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This is the introductory part of my PhD thesis on inverse problems arising in medical and seismic imaging. The topics include X-ray tomography of scalar and vector fields with partial data, higher order fractional Calder\'on problems,…

We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM…

This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) {\em Unique determination of (phase) projections from diffraction patterns}…

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that…

The lattice Sommerfield model, describing a massive vector gauge field coupled to a light fermion in 2d, is an ideal candidate to verify perturbative conclusions. In contrast with continuum exact solutions, we prove that there is no…

Inversion of potential field data is a central technique of remote sensing in physics, geophysics, neuroscience and medical imaging. In spite of intense research, uniqueness theorems for potential-field inversion are scarce. Applied studies…

It is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are…

We consider the reconstruction of the vertex weight in the discrete Gel'fand's inverse boundary spectral problem for the graph Laplacian. Given the boundary vertex weight and the edge weight of the graph, we develop reconstruction…

This paper concerns time-harmonic inverse source problems with a single far-field pattern in two dimensions, where the source term is compactly supported in an a priori given inhomogeneous background medium. For convex-polygonal source…

We study the inverse problem of unique recovery of a complex-valued scalar function $V:\mathcal M \times \mathbb C\to \mathbb C$, defined over a smooth compact Riemannian manifold $(\mathcal M,g)$ with smooth boundary, given the Dirichlet…

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical…

We construct an effective Lagrangian of heavy and light mesons with $1/M_Q$ correction. The Lagrangian is constructed model independent way by using only the information of the symmetry of QCD. Reparameterisation invariance at the meson…

Using pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential…

We theoretically describe the optical computation of the divergence of a two-dimensional vector field, which is composed by the transverse electric field components of an incident light beam. The divergence is computed in reflection at…

In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calder\'{o}n problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian,…

For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well…

We introduce a technique for recovering a sufficiently smooth function from its ray transform over a wide class of curves in a general region of Euclidean space. The method is based on a complexification of the underlying vector fields…

A single-shot method to reconstruct the transverse self-wakefields acting on a beam, based only on screen images, is introduced. By employing numerical optimization with certain approximations, a relatively high-dimensional parameter space…

We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities…

The late-time nonlinear Lagrangian displacement field is highly correlated with the initial field, so reconstructing it could enable us to extract primordial cosmological information. Our previous work [1] carefully studied the displacement…