Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Sombuddha Bhattacharyya

IISER Bhopal

6

Introduction to Calderón inverse problem:

We will discuss on the Calderón inverse problem, a mathematical basis for several imaging modalities.
Brief plan for the lectures:

• We will start with defining the conductivity equation and study the existence uniqueness and stability of the solution of the equation given boundary Dirichlet measurement. Finally, we will define the Dirichlet to Neumann map for the conductivity equation and connect it to the boundary current to voltage measurement.
• In this lecture we will formally state Calderón inverse problem. We will consider the inverse problem for a special class of conductivities and relate it with the inverse problem of recovering a lower order perturbations of the Schrödinger equation.
• In this lecture we will prove a Carleman estimate and construct Complex Geometric Optics (CGO) solutions for the Schrödinger equation, which will be helpful for solving the inverse problem.
• We will solve Calderón inverse problem using the CGO solutions for the Schrödinger equation constructed in the last lecture. For the proof we will follow the approach of Sylvester-Uhlmann (1987).
• Next we will discuss the problem of recovering lower order perturbation of the Schrödinger equation from the boundary measurements, but now we restrict measurements over only a part of the boundary.
• Finally, we will discuss the partial data inverse problem for the Magnetic Schrödinger equation. We will follow the work of D.D.S. Ferreira, C.E. Kenig, J. Sjöstrand, G. Uhlmann.

References:

1. Calderón problem Lecture notes, Spring 2008, M. Salo, http://users.jyu.fi/~salomi/lecturenotes/calderon_lectures.pdf

2. A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73 (1980).

3. J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 125 (1987), 153– 169.

4. G. Uhlmann. Inverse problems: seeing the unseen. Bull. Math. Sci., 4(2):209–279, 2014.

5. D. dos Santos Ferreira, C. Kenig, J. Söjstrand, and G. Uhlmann, Determining a Magnetic Schrödinger Operator from Partial Cauchy Data. Comm. Math. Phys.271, 467-488 (2007).

Thamban Nair,

IIT Madras

Ill-posed operator equations and their regularization

The syllabus will be selected contents of the book by Thamban Nair titled "Linear Operator Equations: Approximation and Regularization"  World Scientific, 2009.

Reference:

Thamban Nair, Linear Operator Equations: Approximation and Regularization, World Scientific, 2009.

Rohit Kumar Mishra

IIT Gandhinagar

Integral geometry in inverse problems:

The aim of this course is to motivate and introduce some integral transforms that are crucial from the application point of view. For instance, the inversion of X-ray transform helps to reconstruct optical properties of a human body by probing it with X-rays and many more. In simple words the problem of interest is the following: What kind of information can be recovered about a non- transparent object from boundary measurement. While studying these integral transforms, one must ask whether the transform considered is injective, how stable its inversion is, what are the mapping properties, etc. We plan to discuss these questions for the Radon transform and some of its generalizations.

Course plan

• Motivation, the definition of the Radon transform and X-ray transform, the interplay between Radon transform (X-ray transform), Fourier transform, and convolution.

• Back projection operator and inversion algorithms for the integral operators introduced in the first lecture.

• More discussion on inversion schemes, stability analysis.

• Support theorems and injectivity questions for both transforms.

• Some mapping properties and range spaces for these transforms.

• Generalizations of these transforms for tensor fields and some open problems in the area.

References:

1. F. Natterer. The Mathematics of Computerized Tomography. Volume 32 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.

2. F. Natterer and F. Wübbeling. Mathematical Methods in Image Reconstruction. SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.

3. T. G. Feeman. The Mathematics of Medical Imaging. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham, second edition, 2015. A beginner’s guide.

4. V. A. Sharafutdinov. Integral Geometry of Tensor Fields. Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994.

Manas Kar

IISER Bhopal

In this set of lectures, we plan to consider the unique continuation principle (UCP) for the linear equation conductivity equation with a potential term.

• In this lecture, I will mainly review Holmgren's theorem, which is precisely the UCP for solutions to the conductivity equation with real analytic coefficients.

• L2-Carleman estimates and its application to UCP will be discussed. I will start with proving UCP across a hyperplane for solutions to the conductivity equation in an infinite strip.

• This will be a continuation of Lecture 2. We will continue with UCP along hypersurface and finally will prove the result for the case when the conductivity is 1 and the potential is in bounded.

•  In this lecture, we prove UCP for solutions of conductivity equation when the conductivity is 1 and the potential is in Ln/2. We will prove an Lp Carleman estimate.

•  In this last lecture, we prove UCP for solutions of conductivity equation when the conductivity is Lipschitz regular and the potential is 0.

References:

    1. F. John, Partial differential equations (Section 3.5), 4th edition, Springer-Verlag, 1982.

    2. C. Sogge,Fourier integrals in classical analysis (Section 5.1), Cambridge University Press, 1993.

    3. N. Garofalo, F. Lin, Monotonicity properties of variational integrals,Apweights and unique continuation, Indiana U Math J, 1986.

    4. N. Garofalo, F. Lin, Unique continuation for elliptic operators: A geometric-variational approach,CPAM, 1987.

    5. M. Salo Unique continuation for elliptic equations, Notes, Fall 2014.

    6. L. H ormander,The analysis of linear partial differential operators, vol.1 (Section 8.6).

    7. F. Treves, Basic linear partial differential equations (Section II.21), Academic Press, 1975.

    8. N. Lerner, Carleman inequalities. An introduction and more. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 353. Springer, Cham, 2019, xxvii+557 pp.

Manmohan Vashisth

IIT Jammu

Inverse problems for linear hyperbolic PDEs:

In these lectures we will introduce the inverse problems related to linear hyperbolic PDEs.

• In the first two lectures, we shall introduce the Calderón type inverse problems for linear hyperbolic PDEs and study the BukhgeimKlibanov method for determining the potential in a wave equation from boundary data.

•  In these lectures, we shall focus on the construction of geometric optics solutions and their application to solve inverse problems for 2nd order linear hyperbolic PDEs. Our focus will be on unique determination of lower order perturbations for the wave operator.

• These lectures will be based on studying the inverse problems for wave equation with formally or under determined data. Here we study the inverse problems of determining the density coefficient of a wave equation with point source or receiver data.

References:

    1. A.L. Bukhgeim and M.V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260(2):269--272, 1981.

    2. Rakesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation. Comm. Partial Differential Equations, 13(1):87--96, 1988.

    3. A. G. Ramm and Rakesh; Property C and an inverse problem for a hyperbolic equation. J. Math. Anal. Appl., 156(1):209--219, 1991.

    4. A. G. Ramm and J. Sjostrand; An inverse problem of the wave equation, Math. Z., 206(1):119--130, 1991.

    5. R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem. Inverse Problems, 29(9):095015, 17, 2013.

    6. Rakesh, Inverse problems for the wave equation with a single coincident source-receiver pair. Inverse Problems 24 (2008), no. 1, 015012, 16 pp.

    7. Rakesh and P. Sacks; Uniqueness for a hyperbolic inverse problem with angular control on the coefficients. J. Inverse Ill-Posed Probl. 19 (2011), no. 1, 107--126.

    8. Rakesh and G. Uhlmann; The point source inverse back-scattering problem. Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, 279--289.

    9. M. Vashisth, An inverse problems for the wave equation with source and receiver at distinct points, Journal of Inverse and Ill-posed Problems, Volume 27, Issue 6, Pages 835–-843.

Anupam Pal Choudhury

NISER Bhubaneshwar

An introduction to the mathematical theory of obstacle scattering:

In this course, we shall discuss direct and inverse problems in obstacle scattering theory. The focus will mainly be on the treatment of acoustic
waves. Our goal would be to understand the inverse medium problems in this context.

The course will end with a quick introduction to applications of this theory to imaging techniques and effective medium.

The tentative lecture-wise plan is as follows.

• Introduction to basic ideas of scattering and Helmholtz equation.

• The direct acoustic obstacle scattering-I.

• The direct acoustic obstacle scattering-II.

• The inverse acoustic obstacle scattering.

• Acoustic waves in inhomogeneous medium.

• Application to inverse medium problems and imaging.

The tutorials will be geared towards further discussions on the topics discussed in the lectures.

Reference:

1. D. Colton, and R. Kress, Inverse acoustic and electromagnetic scattering theory, Third edition. Applied Mathematical Sciences, 93.
Springer, New York, 2013. xiv+405 pp.

Venky Krishnan

TIFR CAM

Microlocal analysis and inverse problems:

In this set of lectures, we review the basics of microlocal analysis and give an introduction to microlocal analysis in imaging and tomography problems.

• Introduction of microlocal analysis

• Applications of microlocal analysis in tensor tomography problems and image reconstruction problems.

• Applications of microlocal analysis in Calderón inverse problem.

References:

    1. V. Krishnan and E. T. Quinto, Microlocal analysis in tomography, In Handbook of Mathematical Methods in Imaging, 2e.

    2. P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J., 34(1):67-112, 2008.

    3. C. Kenig, J. Sjöstrand, G. Uhlmann, The Calderón Problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567-591.

    4. G. Ambartsoumian, R. Felea, V. Krishnan, C. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, J. Funct. Anal., 264 (2013), no. 1, 246–269.