plenary and mediav lectures - math at...

Plenary and MediaV Lectures

PL. Plenary and MediaV Lectures Room: ECC B4 Theater

6/27(Mon)10:30–11:20

Chair: Benny HonSpectral theory of the Neumann-Poincare operator and applicationsHyeonbae Kang, Inha University, Department of Mathematics [PL-01, p.6]

6/27(Mon)11:30–12:20

Chair: Benny HonMulti-frequency inverse acoustic source problemsShuai Lu, School of Mathematical Sciences, Fudan University, China [PL-02, p.6]

6/27(Mon)2:00–2:50

Chair: Sungwhan KimSparse Image Modelling and Blind De-convolutionHui Ji, Department of Mathematics, National University of Singapore [PL-03, p.6]

6/28(Tue)9:00–9:50

Chair: Xudong ChenSome inverse problems and numerical inversions for the fractional diffusion equationsGongsheng Li, Shandong University of Technology [PL-04, p.7]

6/28(Tue)2:00–2:50

Chair: Jin Cheng

[MediaV Award 1] TBATBA [PL-05, p.7]

6/29(Wed)9:00–9:50

Chair: Masahiro YamamotoPhase retrieval with one or two diffraction patterns by alternating projections of thenull vectorPengwen Chen, Applied Math., National Chung Hsing University [PL-06, p.7]

6/29(Wed)10:30–11:20

Chair: Jijun LiuCT Metal Artifacts and Reduction AlgorithmsChang-Ock Lee, Mathematical Sciences, KAIST [PL-07, p.8]

6/30(Thur)9:00–9:50

Chair: Hyundae LeeNumerical Differentiation by Kernel-based Probability MeasuresLeevan Ling, Mathematics, Hong Kong Baptist University [PL-08, p.8]

6/30(Thur)2:00–2:50

Chair: Jenn-Nan Wang

[MediaV Award 2] TBATBA [PL-09, p.8]

7/1(Fri)9:00–9:50

Chair: Mikyung LimInverse source problems for hyperbolic-type equationsYikan Liu, Graduate School of Mathematical Sciences, The University of Tokyo [PL-10, p.9]

Program 1

Minisymposium and Contributed Talks

MS01. Reconstruction schemes for evolution equationsJune 27(Mon) Room: ECC B155

3:30-5:30 Chair: Gen Nakamura and Haibing Wang

3:30–3:55 Stability analysis for inverse coefficient problems in the diffusion equation using the adjointmethodGongsheng LI, Shandong University of Technology [MS01-01, p.10]

4:00–4:25 Algebraic reconstruction scheme of moving dipole wave sources from boundary measure-mentsTakashi Ohe, Department of Applied Mathematics, Faculty of Science, Okayama University of Sci-ence [MS01-02, p.10]

4:30–4:55 Semigroup-theoretic approach to identification of linear diffusion coefficientsNoboru Okazawa, Tokyo University of Science [MS01-03, p.11]

5:00–5:25 Recovery of thermal conductivity in two-dimensional media with nonlinear source by opti-mizationsBingxian Wang, Department of Mathematics, Southeast University,China [MS01-04, p.11]

CP02. Contributed Talks 1/2June 27(Mon) Room: ECC B157

3:30–5:30 Chair: Kiwan Jeon

3:30–3:55 Numerically Solving Inverse Scattering Problems using Sparsity Regularization and TotalVariationFlorian Buergel, Center for Industrial Mathematics, University of Bremen [CP02-01, p.12]

4:00–4:25 Reconstruction of the shear modulus of viscoelastic systems in a thin cylinder: an inversionscheme and experimentsJunyong Eom, Mathematics, Inha University [CP02-02, p.12]

4:30–4:55 Non-linear Tikhonov Regularization in Banach Spaces for Inverse Scattering fromAnisotropic Penetrable MediaMarcel Rennoch, University of Bremen, Center for Industrial Mathematics [CP02-03, p.12]

5:00–5:25 On the notion of elastic scattering coefficientsAbdul Wahab, BISPL, Department of Bio & Brain Engineering, Korea Advanced Institute of Science &Technology, Daejeon [CP02-04, p.13]

0:00–0:00 On a geometrical inverse problem and its applications to scattering theory and cloakingHongyu LIU, Department of Mathematics, Hong Kong Baptist University [CP02-05, p.14]

MS03. Qualitative and quantitative Inverse Scattering algorithms 1/3June 28(Tue) Room: ECC B155

10:30-12:30 Chair: Xudong Chen and Hongyu Liu

10:30–10:55 A Multilevel Sampling Method for Detecting Sources in a Stratified Ocean WaveguideKeji Liu, Shanghai University of Finance and Economics [MS03-01, p.14]

11:00–11:25 Inverse Random Source Scattering ProblemsPeijun Li, Department of Mathematics, Purdue University [MS03-02, p.14]

11:30–11:55 The direct and inverse scattering problems with oblique derivative boundary conditionHabing Wang, Dept of Mathematics, Southeast University, Nanjing [MS03-03, p.15]

2 The 8th International Conference on Inverse Problems and Related Topics

MS06. June 28(Tue)

12:00–12:25 MUSIC algorithm for imaging perfectly conducting crack in limited-view inverse scatteringproblemWon-Kwang Park, Department of Mathematics, Kookmin University [MS03-04, p.15]

MS04. Industrial and medical applications of electrical impedance tomography 1/2June 28(Tue) Room: ECC B157

10:30-12:30 Chair: Hyeuknam Kwon

10:30–10:55 PCA-based Characterization in Lung Electrical Impedance TomographyLiangdong Zhou, Department of Computational Science & Engineering, Yonsei University [MS04-01, p.15]

11:00–11:25 A robust EIT reconstruction for lung imaging based on sensitivity-data correlationKyounghun Lee, Dept of Computational Sci and Eng, Yonsei University [MS04-02, p.16]

11:30–11:55 3D EIT model construction and image reconstructionXiaoyan Chen, Tianjin University of Science and Technology, Tianjin, China [MS04-03, p.16]

12:00–12:25 Identification of the boundary heat transfer coefficient from interior measurement of tem-perature fieldLiyan Wang, Department of Mathematics, Southeast University,China [MS04-04, p.16]

MS05. Inverse problems for fractional partial differential equationsJune 28(Tue) Room: ECC B155

3:30-5:30 Chair: Manabu Machida

3:30–3:55 Unique continuation property for anomalous diffusion equationGen Nakamura, Department of Mathematics, Hokkaido University [MS05-01, p.17]

4:00–4:25 An inverse problem for distributed order time-fractional diffusion equationZhiyuan Li, The University of Tokyo [MS05-02, p.17]

4:30–4:55 Determination of the temporal component in the source term of a fractional diffusion equa-tionYikan LIU, Graduate School of Mathematical Sciences, The University of Tokyo [MS05-03, p.18]

5:00–5:25 Half-order fractional inverse transport problems by Carleman estimatesManabu Machida, Institute for Medical Photonics Research, Hamamatsu University School ofMedicine [MS05-04, p.18]

MS06. Image and Image ProcessingJune 28(Tue) Room: ECC B157

3:30–6:00 Chair: Hyoung Suk Park and Jae Kyu Choi

3:30–3:55 Limited Tomography Reconstruction via Tight Frame and Simultaneous Sinogram Extrap-olationJae Kyu Choi, Institute of Natural Sciences, Shanghai Jiao Tong University [MS06-01, p.19]

4:00–4:25 Various methods for ill-posed inversion in Quantitative Susceptibility MappingShuai Wang, School of Electronic Engineering, University of Electronic Science and Technology ofChina [MS06-02, p.19]

4:30–4:55 Intensity Nonuniformity Correction Method in brain MR imagesYunho Kim, Dept. of Mathematical Sciences, UNIST [MS06-03, p.19]

5:00–5:25 The optimal methods for inversion of the noisy k-plane transformTigran Bagramyan, Samsung Electronics [MS06-04, p.20]

Program 3

MS06. June 28(Tue)

5:30–5:55 Towards Beam Hardening Correction in X-ray CTHyoung Suk Park, Dept. of Computational Science and Engineering, Yonsei University [MS06-05, p.21]

MS07. Qualitative and quantitative Inverse Scattering algorithms 2/3June 30(Thur) Room: ECC B155


10:30–10:55 A Fast and Robust Sampling Method for Shape Reconstruction in Inverse Acoustic Scatter-ing ProblemsXiaodong Liu, Chinese Academy of Sciences [MS07-01, p.21]

11:00–11:25 Regularized full and partial cloak for thin and plat objects in scattering problemsYoujun Deng, Department of Applied Mathematics, Central South University, China [MS07-02, p.21]

11:30–11:55 Detection and classification from electromagnetic induction dataJunqing Chen, Department of Mathematical Sciences, Tsinghua University [MS07-03, p.22]

12:00–12:25 Mathematical design of a novel gesture-based instruction/input device using wave detectionYuliang Wang, Department of Mathematics, Hong Kong Baptist University [MS07-04, p.22]

MS08. Industrial and medical applications of electrical impedance tomography 2/2June 30(Thur) Room: ECC B157

10:30-12:00 Chair: Hyeuknam Kwon

10:30–10:55 Bioimpedance imaging to assess abdominal fatness using EITHyeuknam Kwon, Dept. Computational Science & Engineering, Yonsei University [MS08-01, p.23]

11:00–11:25 An application of a novel optimization method in EIT imagingMing Zhang, School of Electrical Engineering and Automation, Tianjin University [MS08-02, p.23]

11:30–11:55 Image Reconstruction Algorithm Based on Compressed Sensing for Electrical CapacitanceTomographyLifeng Zhang, North China Electric Power University [MS08-03, p.23]

MS09. Inverse problems for system of equationsJune 30(Thur) Room: ECC B155

3:30-5:30 Chair: Jenn-Nan Wang

3:30–3:55 Solving Electrical Impedance Tomography via Subspace-Based Optimization MethodXudong Chen, Dept. of Electrical & Computer Engr, National University of Singapore [MS09-01, p.24]

4:00–4:25 Decoupling elastic waves and its applicationsHongyu Liu, Department of Mathematics, Hong Kong Baptist University [MS09-02, p.24]

5:00–5:25 Strong Unique Continuation for a Residual Stress System with Gevrey CoefficientsYi-Hsuan Lin, Institute of Mathematics, National Taiwan University [MS09-03, p.25]

CP10. Contributed Talks 2/2June 30(Thur) Room: ECC B157


MS11. July 1(Fri)

3:30–5:30 Chair: Eunjung Lee

3:30–3:55 Finite Integration Method for Inverse Heat Conduction ProblemYan Li, Math Dept, City University of Hong Kong [CP10-01, p.25]

4:00–4:25 Analysis of a numerical method for radiative transfer equation based bioluminescence to-mographyRongfang Gong, Department of Mathematics, Nanjing University of Astronautics and Aeronau-tics [CP10-02, p.25]

4:30–4:55 Recovery of the heat equation from a single boundary measurementKim Tuan Vu, Dept of Math, University of West Georgia, GA, USA [CP10-03, p.26]

5:00–5:25 Biomedical inverse problemsDarya Yermolenko, Novosibirsk State University [CP10-04, p.26]

0:00–0:00 Monotonicity-based regularization of inverse coefficient problemsBastian Harrach, Goethe University Frankfurt, Institute of Mathematics [CP10-05, p.27]

MS11. Qualitative and quantitative Inverse Scattering algorithms 3/3July 1(Fri) Room: ECC B4 Theater


10:30–10:55 A novel integral equation for scattering by locally rough surfaces and application to theinverse problem: the Neumann caseHaiwen Zhang, Institute of Applied Mathematics, Academy of Mathematics and Systems, Science ChineseAcademy of Sciences [MS11-01, p.28]

11:00–11:25 A direct method for inverse acoustic scattering in the time domainYukun Guo, Dept of Mathematics, Harbin Institute of Technology [MS11-02, p.28]

11:30–11:55 Resonance and scattering in a bubbly mediaHyundae Lee, Mathematics, Inha University [MS11-03, p.29]

Program 5

Plenary and MediaV Lectures

PL. Plenary and MediaV Lectures

PL-01. Spectral theory of the Neumann-Poincare operator and applications

Hyeonbae Kang, Inha University, Department of Mathematics

The Neumann-Poincare (NP) operator is a boundary integral operator which appearsnaturally when solving boundary value problems using layer potentials. The study onthe NP operator goes back to C. Neumann and Poincare as the name suggests. Thistalk surveys a history of the NP operator and recent development on the spectral theory,especially symmetrization of the NP operator, distribution of eigenvalues, and the studyon the continuous spectrum. It also surveys an application on mathematical modeling ofthe plasmon resonance.

E-mail: [email protected]

Time: 6/27(Mon) 10:30–11:20, Room: ECC B4 Theater

PL-02. Multi-frequency inverse acoustic source problems

Shuai Lu, School of Mathematical Sciences, Fudan University, China

In this talk, we investigate an interior Helmholtz inverse source problem with multi-ple frequencies. By implementing sharp uniqueness of the continuation results and exactobservability bounds for the wave equation, a (nearly Lipschitz) increasing stability es-timate is explicitly obtained for Cauchy measurements in a non-empty wave-number in-terval. With a specific geometric domain, an iterative/recursive reconstruction algorithmis proposed aiming at recovering unknown sources by the multifrequency boundary mea-surement. Both convergence and error estimates are derived to guarantee its reliability.Numerical examples verify the efficiency of our proposed algorithm. It is a joint workwith Gang Bao (Zhejiang University), Jin Cheng, Boxi Xu (Fudan University), VictorIsakov (Wichita University) and William Rundell (TAMU).



PL-03. Sparse Image Modelling and Blind De-convolution

Hui Ji, Department of Mathematics, National University of Singapore

Image de-convolution is to reconstruct the original scene from one blurry observationin which many high-frequency components are significantly attenuated or removed. Thede-convolution process is critical to many applications in optics and other imaging. Onechallenging image de-convolution problem is the so-called blind de-convolution problem,a difficult ill-posed non-linear inverse problem. Blind image de-convolution aims at re-covering the clear image from one blurred observation without knowing how it is blurred.In this talk, I will present several mathematical models and techniques that provide astrong foundation for resolving this challenging problem. Our work is built upon wavelettight frame theory, sparse image modelling and the relating L1 norm based optimiza-tion techniques. The main results include (1) a sparse approximation based variational



model and numerical solver for blind image deconvolution; (2) a robust linear image de-convolution technique in the presence of kernel error; (3) a two-stage approach for blindde-convolution when blurring process is non-stationary; and (4) a data-driven tight framefor optimal sparse image modelling.



PL-04. Some inverse problems and numerical inversions for the fractional diffusion equa-tions

Joint work with Jia Xianzheng and Sun ChunGongsheng Li, Shandong University of Technology

In this talk, we firstly consider some simultaneous inverse problems in the time/spacefractional diffusion equations, including the inverse problem of determining the diffusioncoefficient and the fractional order in the time FDE, the inverse problem of identifying thefractional orders and the initial function in the time-space FDE, and the inverse problem ofdetermining the diffusion coefficient and the source term in the space FADE, etc. We givesome theoretical analysis for the inversion algorithm and present numerical inversionsusing the optimal perturbation regularization algorithm or the homotopy regularizationalgorithm.

Next we are concerned with a backward problem for the multi-term time fractionalhomogeneous diffusion equation. Based on the analytical properties of the multivariateMittag-Leffler function, an explicit expression of the solution to the backward problem isobtained, and its instability is proved. Numerical inversions with noisy data are presented.


Time: 6/28(Tue) 9:00–9:50, Room: ECC B4 Theater

PL-05. [MediaV Award 1] TBA

TBA

E-mail: TBA

Time: 6/28(Tue) 2:00–2:50, Room: ECC B4 Theater

PL-06. Phase retrieval with one or two diffraction patterns by alternating projections of thenull vector

Joint work with Albert Fannjiang and Giren LiuPengwen Chen, Applied Math., National Chung Hsing University

Two versions of alternating projection (AP), the parallel alternating projection (PAP)and the serial alternating projection (SAP), are proposed to solve phase retrieval withat most two coded diffraction patterns. The proofs of geometric convergence are givenwith sharp bounds on the rates of convergence in terms of a spectral gap condition. Tocompensate for the local nature of convergence, the null vector method is proposed forinitialization and proved to produce asymptotically accurate initialization for the Gaussiancase. Extensive numerical experiments are performed to show that the null vector methodproduces more accurate initialization than the spectral vector method and that PAP/SAP

Abstract 7


converge faster to more accurate solutions than other iterative schemes for non-convexoptimization such as the Wirtinger flow. Moreover, SAP converges still faster than PAP.In practice AP and the null vector method together produce globally convergent iteratesto the true object.


Time: 6/29(Wed) 9:00–9:50, Room: ECC B4 Theater

PL-07. CT Metal Artifacts and Reduction Algorithms

Chang-Ock Lee, Mathematical Sciences, KAIST

There are several types of artifacts in CT images. The streaking artifact caused by themetallic objects (dental implants, surgical clips, or steel-hip) is one of the major artifactsin CT image and it limits the applications of CT. In this talk we investigate some mathe-matics for the metal artifacts in CT and review previous studies for the the metal artifactreduction. Then we propose our algorithm based on sinogram surgery, which iterativelyremoves the metallic effect in the sinogram using the basic principle of CT image recon-struction. The numerical experiments show that our algorithm reduces the metal artifactseffectively. We analyze the simulation results both quantitatively and qualitatively.


Time: 6/29(Wed) 10:30–11:20, Room: ECC B4 Theater

PL-08. Numerical Differentiation by Kernel-based Probability Measures

Joint work with Ye, QiLeevan Ling, Mathematics, Hong Kong Baptist University

We combine techniques in meshfree methods and stochastic regressions to constructkernel-based estimators for numerical derivatives from noisy data. By constructing Bayesianestimators from normal random variables defined on some (symmetric positive definite)kernel-based probability measures, Tikhonov regularization naturally arise in the formu-lation and the kernels shape parameter also plays the role of the regularization parameter,which reduces the number of parameters to only one. Our analysis provides two impor-tant features to this novel approach. First, we show that the conditional mean square errorof any estimator is computable without knowing the exact derivative and can be used asan a posteriori error bound. This allows user to evaluate the approximation quality of anygiven kernel-based estimator, and hence, select the best one (in the sense of kernel-basedprobability) out of many resulted from some brute-force approaches. Next, we focus onthe Gaussian kernel and identify a quasi-optimal (shape and regularization) parameter de-pending on noise level, data density, dimension, and order of differentiation. An examplein threedimensions is included to numerically verify our proven theories and proposedformulations.


Time: 6/30(Thur) 9:00–9:50, Room: ECC B4 Theater

PL-09. [MediaV Award 2] TBA

TBA



E-mail: TBA

Time: 6/30(Thur) 2:00–2:50, Room: ECC B4 Theater

PL-10. Inverse source problems for hyperbolic-type equations

Joint with Daijun Jiang and Masahiro YamamotoYikan Liu, Graduate School of Mathematical Sciences, The University of Tokyo

Inverse source problems for hyperbolic equations have been investigated for severaldecades mainly as by-products of the corresponding coefficient inverse problems, and thespecialized numerical methods have not been well-developed. Recently, we discovered aclass of hyperbolic-type equations describing the time cone model for phase transforma-tion in odd spatial dimensions, by which the determination of the nucleation rate reducesto an inverse source problem. In this talk, we consider the reconstruction of the spatialcomponent of the source term in three cases. For the wave equation with final observa-tion, we employ the analytic Fredholm theory to show the generic well-posedness. Forthe double hyperbolic equation with partial interior observation, we establish the globalLipschitz stability on basis of Carleman estimates. For the hyperbolic equation with atime- dependent principal part and partial interior / boundary observation, we prove alocal stability result of Holder type based on a newly established Carleman estimate forgeneral hyperbolic operators. Numerically, we develop a universal iterative thresholdingalgorithm for all cases by utilizing the corresponding adjoint systems. Extensive numer-ical experiments demonstrate the efficiency and accuracy of the algorithm. This talk isbased on joint works with Prof. Masahiro Yamamoto (The University of Tokyo) and Dr.Daijun Jiang (Central China Normal University).


Time: 7/1(Fri) 9:00–9:50, Room: ECC B4 Theater

Abstract 9

Minisymposium and Contributed Talks

MS01. Reconstruction schemes for evolution equations

MS01-01. Stability analysis for inverse coefficient problems in the diffusion equation using theadjoint method

Joint work with Jia Xianzheng and Sun ChunlongGongsheng LI, Shandong University of Technology

In this talk, we consider tow inverse coefficient problems in the parabolic equation

ut +A[u]+q(x)u = 0, (x, t) ∈ΩT = Ω× (0,T )

where A denotes a symmetric, uniformly elliptic operator, q(x) is the first-order coeffi-cient, and Ω denotes a bounded domain in Rn with smooth boundary. (1) To determinethe coefficient q = q(x) given the Dirichelet-Neumann data at the boundary. We introducethe adjoint method to prove the uniqueness of the inverse problem. (2) To determine thecoefficient q = q(x) given the final observations at t = T . We establish a conditional Lip-schitz stability for the inverse problem using the adjoint method in 1D and 2D/3D cases.(i) In 1D case with Robins boundary condition; (ii) In 2D/3D case in regular domains.The key point for using the adjoint method to prove uniqueness of an inverse problemlies in approximate controllability of the adjoint problem, and it lies in data compatibilityanalysis and construction of a variational identity for the stability analysis.


Time: June 27(Mon) 3:30–3:55, Room: ECC B155

MS01-02. Algebraic reconstruction scheme of moving dipole wave sources from boundarymeasurements

Takashi Ohe, Department of Applied Mathematics, Faculty of Science, Okayama Univer-sity of Science

Inverse source problem for wave equation is an important mathematical model of manyproblems in science, engineering, and medical fields. In this talk, we consider this prob-lem for three dimensional cases, and discuss a real-time algebraic reconstruction schemefor wave sources. Let Ω be a convex bounded domain in R3 with smooth boundary Γ.Let u(r, t) be the solution of the initial- and boundary-value problem of the scalar waveequation:

(1)

1c2

∂ 2u∂ t2 u(r, t)−∆u(r, t) = F(r, t), (r, t) ∈Ω× (0,T ),

u(r, t) = 0, (r, t) ∈ Γ× (0,T ),u(r,0) = 0, r ∈Ω,∂u∂ t (r,0) = 0, r ∈Ω,

where c > 0 and T > 2 · diag Ω > 0 are given constant, and F(r, t) is unknown sourceterm defined in Ω× (0,T ). We assume that unknown source term F is expressed by alinear combination of moving dipole sources as follows:

F(r, t) =K

∑k=1

mk(t) ·∇rδ (r−pk(t)),


MS01. June 27(Mon)

where δ denotes the three-dimensional Diracs delta distribution, K the number of dipolesources, and pk(t) = (pk,x(t), pk,x(t), pk,x(t)) and mk(t) = (mk,x(t),mk,y(t),mk,z(t)) the lo-cation and the dipole moment of k-th source, respectively. We also assume that the speedsof moving dipoles are slower than the wave propagation speed c. Our problem is toreconstruct all parameters of unknown source term, i.e. K,pk(t) and mk(t) from bound-ary measurements of the normal derivative φ = ∂u

∂νon Γ× (0,T ). For this problem, we

propose a real-time algebraic reconstruction scheme using the reciprocity gap functionaldefined on the space of solutions of adjoint problem of (1). We also give some numericalexperiments to show the effectiveness of our reconstruction scheme.



MS01-03. Semigroup-theoretic approach to identification of linear diffusion coefficients

Noboru Okazawa, Tokyo University of Science

Let A be a quasi-m-sectorial operator with domain D(A) and range in a complex Ba-nach space X . This talk is concerned with the identification of diffusion coefficients ν > 0in the parabolic-type initial-value problem:

(d/dt)u(t)+νAu(t) = 0, t ∈ (0,T ), u(0) = x ∈ X ,

with additional condition ‖u(T )‖ = ρ , where ρ > 0 is known. Except for the addi-tional condition, the solution to the initial-value problem is given by u(t) := e−t νAx ∈C([0,T ];X)∩C1((0,T ];X). Therefore, the identification of ν is reduced to solving theequation ‖e−νTAx‖ = ρ . It will be shown that the unique root ν = ν(x,ρ) depends on(x,ρ) locally Lipschitz continuously if the datum (x,ρ) fulfills the restriction ‖x‖ > ρ .This extends those results in Mola (2011).



MS01-04. Recovery of thermal conductivity in two-dimensional media with nonlinear sourceby optimizations

Joint work with Liu JijunBingxian Wang, Department of Mathematics, Southeast University,China

Consider the heat conduction process with the temperature-depended source modeledby a nonlinear parabolic equation. We aim to identify the thermal conductivity fromthe extra measurement. By introducing the cost functional with regularization terms, theinverse problem is reformulated as an optimization problem. We prove the existence of themini- mizers of the cost functional, with a rigorous analysis on the convergence propertyof the minimizing sequence. An iteration algorithm solving the optimization problem isproposed. Keywords: Inverse problem, parabolic equation, regularization, optimization,convergence.



Abstract 11

MS01. June 27(Mon)

CP02. Contributed Talks 1/2

CP02-01. Numerically Solving Inverse Scattering Problems using Sparsity Regularization andTotal Variation

Joint work with Armin Lechleiter and Kamil S. Kazimierski-HentschelFlorian Buergel, Center for Industrial Mathematics, University of Bremen

We consider inverse medium scattering in two or three dimensions modelled by theHelmholtz equation using incident point sources or plane waves and either near or far fieldmeasurements. To this end, we set up an efficient minimization-based inversion schemethat follows on the one hand the paradigm to, roughly speaking, minimize the discrep-ancy but on the other hand takes into account various structural a-priori information viasuitable penalty terms. This allows for instance to combine sparsity-promoting with total-variation based regularization, while at the same time respecting physical bounds for theinhomogeneous medium. The exibility of our approach is due to a primal-dual algorithmthat we employ to minimize the corresponding Tikhonov functional. We show feasibilityand performance of the resulting inversion scheme via reconstructions from synthetic andmeasured data in two and three dimensions. This is a joint work with Prof. Dr. ArminLechleiter and Dr. Kamil S. Kazimierski-Hentschel.



CP02-02. Reconstruction of the shear modulus of viscoelastic systems in a thin cylinder: aninversion scheme and experiments

Joint work with Hyeonbae Kang, Gen Nakamura and Yun-Che WangJunyong Eom, Mathematics, Inha University

We consider a problem of reconstructing the shear modulus of an viscoelastic system ina thin cylinder from the measurements of displacements induced by torques applied at thebottom of the cylinder. The viscoelastic system is a mathematical model of a pendulum-type viscoelastic spectrometer (PVS).We first compute in an explicit form the solution ofthe viscoelastic system, and then derive with an error estimate the leading order term ofthe average of the solution. This leading order term yields a nonlinear inversion schemeto determine the shear modulus from the measurements of displacements.We apply theinversion scheme to determine the shear modulus using experimental data acquired froma PVS system.



CP02-03. Non-linear Tikhonov Regularization in Banach Spaces for Inverse Scattering fromAnisotropic Penetrable Media

Joint work with Armin LechleiterMarcel Rennoch, University of Bremen, Center for Industrial Mathematics

We consider Tikhonov and sparsity-promoting regularization in Banach spaces for in-verse scattering from penetrable anisotropic media. To this end, we equip an admissi-ble set of material parameters with the Lp-topology and use Meyers’ gradient estimate


CP02. June 27(Mon)

for solutions of elliptic equations to analyze the dependence of scattered fields and theirFr’echet derivatives on the material parameter. This allows to show convergence of a non-linear Tikhonov regularization against a minimum-norm solution to the inverse problem,but also to set up sparsity-promoting versions of that regularization method. For both ap-proaches, the discrepancy is defined via a q-Schatten norm or an Lq-norm with 1 < q < ∞.Numerical reconstruction examples indicate the reconstruction quality of the method, aswell as the qualitative dependence of the reconstructions on q. This is a joint work withProf. Dr. Armin Lechleiter.



CP02-04. On the notion of elastic scattering coefficients

Joint work with Guanghui Hu and Jong Chul YeAbdul Wahab, BISPL, Department of Bio & Brain Engineering, Korea Advanced Insti-tute of Science & Technology, Daejeon

The notion of scattering coefficients for acoustic and electromagnetic inclusions emergedin an effort to design enhanced near invisibility cloaks [4, 5]. These mathematical objectscontain rich information of the contrast of material parameters, high order shape oscilla-tions, frequency profile, and the maximum resolving power in inverse scattering. Theyhave been effectively used for inverse medium scattering [3], echo-location and shape de-scription[6], and mathematical understanding of super-resolution phenomena in imaging[2]. In electromagnetic or acoustic media, scattering coefficients provide a natural exten-sion to the concept of contracted polarization tensors with respect to frequency depen-dence. They are defined in terms of the Fourier-Bessel coefficients (in 2D) or sphericalharmonic coefficients (in 3D) of the far-field scattering amplitude and can be retrievedwith high accuracy from the multi-static response data by solving a least-squares opti-mization problem. The impetus behind this study is the mathematical imaging of smallelastic inclusions of diminishing characteristic size. In this talk, the notion of elastic scat-ting coefficients will be introduced using cylindrical and spherical eigen-vectors of theLam’e equation [1]. A reconstruction framework to recover elastic scattering coefficientsfrom the multi-static response matrix will be presented. Moreover, their role in directand inverse elastic scattering and enhancement of the nearly-elastic cloaking will also bediscussed.

References

[1] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee, A. Wahab, MathematicalMethods in Elasticity Imaging (Princeton University Press, 2015).

[2] H. Ammari, Y. T. Chow, J. Zou, Super-resolution in imaging high contrast targetsfrom the perspective of scattering coefficients, arxiv.org/pdf/1410.1253.pdf.

[3] H. Ammari, Y. T. Chow, J. Zou, The concept of heterogeneous scattering coeffi-cients and its application in inverse medium scattering, SIAM J. Math. Anal. 46,pp. 2905-2935 (2014).

[4] H. Ammari, H. Kang, H. Lee, M. Lim, Enhancement of near-cloaking. Part II:the Helmholtz equation, Commun. Math. Phys. 317, pp. 485-502 (2013).

[5] H. Ammari, H. Kang, H. Lee, M. Lim, S. Yu, Enhancement of near cloaking forthe full Maxwell equations, SIAM J. Appl. Math. 73, pp. 2055-2076 (2013).

[6] H. Ammari, M. P. Tran, H. Wang, Shape identification and classification in echolo-cation, SIAM J. Imaging Sci. 7, pp.1883-1905 (2014).

Abstract 13

CP02. June 27(Mon)



CP02-05. On a geometrical inverse problem and its applications to scattering theory and cloak-ing

Hongyu LIU, Department of Mathematics, Hong Kong Baptist University

In this talk, I shall report our recent progress on the shape determination of an in-homogeneous penetrable or impenetrable scatterer by a minimal number of scatteringmeasurements, including uniqueness, stability and numerical reconstruction algorithms.I shall also discuss its applications and implications to scattering theory and invisibilitycloaking.



MS03. Qualitative and quantitative Inverse Scattering algorithms 1/3

MS03-01. A Multilevel Sampling Method for Detecting Sources in a Stratified Ocean Waveg-uide

Keji Liu, Shanghai University of Finance and Economics

We consider two regularized transformation-optics cloaking schemes for scatteringproblems. Both schemes are based on the blowup construction with the generating setsbeing, respectively, a generic curve and a planar subset. We derive sharp asymptotic es-timates in assessing the cloaking performances of the two constructions in terms of theregularization parameters and the geometries of the cloaking devices. The first construc-tion yields an approximate full-cloak, whereas the second construction yields an approx-imate partial-cloak. Moreover, by incorporating properly chosen conducting layers, bothcloaking constructions are capable of nearly cloaking arbitrary scattering contents.


Time: June 28(Tue) 10:30–10:55, Room: ECC B155

MS03-02. Inverse Random Source Scattering Problems

Peijun Li, Department of Mathematics, Purdue University





MS04. June 28(Tue)

MS03-03. The direct and inverse scattering problems with oblique derivative boundary condi-tion

Habing Wang, Dept of Mathematics, Southeast University, Nanjing




MS03-04. MUSIC algorithm for imaging perfectly conducting crack in limited-view inversescattering problem

Won-Kwang Park, Department of Mathematics, Kookmin University

This study examines mathematical representation of well-knownMUltiple SIgnal Clas-sification (MUSIC) algorithm to image the shape of arc-like perfectly conducting crackfrom scattered field data collected within the so-called Multi-Static Response (MSR) ma-trix in limited-view inverse scattering problem. For this purpose, we establish a relation-ship between the MUSIC imaging functional and an infinite series of Bessel functions ofinteger order of the first kind. This relationship is based on the so-called physical factor-ization ofMSR matrix. Various results of numerical simulation are presented in order tosupport the identified structure of MUSIC. Although a priori information of the target isneeded, we examine a least condition of range of incident and observation directions toapply MUSIC in the limited-view problem.



MS04. Industrial and medical applications of electrical impedance tomography 1/2

MS04-01. PCA-based Characterization in Lung Electrical Impedance Tomography

Joint work with Jin Keun SeoLiangdong Zhou, Department of Computational Science & Engineering, Yonsei Univer-sity

As is well known that inverse problem of EIT is highly nonlinear and ill-posed. Time-varying lung EIT data is driven by various dynamic factors including pulmonary activi-ties, cardiac activities, diaphragm movement, boundary movement, etc. Imaging methodproducing robust and reliable reconstruction is on need.We proposes a PCA-based EITtechnique, which aims to provide characteristic features for diagnosis of lung conditionincluding lung collapse detection and lung condition monitoring. The proposed method isused to separate the pulmonary and cardiac signals and extract the lung collapse character-istics by investigating the spatial and temporal features of the time series boundary data.

Abstract 15

MS04. June 28(Tue)

Numerical experiments and real human experiments show the validity of the proposedmethod.



MS04-02. A robust EIT reconstruction for lung imaging based on sensitivity-data correlation

Joint work with Jin Keun SeoKyounghun Lee, Dept of Computational Sci and Eng, Yonsei University

Electrical impedance tomography (EIT) for lung imaging has been suffered from deal-ing with the forward modeling error caused by the thorax movements during ventilation.The inherent ill-posed nature of EIT combined with the boundary geometry uncertain-ties produces serious boundary artifacts when using conventional image reconstructionmethods. This work proposes a new reconstruction method that effectively deals withthe boundary geometry uncertainties using the correlations between the columns of thesensitivity matrix and the EIT-data, and that does not require to solve the linearized EITsystem.



MS04-03. 3D EIT model construction and image reconstruction

Xiaoyan Chen, Tianjin University of Science and Technology, Tianjin, China

is 3D EIT reconstruction for lungs. In the research, a 3D EIT simulation phantom wasbuilt with the body thorax images scanned by 3D optical cloud point technology con-sidering the structure information of the lungs offered by X-ray images. According tothe prior conductivity information of inflated and deflated lungs, the sensitivity matrix issolved by COMSOL software, and EIT images of six different positions are reconstructedby CG iteration algorithm. In order to improve the sensitive field uniformity, four simula-tion experiments are completed differently in interval space between the electrode layers.The simulation results showed that, the imaging quality is superior if the interval is 8cmwhen the region of interesting is 33cm, the evaluation index of which is satisfied also, themaximum correlation coefficient is 0.8103, the sensitive field uniformity is 1.8696*103and the structure similarity is 0.4825.Obviously, the uniformity of the sensitive field isimproved, image reconstruction quality significantly enhanced.



MS04-04. Identification of the boundary heat transfer coefficient from interior measurementof temperature field

Joint work with Jijun LiuLiyan Wang, Department of Mathematics, Southeast University,China

Consider the heat conduction process for a homogeneous solid rod with one endpointcontacted with some liquid media. The aim is to identify the boundary heat transfer coef-ficient from the measured temperature field, which is essentially nonlinear and ill-posed.Based on the 1-dimensional heat equation model with Robin boundary condition, we


MS05. June 28(Tue)

prove the recognizability of the time-dependent Robin coefficient from the temperaturefield specified at only one interior point. To give a stable reconstruction for noisy inversioninput data, we propose a regularizing scheme by minimizing a cost functional includingpenalty term with rigorous mathematical analysis. A Newton type iterative scheme isestablished to solve this non-quadratic optimizing problem.



MS05. Inverse problems for fractional partial differential equations

MS05-01. Unique continuation property for anomalous diffusion equation

Joint work with Ching-Lung LinGen Nakamura, Department of Mathematics, Hokkaido University

In this talk a Carleman estimate and the unique continuation property of solutions(UCP) for an anomalous diffusion equation with fractional time derivative of order a(0 < a < 1 or 1 < a < 1) will be given. The estimate is derived via some subellipticestimate for an operator associated to the anomalous diffusion equation using calculusof pseudo- differential operators. Recently a strong inertia to the study of anomalousdiffusion equation came from a study in environmental science. It was shown by anexperiment that the spread of pollution in soils cannot be modeled correctly by the usualdiffusion equation, but it can be modeled by an anomalous diffusion equation. UCP foranomalous diffusion equation is a key to the study of control problem and inverse problemfor this equation. It can give the approximate boundary controllability for the controlproblem and it is very important for inverse problem if one wants to develop for instancelinear sampling type reconstruction scheme to identify unknown objects such as cracks,cavities and inclusions inside an anomalous diffusive medium. This is a joint work withChing-Lung Lin.



MS05-02. An inverse problem for distributed order time-fractional diffusion equation

Joint work with Yuri, LUCHKO and Masahiro YAMAMOTOZhiyuan Li, The University of Tokyo

Recently, a fractional diffusion equation with distributed order time-fractional deriva-tives

(2)∫ 1

0µ(α)∂ α

t u(x, t)dα = ∆u(x, t)+b(x)u(x, t),

where the non-negative weight function µ ∈C[0,1] does not vanish on the interval [0,1],has been proposed and successfully used for modelling several anomalous diffusion pro-cesses with the logarithmic growth of the mean squared displacement of the diffusiveparticles. As is known, when we consider (1) as model equation for describing e.g.,anomalous diffusion in inhomogeneous media, the weight function µ in distributed orderfractional derivatives should be determined by the inhomogeneity of the media. How-ever, it is not clear which physical law can correspond the inhomogeneity to the weight

Abstract 17

MS05. June 28(Tue)

function. Thus one reasonable way for estimating µ is an inverse problem of determiningµ in order to match available data. For this, we first consider an initial-boundary valueproblem for a distributed order time-fractional diffusion equation with the fractional de-rivative. The method of the eigenfunctions expansion in combination with the Laplacetransform is first employed to prove the uniqueness and existence of the solution to theinitial-boundary value problem and then to show its analyticity in time. As an appli-cation of the analyticity of the solution, a uniqueness result for the inverse problem ofdetermination of the weight function in the distributed order derivative contained in thetime-fractional diffusion equation from one interior point observation is obtained.



MS05-03. Determination of the temporal component in the source term of a fractional diffu-sion equation

Yikan LIU, Graduate School of Mathematical Sciences, The University of Tokyo

In this talk, we investigate the determination of the temporal component in the sourceterm of a time-fractional diffusion equation by the single point observation. Such an in-verse source problem coincides with the situation that the location of the contaminantsource is known in advance while the evolution pattern is unknown, which is of prac-tical importance when the observation point is far from the source. Based on a newlyestablished strong maximum principle and a reverse convolution inequality, we prove astability result for the inverse problem at the cost of a small extra interval of observation.For the numerical reconstruction, we adopt a fixed point iteration and incorporate the to-tal variation regularization to obtain stabilized solutions. This is a joint work with Mr.Zhidong Zhang (Texas A&M University) and Mr. Keren Yang (Fudan University).



MS05-04. Half-order fractional inverse transport problems by Carleman estimates

Manabu Machida, Institute for Medical Photonics Research, Hamamatsu University Schoolof Medicine

We consider the radiative transport equation in which the time derivative is replacedby the Caputo derivative. Such fractional-order derivatives are related to anomaloustransport and anomalous diffusion. By establishing Carleman estimates, we prove theglobal Lipschitz stability in determining the coefficient for the total attenuation in theone-dimensional time-fractional radiative transport equation of half-order. This is a jointwork with Atsushi Kawamoto (Department of Mathematical Sciences, The University ofTokyo).




MS06. June 28(Tue)

MS06. Image and Image Processing

MS06-01. Limited Tomography Reconstruction via Tight Frame and Simultaneous SinogramExtrapolation

Joint work with Bin Dong and Xiaoqun ZhangJae Kyu Choi, Institute of Natural Sciences, Shanghai Jiao Tong University

X-ray computed tomography (CT) is one of widely used diagnostic tools for medi-cal and dental tomographic imaging of the human body. However, the standard filteredbackprojection reconstruction method requires the complete knowledge of the projectiondata. In the case of limited data, the inverse problem of CT becomes more ill-posed,which makes the reconstructed image deteriorated by the artifacts. In this paper, we con-sider two dimensional CT reconstruction using the projections truncated along the spatialdirection in the Radon domain. Over the decades, the numerous results including the spar-sity model based approach has enabled the reconstruction of the image inside the regionof interest (ROI) from the limited knowledge of the data. However, unlike these existingmethods, we try to reconstruct the entire CT image from the limited knowledge of thesinogram via the tight frame regularization and the simultaneous sinogram extrapolation.Our proposed model shows more promising numerical simulation results compared withthe existing sparsity model based approach.



MS06-02. Various methods for ill-posed inversion in Quantitative Susceptibility Mapping

Joint work with Tian Liu, Weiwei Chen, and Yi WangShuai Wang, School of Electronic Engineering, University of Electronic Science andTechnology of China

This talk is concerned with the well-known ill-posed dipole inversion problem ofQuantitative Susceptibility Mapping (QSM) in Magnetic Resonance Imaging (MRI). Wesystematically analyzed the inversion part of QSM in terms of their treatment of noise inthe data fidelity term and their choice of prior. Based on analysis of the effects of noisetreatment in the data fidelity term and the choice of prior in QSM based on a few pub-lished methods in literature, it was found that noise whitening and structure priors areuseful in improving QSM using regularized minimization. This is a joint work with TianLiu, Weiwei Chen, Yi Wang.



MS06-03. Intensity Nonuniformity Correction Method in brain MR images

Joint work with Hemant TagareYunho Kim, Dept. of Mathematical Sciences, UNIST

Intensity nonuniformity artifact is unavoidable in MR imaging due to RF coil inhomo-geneity, gradient-driven eddy current, interactions within the body, etc., which result innonuniform signals where we expect uniform ones. Especially, we expect three values

Abstract 19

MS06. June 28(Tue)

in T1 weighted brain MR images for GM (gray matter), WM (white matter), CSF (cere-brospinal fluid). However, the measured data contain smooth variation in values overthe region. In this talk, we will describe the problem in detail and propose a rigorousmathematical framework, where we can detect the shape of smooth variation in values re-vealing uniform underlying signal. The only assumption of our model is that the artifacthas a slowly varying characteristic over a bounded region, which is enough tofactor outthe artifact correctly.



MS06-04. The optimal methods for inversion of the noisy k-plane transform

Tigran Bagramyan, Samsung Electronics

In many applied and theoretical problems one needs to recover a function (functionalor operator) from the information, which can be incomplete or given with an error. Suchproblems are investigated in the optimal recovery theory - a modern branch of approxima-tion theory. In general the problem is to find the best approximation of a value of linearoperator U : X → Z on a given set X from values of another linear operator I : X → Y(called information) given with an error in some metric. In present work we consider thek-plane transform− an operator, that maps a function on Rd to the set of its integrals overall k-planes. This operator is widely used in the computerized tomography theory, whichdeals with the numerical reconstruction of functions from their linear integrals. Specialcases are the Radon transform (k = d− 1) and the X-ray transform (k = 1). For the par-ticular classes of functions there exist different inversion formulas that allow to produceexact reconstruction. We consider the case when the k-plane transform is measured withan error δ > 0 in the mean square metric. Consider Gk,d the Grassmann manifold of (non-oriented) k-dimensional subspaces in Rd . Given representation of a point x∈Rd in a formx = x′+x′′, x′ ∈ π, x′′ ∈ π⊥, ∈Gk,d the k-plane transform is defined by the integral alongthe plane parallel to through the point x′′:

P f (π,x′′) = Pπ f (x′′) =∫

π

f (x′+ x′′)dx′, x′′ ∈ π⊥.

Its domain is the manifold of all k-planes in Rd Gk,d = (π,x′′) : π ∈ Gk,d , x′′ ∈ π⊥. Wewill work with the class of functions

W = f ∈ L2(Rd) : ‖Λα f‖L2(Rd) ≤ 1; P f ∈ L2(Gk,d).

The Lambda operator is defined for α > 0 by the equation Λα f (ξ ) = |ξ |α f (ξ ) on the setof functions f ∈ L2(Rd) that satisfy the condition |ξ |α ∈ f (ξ )∈ L2(Rd). Suppose that forevery function f ∈W we know function g ∈ L2(Gk,d) such that ‖P f −g‖L2(Gk,d) ≤ δ . Onthis information we want to recover function Λβ as an element of L2(Rd), where 0≤ β <α . We consider all possible methods or recovery− arbitrary maps m : L2(Gk,d)→ L2(Rd).For every method of recovery m define its error e(δ ,m) by

e(δ ,m) = supf ∈W,g ∈ L2(Gk,d)‖P f −g‖L2(Gk,d)≤δ

‖Λβ f −m(g)‖L2(Rd).

The smallest error among all the methods is called the error of the optimal recovery

E(δ ) = infm:L2(Gk,d)→L2(Rd)

e(δ ,m).


MS07. June 30(Thur)

Method m for which the error of the optimal recovery is attained, i.e. e(δ ,m) = E(δ ), iscalled optimal. We present the explicit construction for the optimal methods and the errorof the optimal recovery. As a consequence, we give one inequality for the norms of thedegree of the Lambda operator and the k-plane transform. Particular cases include newinversion methods and inequalities for the classical Radon and X-ray transforms.



MS06-05. Towards Beam Hardening Correction in X-ray CT

Joint work with Hao Gao, Sung Min Lee, and Jin Keun SeoHyoung Suk Park, Dept. of Computational Science and Engineering, Yonsei University

X-ray computed tomography (CT) is the most widely used tomographic imaging tech-nique in the field of dental and medical radiography. However, its advantage is partlylimited by the metallic object-related artifacts in the images, which appear as shading andstreaking artifacts. In this presentation, we investigate the mathematical characteristics ofthose metal artifacts from the structure of the projection data. We also proposed a newmethod to correct metal artifacts without degrading intact anatomical images.




MS07-01. A Fast and Robust Sampling Method for Shape Reconstruction in Inverse AcousticScattering Problems

Xiaodong Liu, Chinese Academy of Sciences



Time: June 30(Thur) 10:30–10:55, Room: ECC B155

MS07-02. Regularized full and partial cloak for thin and plat objects in scattering problems

Youjun Deng, Department of Applied Mathematics, Central South University, China

We consider two regularized transformation-optics cloaking schemes for scatteringproblems. Both schemes are based on the blowup construction with the generating setsbeing, respectively, a generic curve and a planar subset. We derive sharp asymptotic es-timates in assessing the cloaking performances of the two constructions in terms of the

Abstract 21

MS07. June 30(Thur)

regularization parameters and the geometries of the cloaking devices. The first construc-tion yields an approximate full-cloak, whereas the second construction yields an approx-imate partial-cloak. Moreover, by incorporating properly chosen conducting layers, bothcloaking constructions are capable of nearly cloaking arbitrary scattering contents.

E-mail: dengyijun [email protected]


MS07-03. Detection and classification from electromagnetic induction data

Junqing Chen, Department of Mathematical Sciences, Tsinghua University

I will introduce an efficient algorithm for identifying conductive objects using induc-tion data derived from eddy currents. The method consists of first extracting geometricfeatures from the induction data and then matching them to precomputed data for knownobjects from a given dictionary. The matching step relies on fundamental properties ofconductive polarization tensors and new invariance properties introduced in this paper.A new shape identification scheme is developed and tested in numerical simulations inthe presence of measurement noise. Resolution and stability properties of the proposedidentification algorithm are investigated.



MS07-04. Mathematical design of a novel gesture-based instruction/input device using wavedetection

Joint work with Hongyu Liu and Can YangYuliang Wang, Department of Mathematics, Hong Kong Baptist University

In this paper, we present a conceptual design of a novel gesture-based instruction/inputdevice using wave detection. The device recognizes/detects gestures from a person andbased on which to give the specific orders/inputs to the computing machine that is con-nected to it. The gestures are modelled as the shapes of some impenetrable or penetrablescatterers from a certain admissible class, called a dictionary. The device generates time-harmonic point signals for the gesture recognition/detection. It then collects the scatteredwave in a relatively small backscattering aperture on a bounded surface containing thepoint sources. The recognition algorithm consists of two steps and requires only two in-cident waves of different wavenumbers. The approximate location of the scatterer is firstdetermined by using the measured data at a small wavenumber and the shape of the scat-terer is then identified using the computed location of the scatterer and the measured dataat a regular wavenumber. We provide the mathematical principle with rigorous justifica-tions underlying the design. Numerical experiments show that the proposed device workseffectively and efficiently in some practical scenarios.




MS08. June 30(Thur)

MS08. Industrial and medical applications of electrical impedance tomography 2/2

MS08-01. Bioimpedance imaging to assess abdominal fatness using EIT

Hyeuknam Kwon, Dept. Computational Science & Engineering, Yonsei University

This research presents a static electrical impedance tomography (EIT) technique toevaluate abdominal obesity by estimating thickness of subcutaneous fat and occupationratio of visceral fat. EIT has fundamental drawbacks in the absolute admittivity imaging,because it is very sensitive to the forward modeling errors due to the boundary geometryerrors and the absence of reference data. To probe abdominal fat with reducing effectsof boundary geometry error, we develop a local reconstruction method which allows toidentify the border between subcutaneous fat and muscle by using special current injec-tion patterns to make reference-like data. We develop the local reconstruction method offinding the border between subcutaneous fat and muscle using special current injectionpatterns. We found a linear relation between the reference-like data and a measured datacorresponding to the special current injection patterns. This relation allows to eliminatethe influence of the subcutaneous fat from the measured data under the assumptions re-garding on subcutaneous fat. To find the border between subcutaneous fat and muscle,the local reconstruction method is suggested by taking advantages of the data of removedthe influence of subcutaneous fat. The performance of the proposed method were demon-strated with numerical simulations, phantom and human experiments. As shown in thefigure of numerical simulation result, the border between subcutaneous fat and musclecan be found in the local region corresponding to the special current injection patterns.The proposed local reconstruction method can find border between subcutaneous fat andmuscle.



MS08-02. An application of a novel optimization method in EIT imaging

Ming Zhang, School of Electrical Engineering and Automation, Tianjin University

Research is concerned with EIT imaging, a widely applied technique in industrial de-tecting process. The existing EIT image technique is severely ill-posed, since the numberof available measuring data is much less than that of the unknown variables (e.g., pixelunit, etc). To overcome this problem, many optimization methods have been applied toreformulate the existing imaging process. In this paper, a new optimization is used toEIT imaging process. Different from the existing methods, the new optimization methodcan efficiently better minimize the robust loss function and converge very fast in practice.Both theoretical analysis and experiments result demonstrate the new method could ob-tain better space resolution than the existing ones, especially for small target detection.



MS08-03. Image Reconstruction Algorithm Based on Compressed Sensing for Electrical Ca-pacitance Tomography

Lifeng Zhang, North China Electric Power University

Abstract 23

MS08. June 30(Thur)

In order to improve the sampling rate and the quality of reconstructed images of elec-trical capacitance tomography (ECT) system, a new ECT image reconstruction algorithmbased on compressed sensing (CS) theory is proposed. Firstly, using discrete Fourierorthogonal basis, the original image gray signal can be transformed into a sparse sig-nal. Then, the electrodes are excited randomly and the capacitance values of differentelectrode pairs are also measured in a random order. Thus, the capacitance signals andthe corresponding observation matrix are obtained. After that, using L1 regularizationmodel and primal dual interior point method, the reconstruction of original gray imagecan be obtained. Finally, the simulation experiments are performed. Simulation resultshave shown that the relative error of the reconstructed images obtained by the proposedmethod is smaller than the corresponding images obtained by the LBP algorithm and theLandweber algorithm.



MS09. Inverse problems for system of equations

MS09-01. Solving Electrical Impedance Tomography via Subspace-Based Optimization Method

Xudong Chen, Dept. of Electrical & Computer Engr, National University of Singapore

The Subspace-Based Optimization Method is proposed to solve the electric impedancetomography (EIT) problem. Two versions are preprinted. The first is the new fast Fouriertransform subspace-based optimization method (NFFT-SOM). Instead of implementingoptimization within the subspace spanned by smaller singular vectors in subspace-basedoptimization method (SOM), a space spanned by complete Fourier bases is used in theproposed NFFT-SOM. We discuss the advantages and disadvantages of the proposedmethod through numerical simulations and comparisons with traditional SOM. The sec-ond is the low frequency subspace optimized method (LF-SOM), in which we replace thedeterministic current and noise subspace in SOM with low frequency current and spacespanned by discrete Fourier bases, respectively. The proposed reconstruction methods aretested by numerical simulations, which show that the proposed algorithms are fast androbust.



MS09-02. Decoupling elastic waves and its applications

Hongyu Liu, Department of Mathematics, Hong Kong Baptist University

It is known that the elastic wave field can be decomposed into the shear and com-pressional parts, namely, the pressure and shear waves that are generally coexisting, butpropagating at different speeds. We shall show how to decouple the pressure and shearwaves by using proper geometries and boundary conditions. Then we discuss the appli-cations of the decoupling results to uniqueness and stability in inverse problems, as wellas the partial invisibility cloaking for elastic wave scattering.




CP10. June 30(Thur)

MS09-03. Strong Unique Continuation for a Residual Stress System with Gevrey Coefficients

Yi-Hsuan Lin, Institute of Mathematics, National Taiwan University

We consider the problem of the strong unique continuation for an elasticity system withgeneral residual stress. Due to the known counterexamples, we assume the coefficientsof the elasticity system are in the Gevrey class of appropriate indices. The main tools areCarleman estimates for product of two second order elliptic operators.



CP10. Contributed Talks 2/2

CP10-01. Finite Integration Method for Inverse Heat Conduction Problem

Joint work with Y. C. HonYan Li, Math Dept, City University of Hong Kong

We present in this talk the application of a recently developed Finite integration method(FIM) for solving inverse heat conduction problems (IHCPs). The use of the Laplacetransform technique for temporal discretization and Lagrange quadrature formula for spa-tial integration is shown to be effective and efficient for solving IHCPs under regular do-mains. For problems defined on irregular domains, the meshless radial basis function(RBF) is combined with the FIM to give a highly accurate spatial approximation to theIHCPs. Numerical stability analysis indicates that the convergence of the FIM-RBF isless sensitive to the chosen value of the shape parameter contained in the multiquadricradial basis functions. The optimal choice for this shape parameter is still an open prob-lem, whose value is critical to the accuracy of the approximation. For tackling the ill-conditioned linear system of algebraic equations, the standard regularization methods ofsingular value decomposition and Tikhonov regularization technique are both adopted forsolving IHCPs with data measurement errors.



CP10-02. Analysis of a numerical method for radiative transfer equation based biolumines-cence tomography

Joint work with Joseph Eichholz, Xiaoliang Cheng and Weimin HanRongfang Gong, Department of Mathematics, Nanjing University of Astronautics andAeronautics

In the bioluminescence tomography (BLT) problem, one constructs quantitatively thebioluminescence source distribution inside a small animal from optical signals detectedon the animal’s body surface. The BLT problem is ill-posed and often the Tikhonovregularization is used to obtain stable approximate solutions. In conventional Tikhonovregularization, it is crucial to choose a proper regularization parameter to balance the ac-curacy and stability of approximate solutions. In this talk, a parameter-dependent coupledcomplex boundary method based Tikhonov regularization is applied to the BLT problemgoverned by the radiative transfer equation. By properly adjusting the parameter in the

Abstract 25

CP10. June 30(Thur)

Robin boundary condition, we achieve one important property: the regularized solutionsare uniformly stable with respect to the regularization parameter so that the regularizationparameter can be chosen based solely on the consideration of the solution accuracy. Thediscrete-ordinate finite-element method is used to compute numerical solutions. Numer-ical results are provided to illustrate the performance of the proposed method. This isa joint work with Joseph Eichholz of Rose-Hulman Institute of Technology, XiaoliangCheng of Zhejiang University and Weimin Han of University of Iowa.

E-mail: grf [email protected]


CP10-03. Recovery of the heat equation from a single boundary measurement

Kim Tuan Vu, Dept of Math, University of West Georgia, GA, USA

Consider the heat process in a finite length rod, where the heat source is proportionalto the temperature distribution ut(x, t) = uxx(x, t)−q(x)u(x, t), 0 < x < 1, t > 0,

u(0, t) = 0,u(1, t) = a(t),u(x,0) = 0

We are concerned with the recovery of the heat coefficient q(x) from the measurementof the heat ux(1, t) = b(t), at one end of the rod only. We show that if u(1, t) = a(t) isa nontrivial, nonnegative, and continuously differentiable function with compact supporton (0,T ), then a single measurement of ux(1, t) = b(t) either on (T,T1) or at t = 1,2, · · ·determines q(x) uniquely. This is a joint work with Dr. Amin Boumenir.



CP10-04. Biomedical inverse problems

Joint work with Sergey Kabanikhin, Olga Krivorotko, Victoria Kashtanova, and DmitriyVoronovDarya Yermolenko, Novosibirsk State University

Mathematical modeling is a tool to obtain recommendations that are not available us-ing modern measurement instruments. It is the reason that in our century mathematicalmodels of biology and medicine are developing rapidly. These models are described bysystems of nonlinear ordinary differential equations (ODE) and characterized by a set ofparameters. In epidemiology these parameters are speed of the disease expansion, param-eter of transmissibility, the likelihood of infection expansion, parameter of the mortality,etc., in immunology the set of parameters describes virus natural death rate, target cellproduction rate, treatment efficacy reduction, etc. Parameters of ODE systems have to beidentified to obtain important information about the disease, immune response, suscep-tibility to specific drugs, epidemic spread, etc. For this, an approach of inverse problemtheory in which the unknown model parameters are determined from the available ex-perimental data (observations) is used. In this talk, a numerical algorithm for solvinginverse problems for Cauchy problem of ODE systems X = Φ(X(t), p), X(0) = X0 areconsidered. Here X(t) is a vector function describing the state of the system (in vivodrug concentration, immune system components, cases of a spreading epidemic disease,etc.), and p is a set of nonnegative parameters pm ≥ 0, m = 1, · · · ,M, characterizing the


CP10. June 30(Thur)

model under study. Inverse problem consists in determination the set of parameters andinitial data q = (p,X0)

T from some additional information X(tk) = fk,k = 1, · · · ,K. Theinverse problem is reduced to minimizing a misfit function J(q) =∑

Kk=1 |X(tk; p)− fk|2 by

combination of stochastic (simulated annealing, genetic algorithm) and gradient (minimalerrors, Newton-Kantorovich) methods [1]. To calculate the gradient of the functional, thesolution to a corresponding adjoint problem is used. Some inverse problems of phar-macokinetics [2], immunology, HIV infection dynamics [3], and tuberculosis epidemicspread in Russian Federation are considered as examples.

References

[1] S.I. Kabanikhin, O.I. Krivorotko. Identification of biological models describedby systems of nonlinear differential equations, Journal of Inverse and Ill-PosedProblems, Vol. 23, No. 5, pp. 519-527 (2015)

[2] A. Ilyin, S.I. Kabanikhin, D.B. Nurseitov, A.T. Nurseitova, N.A. Asmanova, D.A.Voronov, D. Bakytov. Analysis of ill-posedness and numerical methods of solv-ing a nonlinear inverse problem in pharmacokinetics for the two-compartmentalmodel with extravascular drug administration, Journal of Inverse and Ill-PosedProblems, Vol. 20, No. 1, pp. 39-64 (2012).

[3] S. Kabanikhin, O. Krivorotko, A. Mortier, D. Voronov, D. Yermolenko. A nu-merical algorithm of parameter estimation for dynamic model for HIV infectionof CD4+ T cells The Siberian Scientific Medical Journal, Vol. 36, No. 1, pp.29-35. (2016).



CP10-05. Monotonicity-based regularization of inverse coefficient problems

Bastian Harrach, Goethe University Frankfurt, Institute of Mathematics

Newly emerging imaging methods lead to the inverse problem of determining oneor several coefficient function(s) in an elliptic partial differential equation from (partial)knowledge of its solutions. A prominent example is electrical impedance tomography(EIT), where electrical currents are driven through an imaging subject to image its interiorconductivity distribution. In the so-called continuum model, EIT requires recovering thecoefficient function σ(x) in

(3) ∇ · (σ(x)∇u(x)) = 0, x ∈Ω,

from knowledge of the Neumann-Dirichlet operator

Λ(σ) : g 7→ u|∂Ω,

where u solves (3) with σ∂ν u|∂Ω = g.

Given noisy difference measurements

Λδmeas ≈ Λ(σ)−Λ(σ0)≈ Λ

′(σ0)(σ −σ0),

a natural and generic approach is to approximate the conductivity difference κ ≈ σ −σ0by minimizing a linearized and regularized data-fit functional

||Λ′(σ0)κ−Λδmeas||2 + regularization→min!

Such algorithms are widely used in practice but their theoretical justification has remainedan open question, and the choice of regularization is often heuristic.

Abstract 27

CP10. June 30(Thur)

In this talk, we will show how to regularize the linearized data-fit functional withmonotonicity-based constraints in such a way that convergence of certain shape propertiesin the reconstructions can be rigorously guaranteed.

References

[1] B. Harrach, M. N. Minh: Enhancing residual-based techniques with shape recon-struction features in Electrical Impedance Tomography, submitted.

[2] B. Harrach, J. K. Seo: Exact shape-reconstruction by one-step linearization inelectrical impedance tomography, SIAM Journal on Mathematical Analysis 42(4),1505–1518.

[3] B. Harrach, M. Ullrich: Monotonicity-based shape reconstruction in electricalimpedance tomography, SIAM Journal on Mathematical Analysis 45 (6), 3382–3403.




MS11-01. A novel integral equation for scattering by locally rough surfaces and application tothe inverse problem: the Neumann case

Joint work with Fenglong Qu and Bo ZhangHaiwen Zhang, Institute of Applied Mathematics, Academy of Mathematics and Systems,Science Chinese Academy of Sciences

This talk is concerned with the direct and inverse acoustic or electromagnetic scatter-ing problems by a locally perturbed, perfectly reflecting, infinite plane (which is calleda locally rough surface) with Neumann boundary condition. We propose a novel inte-gral equation formulation for the direct scattering problem which is defined on a boundedcurve (consisting of a bounded part of the infinite plane containing the local perturba-tion and the lower part of a circle) with two corners. This novel integral equation canbe solved efficiently by using the RCIP method introduced previously by Johan Helsingand is capable of dealing with large wave number cases. For the inverse problem, wepropose a Newton iteration method to reconstruct the local perturbation of the plane frommultiple frequency far-field data, based on the novel integral equation formulation. Nu-merical examples are carried out to demonstrate that our reconstruction method is stableand accurate even for the case of multiple-scale profiles.


Time: July 1(Fri) 10:30–10:55, Room: ECC B4 Theater

MS11-02. A direct method for inverse acoustic scattering in the time domain

Joint work with Dietmar Homberg, Guanghui Hu, Jingzhi Li and Hongyu LiuYukun Guo, Dept of Mathematics, Harbin Institute of Technology

This talk is concerned with the inverse scattering problems of imaging unknown/inaccessiblescatterers by transient acoustic near-field measurements. Based on the analysis of the mi-gration method, we propose efficient and effective sampling schemes for imaging small


MS11. July 1(Fri)

and extended scatterers from knowledge of time-dependent scattered data due to incidentimpulsive point sources. Though the inverse scattering problems are known to be nonlin-ear and ill-posed, the proposed imaging algorithms are totally direct involving only inte-gral calculations on the measurement surface. Theoretical justifications will be presentedand numerical results will be shown to demonstrate the effectiveness and robustness ofour methods. This is a joint work with Dietmar Homberg, Guanghui Hu, Jingzhi Li andHongyu Liu.



MS11-03. Resonance and scattering in a bubbly media

Joint work with Habib Ammari, Brian Fitzpatrick, David Gontiery, and Hai ZhangzHyundae Lee, Mathematics, Inha University

We derive an original formula for the Minnaert resonances of bubbles of arbitraryshapes using layer potential techniques and Gohberg-Sigal theory. Our formula can begeneralized to multiple bubbles. We provide a mathematical justification of the monopoleapproximation and demonstrate the enhancement of the scattering in the far field at theMinneart resonances. Our results show that at the Minnaert resonance it is possible toachieve superfocusing of acoustic waves or imaging of passive sources with a resolutionbeyond the Rayleigh diffraction limit. The main finding of this paper are illustrated witha few numerical examples in two dimensions.



Abstract 29

plenary and mediav lectures - math at...

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