approach is to formulate these sparsity constraints by incorporating L1-type norms. Recently, using similar sparsity-constraints in the Bayesian framework for inverse problems has attracted attention. Using this non-gaussian type or priors has re-vitalized discussions about the relation between regularization theory and Bayesian inference and raised a number of important questions for future research. In this talk, I want to illustrate and summarize our work in this field and point to possible directions of future research. The focus will be on the comparison between maximum a posteriori (MAP) and conditional mean (CM) point estimates under different aspects. This is traditionally the most obvious and, thus, most often discussed question. Based on a fast MCMC sampling algorithm that we recently developed, I will compare MAP and CM estimates for several examples of high-dimensional inverse problems. The results clearly challenge the classical Bayesian view on the relationship between MAP and CM estimates. In the last part of the talk, I will present recent work that examines Bayesian inversion by the use of Bregman distances. The newly developed concepts resolve the discrepancy between computational results and classical theoretical prediction and disprove common beliefs about MAP estimates. First, the MAP estimate is a Bayes estimator with respect to a proper, convex cost function and not only an asymptotic Bayes estimator with respect to the degenerate uniform cost. Second, while the posterior is well centered around the CM estimate with respect to the mean squared error and the CM estimate minimizes the expected variance with respect to the real solution, the posterior is also well centered around the MAP estimate with respect to a convex functional and the MAP estimate minimizes the expected Bregman distance with respect to the real solution. This is joint work with Martin Burger. 31

34 Zhiyuan Li Graduate School of Mathematical Sciences The University of Tokyo, Japan Initial-boundary value problems for diffusion equation with multiple time-fractional derivatives and applications to some inverse problems. Supposing d bounded domain, sufficiently smooth. We consider the following problem For (0,1) 0 : Caputo derivative [1], l 1 1, t. r 0 is a diffusion parameter. In this talk, firstly, we are to consider the forward problem for our initial-boundary value problem. Via the Mittag-Leffler function and the eigenfunction expansion, we apply the general Gronwall's inequality and fixed point method to prove unique existence as well as regularity of solution. Moreover, based on the existence result, the regularity of the solution for the time t can be raised up step by step till the Hölder regularity. For the case of homogeneous equation, the solution can be analytically extended to a sector { ; 0, arg z z z 2 }. In the case when all of the coefficients of the time-fractional derivatives are positive constants, the use of Laplace transform, Watson's lemma and the analyticity leads to some further properties, such as, the decay rate of the solution is shown to be determined by the lowest order of fractional time-derivative, and the initial-boundary value problem under consideration possesses the so-called weak unique continuation property. Next we turn to the discussion of the inverse problems. (i) In the case when the source term is represented as f ( xhxt ) (, ), where h is given. The well-posedness of the inverse source problem by final overdetermining data was proved except for a finite set of r on the basis of the analytic Fredholm theory. (ii) Let the source term be in the form f ( x) ( t), by the unique continuation principle, coupled with the Duhamel's principle, we deduced the uniqueness of determination of source term f by the 32

35 partial Neumann boundary data. (iii) We are required to determine the multiple orders and one coefficient for 1D fractional diffusion equation by boundary data of the solution of the forward problem. [1]. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego,

36 Shuai Lu School of Mathematical Sciences Fudan University, China Parameter identification in non-isothermal nucleation and growth processes We study nonisothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an ener gy balance to account for recalescence effects. The main novelty of our work is the id entification of temperature dependent nucleation rates. We prove that such rates can b e uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation. 34

37 Sunday: September 29 th Robert Plato Department of Mathematics University of Siegen, Germany Quadrature methods for linear first kind Volterra integral equations with noisy data Abstract: The subject of this talk is the regularization of linear Volterra integral equations of the following form,,, 0, (1) Γ with 0 1 and 0, and Γ denotes Euler's gamma function. It is assumed that the kernel function :, 0 y x is sufficiently smooth and satisfies, 1 for 0. The function : 0, is supposed to be approximately given, and a function : 0, satisfying equation (1) has to be determined. Note that in (1), the cases of smooth kernels 1 as well as weakly singular kernels 1 are covered. Quadrature methods for the approximate solution of equation (1) are well studied if the right-hand side is exactly given. In the present talk we consider perturbed given right-hand sides in equation (1). More precisely, we assume that only approximations with for 1, 2,, (2) are available, where n, 1, 2,,, are uniformly distributed nodes, with /, and is a positive integer. In addition, 0 is a given noise level. In this talk we present a survey of recent results about the regularizing properties of quadrature methods for solving (1) with data as in (2). Some numerical results are also presented. Reference: [1] R. Plato. The regularizing properties of linear multistep methods for perturbed first kind Volterra integral equations with smooth kernels. In preparation. [2] R. Plato. The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind. Adv. Comput. Math., 36(2): ,

38 Kenichi Fujishiro Graduate School of Mathematical Sciences The University of Tokyo, Japan Non-homogeneous boundary value problems for fractional diffusion equations and their approximate controllability We consider diffusion equations with time fractional derivatives, which are thought to be a new model describing anomalous diffusion phenomena. Here the fractional derivatives are defined in the Caputo sense. In this talk, we attach the non-homogeneous Dirichlet boundary condition and mainly discuss the regularity of the solution and the approximate controllability. To this end, we also consider the system with Riemann-Liouville fractional derivatives to apply the duality argument. 36

39 Jijun Liu TBA Department of Mathematics Southeast University, China TBA 37

40 Related Information Local Contact: Jin Cheng (GuangHua Eastern Main Buidling, Room 1812) Mobile Phone: Shuai Lu (GuangHua Eastern Main Buidling, Room 1815) Mobile Phone: Wireless connection in Fudan Campus: Please choose the wireless network with the name fduwireless. User ID: ip2013 Password: ip1234 Lunch and Supper: September 25 (Welcome Supper at 18:00 PM) September 26, 28 (Lunch and Supper) September 27, 29 Danyuan Restaurant, 3rd floor A: The conference venue: Room 1801, East Guanghualou B: Danyuan Restaurant C: East gate to Fudan Campus D & Black line: Road Guoding 38

41 Conference Banquet (September 27 Gd 365 Restaurant No. 365, Road Guoding, Shanghai China A: Gd 365 Restaurant B: Fuxuan Hotel C: Fortune Hotel Other Hotels A: The conference venue B: Danyuan Restaurant C: Fuxuan Hotel D: Fortune Hotel E: Crowne Plaza Shanghai Fudan (All directions in 10 minutes by foot) 39

42 List of Participants Name Institute Address Stephan W. Anzengruber Wenbin Chen Jin Cheng Jens Flemming Kenichi Fujishiro Dinh Nho Hào Markus Hegland Bernd Hofmann Yu Jiang Qinian Jin Hanne Kekkonen Zhiyuan Li Kui Lin Fakultät für Mathematik TU Chemnitz, Germany School of Mathematical Sciences Fudan University, China School of Mathematical Sciences Fudan University, China Fakultät für Mathematik TU Chemnitz, Germany Graduate School of Mathematical Sciences The University of Tokyo, Japan Hanoi Institute of Mathematics Vietnam Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Australia Fakultät für Mathematik TU Chemnitz, Germany Department of Applied Mathematics Shanghai University of Finance and Economics, China Centre for Mathematics and its Applications Mathematical Sciences Institute Australian National University, Australia Department of Mathematics and Statistics University of Helsinki, Finland Graduate School of Mathematical Sciences The University of Tokyo, Japan School of Mathematical Sciences Fudan University, China hematik.tu-chemnitz.de k.tu-chemnitz.de au ik.tu-chemnitz.de n hanne.kekkonen@helsinki. fi linkui26@hotmail.com 40

43 Jijun Liu Yikan Liu Shuai Lu Felix Lucka Xiliang Lü Peter Mathé Valeriya Naumova Sergei Pereverzev Robert Plato Ronny Ramlau William Rundell Samuli Siltanen Thomas Schuster Dana Uhlig Wei Wang Department of Mathematics Southeast University, China Graduate School of Mathematical Sciences The University of Tokyo, Japan School of Mathematical Sciences Fudan University, China Institute for Computational and Applied Mathematics University of Münster, Germany School of Mathematics and Statistics Wuhan University, China Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Austria Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Austria Department of Mathematics University of Siegen, Germany Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Austria Department of Mathematics Texas A&M University, USA Department of Mathematics and Statistics University of Helsinki, Finland Working group for Numerical Analysis and Applied Mathematics Saarland University, Germany Fakultät für Mathematik TU Chemnitz, Germany College of mathematics physics and information engineering, Jiaxing University, China de c.at ac.at gen.de -chemnitz.de 41

44 Frank Werner Gunnar Wilken Xiang Xu Min Zhong Institute for Numerical and Applied Mathematics University of Göttingen, Germany Structural Biology Unit Okinawa Institute of Science and Technology Japan Department of Mathematics Zhejiang University, China School of Mathematical Sciences Fudan University, China gen.de n 42

45 About Fudan University Fudan University, located in Shanghai, is one of the oldest and most selective universities in China. Its institutional predecessor was founded in 1905, shortly before the end of China's imperial Qing dynasty. Fudan University is composed of four campuses, including Handan (Main Campus), Fenglin, Zhangjiang, and Jiangwan. Fudan was initially known as Fudan College in The two Chinese characters Fu and Dan, literally meaning "(heavenly light shines) day after day", were chosen by the distinguished educator in modern Chinese history, Father Ma Xiangbo S.J. from the Confucian Classics: "Itinerant as the twilight, sun glows and moon luminesces". In 1911 during the Xinhai Revolution the college was taken up as the headquarters of the Guangfu Army and closed down for almost one year. The university motto comes from Analects Book 19.6, which means "to learn extensively and adhere to aspirations, to inquire earnestly and reflect with self application". Fudan University comprises 17 full-time schools, 69 departments, 73 bachelor's degree programs, 22 disciplines and 134 sub-disciplines authorized to confer Ph.D. degrees, 201 master degree programs, 6 professional degree programs, 7 key social science research centers of Ministry of Education P.R.C, 9 national basic science research and training institutes and 25 post-doctoral research stations. It has 40 national key disciplines granted by the Ministry of Education, nationally third. At present, it has 77 research institutes, 112 cross-disciplinary research institutes and 5 national key laboratories. Fudan University enrolls over 45,000, including full-time students and students in continuing education and online education. Additionally, there are nearly 1,760 students from overseas, ranking second nationally.