SEMINAR : Some Theoretical and Computational Aspects of the Inverse Boundary Value Problems for the Wave Equation

 Yang Yang

Michigan State University

15:30pm-16:30pm, 27 July 2023, Shanghai  (光华东主楼1601)

Inverse boundary value problems (IBVPs) concern recovery of parameters in partial differential equations from boundary data. This talk will focus on the IBVPs for the wave equation. On the theoretical side, we consider the IBVP on a cylinder-like Lorentzian manifold for the Lorentzian wave equationwith lower order terms. We show that local knowledge of the Dirichlet-to-Neumann map stably determines the jets of the wave parameters up to gauge transformations, and global knowledge of the map stably determines the lens relation as well as the light ray transforms of the lower order terms. On the computational side, we present a non-iterative algorithm for the acoustic IBVP to reconstruct the sound speed. The algorithm is based on the boundary control method and validated with both full and partial boundary data. The talk is based on the joint work with Plamen Stefanov and Tianyu Yang.

Seminar on Microlocal Analysis and Applications: 

Analytic double fibration transforms

 Mikko Salo

University of Jyväskylä

20pm-21pm, 9 June 2023, Shanghai  (Virtual)

We study integral transforms associated with a double fibration. This class includes various transforms encountered in tomography problems, such as (magnetic) geodesic X-ray transforms, generalized Radon transforms, and (Lorentzian) light ray transforms. If the underlying curve or surface family is real-analytic and a Bolker condition holds, we show that certain analytic singularities of a function can be determined from its transform which is treated as an analytic elliptic Fourier integral operator. This leads to local and global uniqueness results and Helgason type support theorems for these transforms. This is joint work with Marco Mazzucchelli (ENS Lyon) and Leo Tzou (Amsterdam).

Seminar on Microlocal Analysis and Applications: 

The Lorentzian scattering rigidity problem and rigidity of stationary metrics 

 Plamen Stefanov 

Purdue University

9am-10am, 12 May 2023, Shanghai  (Virtual)

We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation known on a lateral boundary.  We show that, under a non-conjugacy assumption,  every defining function r(x,y) of the submanifold of pairs of boundary points which can be connected by a lightlike geodesic plays the role of the boundary distance function in the Riemannian case in the following sense. Its linearization is the light ray transform of tensor fields of order two which are the perturbations of the metric. Next, we study scattering rigidity of stationary metrics in time-space cylinders and show that it can be reduced to boundary rigidity of magnetic systems on the base; a problem studied previously. This implies several scattering rigidity results for stationary metrics.

Seminar on Microlocal Analysis and Applications: 

Resonant forms at zero for dissipative Anosov flow 

 Gabriel Paternain 

University of Cambridge

20pm-21pm, 17 March 2023, Shanghai  (Virtual)

The Ruelle zeta function is a natural function associated with the periods of closed orbits of an Anosov flow, and it is known to have a meromorphic extension to the whole complex plane. The order of vanishing of the Ruelle zeta function at zero is expected to carry interesting topological and dynamical information and can be computed in terms of certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity. In this talk I will explain how to compute these resonant spaces for any transitive Anosov flow in 3D, with particular emphasis in the dissipative case, that is, when the flow does not preserve any absolutely continuous measure. A prototype example is given by the geodesic flow of an affine connection with torsion and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle. This is joint work with Mihajlo Cekić.

Seminar on Microlocal Analysis and Applications: 

Microlocal Analysis and Inverse problems 

 Gunther Uhlmann 

University of Washington, Hong Kong University of Science and Technology

10am-11am, 3 March 2023, Shanghai  (Virtual)

We will discuss some applications of microlocal analysis to inverse problems, in particular the back scattering problem, Calderon's problem and inverse problems for nonlinear equations. 

Seminar on Microlocal Analysis and Applications: 

A new approach to the nonlinear Schrödinger equation 

 Andrew Hassell 

Australian National University

10am-11am, 4 Novemer 2022, Shanghai  (Virtual)

With collaborators Jesse Gell-Redman and Sean Gomes, we have begun to set up an entirely new framework for tackling the linear and nonlinear Schrödinger equation. I will describe this setup and explain why I believe it is a more powerful framework than existing approaches for studying nonlinear scattering and soliton dynamics. 

Seminar on Microlocal Analysis and Applications: 

Fredholm approach to the Schrödinger equation 

 Jesse Gell-Redman 

University of Melbourne

10am-11am, 21 October 2022, Shanghai  (Virtual)

We discuss a new approach, inspired by work of Hintz and Vasy, to solving the Schrödinger equation $(i \partial_t - \Delta) u = f$ using the Fredholm method. Specifically, we use 'parabolic' pseudodifferential operators (reflecting the parabolic nature of the symbol of $P = i \partial_t - \Delta$) to obtain families of function spaces $X, Y$ for which $P : X \to Y$ is an isomorphism. The spaces further allow us to read off precise regularity and decay information about $u$ directly from that of $f$. We discuss applications to the nonlinear Schrödinger equation, and extensions of this method to equations with compact spatial perturbations, such as smooth decaying potential functions, using the N-body calculus of Vasy. This includes joint work with Dean Baskin, Sean Gomes, and Andrew Hassell. 

Seminar on Microlocal Analysis and Applications: 

Scattering theory in models for Conformal Field Theory

 Colin Guillarmou 

Université Paris-Saclay

20pm-21pm, 7 October 2022, Shanghai  (Virtual)

I will explain a famous model of 2 dimensional Conformal FieldTheory called the Liouville CFT and discuss several aspects relatedto it, including the scattering analysis of its Hamiltonian. This is based on joint work with Kupiainen, Rhodes and Vargas. 

Seminar on Microlocal Analysis and Applications: 

The Feynman propagator in some model singular settings

 Dean Baskin 

Texas A&M University

10am-11am, 7 October 2022, Shanghai  (Virtual)

In this talk I will describe the existence and asymptotic properties of the Feynman propagator in three model singular settings: the scalar wave equation on cones, the scalar wave equation on Minkowski space with an inverse square potential, and the massless Dirac equation in 3 dimensions coupled to a Coulomb potential. The proof combines techniques of Gell-Redman–Haber–Vasy as well as prior work with Booth, Gell-Redman, Marzuola, Vasy, and Wunsch. One novelty of the proof is that it does not rely on Wick rotation (though a shadow of it survives in some special function analysis at infinity).

Seminar on Microlocal Analysis and Applications: 

Semiclassical oscillating functions of isotropic type and their applications 

Zuoqing Wang 

China University of Science and Technology

10am-11am, 8 April 2022, Shanghai  (Virtual)

Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semiclassical analysis. In this talk I will describe how to associate classes of semiclassical oscillating functions to isotropic submanifolds of phase space, and show that these classes are invariant under the action of Fourier integral operators (modulo the usual clean intersection condition). Some sub-classes (coherent states, Hermite states) and applications will also be discussed. This is based on joint works with V. Guillemin (MIT) and A. Uribe (U. Michigan).

Seminar on Microlocal Analysis and Applications: 

Semiclassical analysis and the convergence of the finite element method

Jared Wunsch 

Northwestern University

10am-11am, 11 March 2022, Shanghai  (Virtual)

An important problem in numerical analysis is the solution of the Helmholtz equation in exterior domains, in variable media; this models the scattering of time-harmonic waves.  The Finite Element Method (FEM) is a flexible and powerful tool for obtaining numerical solutions, but difficulties are known to arise in obtaining convergence estimates for FEM that are uniform as the frequency of waves tends to infinity.  I will describe some recent joint work with David Lafontaine and Euan Spence that yields new convergence results for the FEM which are uniform in the frequency parameter.  The essential new tools come from semiclassical microlocal analysis.  No knowledge of either FEM or semiclassical analysis will be assumed in the talk, however.

Joint Fudan-RICAM Seminar on Inverse Problems: 

Statistical aspects of the non-abelian X-ray transform 

Gabriel Paternain 

University of Cambridge

7pm-8pm, 3 March 2022, Shanghai (Virtual)

7pm-8pm, 8 March 2022, Shanghai (Virtual)

I will discuss some recent developments providing theoretical guarantees for Bayesian inversion with Gaussian priors for a class of nonlinear inverse problems. These developments will be illustrated in a specific inverse problem: the non-abelian X-ray transform.This problem underpins new experiments designed to measure magnetic fields inside materials by shooting them with neutron beams from different directions, like in a CT scan. The theoretical guarantees will be of two types: first I will discuss consistency and then a suitable Bernstein von-Mises theorem. This is joint work with F. Monard and R. Nickl. 

SEMINAR : Space time finite element methods for inverse and control problems subject to the wave equation 

Lauri Oksanen 

University of Helsinki

8pm-9pm, 15 November 2021, Shanghai (Virtual)

There is a well-known duality between inverse initial source problems and control problems for the wave equation, and analysis of both these boils down to so-called observability estimates. I will present recent results on numerical analysis of these problems. The inverse initial source problem gives a model of photoacoustic imaging, while, in addition to being interesting in its own right, the control problem arises as a subproblem in several solution methods for non-linear inverse problems, for instance, in the Boundary Control method. The talk is based on joint work with Erik Burman, Ali Feizmohammadi and Arnaud Münch.