Deterministic and statistical methods for inverse problems with partial data

Inverse problems with partial data have many applications in science and engineering. They are more challenging than the complete data cases since the lack of data increases ill-posedness and nonlinearity. The use of only deterministic or statistical methods might not provide satisfactory results. We propose to combine the deterministic and statistical methods to treat such inverse problems. The thesis is organized as follows. In Chapter 1, we briefly introduce the inverse problems and their applications. The classical deterministic methods and Bayesian inversion are discussed. The chapter is concluded with a summary of contributions. Chapter 2 considers the reconstruction of the unknown acoustic sources using partial data. First, we extend the direct sampling method to approximate the source locations. Second, the inverse problem is formulated as a statistical inference problem using the Bayes\u27 formula. The source locations obtained in the first step are coded in the priors. Then a Metropolis-Hastings algorithm is used to explore the posterior density. In Chapter 3, a two-step deterministic-statistical approach is proposed to recover the trajectories of moving sources using partial measured data. In the first step, an approximate direct sampling method is developed to obtain the locations of the sources at different times. Such information is coded in the priors, which is critical for the success of the Bayesian method in the second step. The combined approach inherits the merits of the deterministic method and Bayesian inversion as demonstrated by the numerical examples. Chapter 4 studies the reconstruction of Stekloff eigenvalues and the index of refraction of an inhomogeneous medium from Cauchy data. The inverse spectrum problem of Stekloff eigenvalues is investigated by the reciprocity gap method. Then a Bayesian approach is proposed to estimate the index of refraction using a few reconstructed eigenvalues. In Chapter 5, we consider the inverse spectral problem to determine the material properties given a few transmission eigenvalues. The lack of theoretical results motivates us to propose a Bayesian approach to formulate a statistical inference problem. The MCMC algorithm is used to explore the posterior density. Due to the non-uniqueness nature of the problem, we adopt the local conditional means (LCM) to characterize the posterior density function

On an interior Calder\'{o}n operator and a related Steklov eigenproblem for Maxwell's equations

We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calder\'{o}n operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. With an approach similar to that developed by Auchmuty for the Laplace operator, we provide natural spectral representations for the appropriate trace spaces, for the Calder\'{o}n operator itself and for the solutions of the corresponding boundary value problems subject to electric or magnetic boundary conditions on a cavity.Comment: Submitted for publication to Siam Journal on Mathematical Analysis on 21 March 2019, revised on 12 May 2020, accepted for publication on 16 July 202

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges