Schedule for: 23w5036 - Inverse Problems and Nonlinearity

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk I will explain recent results in joint work with Evangelie Zachos and Qiuye Jia on the geodesic X-ray transform on asymptotically conic spaces, asymptotic to the `large’ end of a cone, both on functions and on symmetric 2-tensors. This includes perturbations of Euclidean space and certain kinds of conjugate points are allowed. The key analytic tool, beyond the artificial boundary approach introduced by Uhlmann and the speaker, is the introduction of a new pseudodifferential operator algebra, the 1-cusp algebra, and its semiclassical version.

I will first review known results on the X-ray transform on the Euclidean disk, in particular the SVD of self-adjoint realizations on weighted spaces, and its relations with distinguished differential operators of Keldysh type. I will then discuss more facts about those differential operators, in particular what kind of Sobolev spaces and Green's identities can be derived for them. This also helps put in perspective which of their self-adjoint realizations is related to the X-ray transform. Finally, I will discuss ongoing work on the hyperbolic X-ray transform, including range characterization and relations with special differential operators. Joint works with Mishra-Zou; Zou-Eptaminitakis-Zou.

Due to beam-hardening effects, metal objects in X-ray CT often produce streaking artefacts which cause degradation in image reconstruction. It is known that the nature of the phenomena is nonlinear. An outstanding inverse problem is to identify the nonlinearity which is crucial for reduction of the artefacts. In this talk, we show how to use microlocal techniques to extract information of the nonlinearity from the artefacts. Our analysis also reveals the interesting connection between artefacts and the geometry of metal objects.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will briefly describe the main ideas that go into proving that two Anosov metrics on a surface having the same marked length spectrum are isometric via an isometry isotopic to the identity. This is joint work with Colin Guillarmou and Thibault Lefeuvre.

Numerical studies of random plane waves, functions $$u=\sum_{j}c_{j}e^{\frac{i}{h}\langle x,\xi_{j}\rangle}$$ where the coefficients $c_{j}$ are chosen ``at random'', have detected an apparent filament structure. The waves appear enhanced along straight lines. There has been significant difference of opinion as to whether this structure is indeed a failure to equidistribute, numerical artefact or an illusion created by the human desire to see patterns. In this talk I will present some recent results that go some way to answering the question. We study the behaviour of a random variable $G(x,\xi)=||P_{(x,\xi)}u||_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ and show that $G(x,\xi)$ fails to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.

In this talk I will introduce several geometric data sets related to the distance function on a compact Riemannian manifold with boundary. For each of these data sets I will provide geometric conditions which are sufficient to determine the isometry class of the manifold producing the data. Also, some stability results for simple Riemannian manifolds will be considered. This talk is based on joint works with Maarten V. de Hoop, Joonas Ilmavirta, Matti Lassas, Boya Liu, and Ella Pavlechko.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will provide a constructive connection between the classical and fractional Calder\'on problems by making use of the Caffarelli-Silvestre extension. As a consequence, it will be possible to prove that uniqueness results for the classical Calder\'on problem lead to (partial data) uniqueness results for the fractional Calder\'on problem. I will also discuss obstructions to inverting this relation. This is based on joint work with G. Covi, T. Ghosh and G. Uhlmann.

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of smooth, compactly supported perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calder \'{o}n problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior of the double null cone emanating from a point. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed. The talk is based on joint works with Spyros Alexakis and Lauri Oksanen.

Many areas of interest within the Earth are anisotropic, meaning that the speed of sound is different in different directions. It turns out that pressure waves are far better behaved than shear waves, but fortunately the different polarizations are coupled together through algebraic geometry. I will explain the surprising power of algebraic geometry in the study of anisotropic inverse problems.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the first part of the talk we shall discuss some recent progress for inverse boundary problems for nonlinear elliptic PDE. Our focus will be on inverse problems for isotropic quasilinear conductivity equations, as well as nonlinear Schrodinger and magnetic Schrodinger equations. In particular, we shall see that the presence of nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart is open. In the second part of the talk, we shall discuss the fractional anisotropic Calderon problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we show that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderon problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher. This talk is based on joint works with Catalin Carstea, Ali Feizmohammadi, Tuhin Ghosh, Yavar Kian, and Gunther Uhlmann.

We study an inverse problem of determining the time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely.

I will discuss joint work with Gunther Uhlmann regarding the anisotropic fractional Calderon problem for Dirac operators on closed manifolds; these give fractional analogues of Maxwell systems. Namely we show that knowledge of the source-to-solution map of the fractional Dirac operator, for data sources sup- ported in an arbitrary open set in a Riemannian manifold allows one to reconstruct the Riemannian manifold, its Clifford module structure, and the associated connection (up to an isometry fixing the initial set). Time permitting I will discuss on-going work regarding Caffarelli-Silvestre type extensions for fractional systems.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the inverse problem of reconstructing the linear and nonlinear susceptibilities of nonlinearities of Kerr type. This problem is studied within the framework of the inverse Born series, an approach that so far has only been applied to inverse problems for linear PDEs. This is joint work with N. Defellipis and S. Moskow.

n this talk, we discuss a microlocal approach to scattering for the nonlinear Schrödinger equation with a compactly supported potential and a compactly supported perturbation of the Euclidean metric. Inspired by work of Hintz and Vasy, we realise the Schrödinger operator $P=D_t+\Delta_g+V$ as a Fredholm (in fact invertible) operator between suitably defined anisotropic Sobolev spaces and leverage this to solve NLS with prescribed asymptotic incoming data (the so-called ``final state problem"). This talk is based on joint works with Jesse Gell-Redman and Andrew Hassell.

In this talk, we will discuss the uniqueness and stability in determining a time-dependent nonlinear coefficient in the Schrodinger equation from the boundary Dirichlet-to-Neumann map. We will present two results: local uniqueness of the coefficient at the points where certain type of geometric optics (GO) solutions can reach; and a stability estimate based on the unique continuation property for the linear equation.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Possible options: **1. Seasonal wildflower blooms:** https://www.alltrails.com/trail/canada/alberta/sunshine-meadows--3, https://www.banffsunshinemeadows.com/hours-directions. (EDIT: looks like this trail is only accessible by gondola, cost $65. However last bus is at 5:45... might not fit time-wise in an afternoon; will inform wed morning. EDIT2: leaving Lloyd Hall at 1pm sharp on wed) **2. Hike to waterfall:** https://www.alltrails.com/trail/canada/alberta/sundance-canyon-trail--2 **3. Hoodoo rock formation hike: ** https://www.alltrails.com/trail/canada/alberta/hoodoos-from-bow-falls-trail **4. Lake Louise and/or Moraine Lake sightseeing and walk:** https://www.banfflakelouise.com/explore-the-park/lake-louise-shuttle, https://www.banfflakelouise.com/explore-the-park/moraine-lake-shuttle **5. Bow river hike:**

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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. These transforms also arise in various nonlinear inverse problems via suitable reductions. 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).

The photoacoustic tomography is a biomedical imaging modality whose purpose is to determine absorption properties of a biological medium in order to identify tissues and check if they are safe or not. This modality has many applications including detection of tumors diagnostic of breast cancer. In this talk, we will consider the acoustic inversion, which is one of the steps of the photoacoustic tomography problem, when the sound speed is unknown. Mathematically, our problem can be stated as the simultaneous determination of a coefficient and an initial data of an initial value problem associated with wave equations from measurement on a surface of the solution. This talk is based on a joint work with Gunther Uhlmann.

The topic of this talk is the scattering problem for non-linear wave equations, namely the problem of finding global solutions backwards in time from asymptotic data. For wave equations satisfying the null condition or weak null condition the scattering problem from infinity is non-trivial due to slow interior decay, and the presence of logarithmic terms in the asymptotic expansion. I will discuss the importance of homogeneous solutions of degree -1 and -2 in this context, both in the interior, and exterior of the lightcone. I will present several recent results of my joint work with Hans Lindblad (Johns Hopkins), on the construction of scattering solutions with non-decaying radiation fields, which satisfy novel matching conditions to homogeneous asymptotics in the exterior and interior. The proof invokes the Funk transform on the sphere.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will discuss the determination of coefficients in time-dependent nonlinear transport equations. We consider both cases of time-independent and time-dependent coefficients.

Nonlinear ultrasound waves are widely used in medical imaging. The propagation of high-intensity ultrasound waves can be modeled by nonlinear wave equations. In this talk, we consider an inverse problem for a nonlinear wave equation with a general nonlinearity. We show the Dirichlet-to-Neumann map (DN map) determines the nonlinearity. We also consider inverse problems for nonlinear wave equations with a general nonlinear term and a damping term.

In the previous paper (Inverse Problems, 32, 015010, 2016), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1-2 trading days ahead of the present one. This new technique uses the Black-Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black-Scholes equation was solved in the positive direction of the time variable, This ill-posed initial boundary value problem was solved by the so-called Quasi-Reversibility Method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. In the current paper, we use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.