Analysis and numerical computation on the transmission problem based on the geometric function theory

Abstract

This thesis is devoted to provide the novel mathematical framework for analyzing phenomena related to the transmission problems. Conducting objects placed in applied electric field induce the perturbation in them. Understanding such phenomena is of great interest in various physical applications such as surface plasmon resonance, cloaking, and stress concentrations. Mathematically, they are formulated in the form of interface problems on which the potential has to satisfy the transmission conditions. The resultant potential field can be represented using layer potential operators. In such problems, it is known that the shape of an underlying object dramatically affects the resulting field perturbation. The contents of this thesis mainly consist of three parts.

In the first part, we construct a family of harmonic basis functions, associated with the shape of the domain, based on the complex geometric function theory. The constructed basis leads to explicit series expansions for single layer potential, double layer potential, and the Neumann-Poincar\'{e} (NP) operators. In particular, the NP operator becomes a doubly infinite, self-adjoint matrix operator, whose entry is given by the Grunsky coefficients corresponding to the domain shape. This matrix formulation, along with the finite section method, provides us with a simple numerical scheme to solve the transmission problem and, also, to compute the spectrum of the NP operator for a smooth domain. The proposed series solution method requires us to know the exterior conformal mapping associated with the domain. To complement, we derive an explicit boundary integral formula, with which the exterior conformal mapping can be numerically computed so that one can apply the method for inclusion of arbitrary shape. We present both numerical and theoretical applications to prove the effectiveness of the proposed method. As numerical demonstrations, we compute exterior conformal mapping coefficients, eigenvalues of the NP operator, and we also solve transmission problems. As a theoretical application, we investigate the decay property of the eigenvalues of the NP operator for arbitrary simply connected domain with $C^{1+p,\alpha}$ boundary in two dimensions, with $p\geq 0,$ $\alpha\in (0,1),p+\alpha>\frac{1}{2}.$ We show that the eigenvalues $\lambda_k$ of the NP operator (ordered in size) satisfy $|\lambda_k|=O(k^{-(p+\alpha)+1/2}).$

In the second part of the thesis, we derive the complete spectral resolution of the NP operator defined on domains generated by two touching disks. There are two types of such domains, each of which has a cusp point, and hence they are not Lipschitz domains. On each example, it is not clear that the NP operator is a continuous linear operator on the space $H^{-1/2}$

accordingly, the analysis based on the Lipschitz property is not directly available. By adopting the M\"{o}bius transform, which maps two touching circles onto two parallel vertical lines, we define a new Hilbert space and extend to this space the NP operator defined on $L^2$ as self-adjoint and continuous linear operator. Then we compute the spectral resolution of the NP operator and characterize the spectral nature of the operator. The NP operator shows only the absolutely continuous spectrum on the whole interval $[-1/2,1/2]$. As an application, we analyze the surface plasmon resonance of a crescent domain and find the blow-up order of resonance.

The last part of this thesis provides an additional topic: 'A Joint Sparse Recovery Framework for Accurate Reconstruction of Inclusions in elastic Media'. A robust algorithm is proposed to reconstruct the spatial distribution of the Lam\'e parameters of multiple inclusions in a homogeneous background elastic body. The proposed method uses a few measurements of the displacement field over a finite collection of boundary points, while it does not require any linearization or iterative update of Green's function. The algorithm relies on an integral representation of the displacement field and sparsity of inclusions distribution. Our method consists of two steps. First, a joint sparse recovery problem has formulated to locate the support on inclusions. Then, a noise robust constrained optimization problem is solved to reconstruct elastic parameters. For numerical simulation, we employed the Multiple Sparse Bayesian Learning (M-SBL) for joint sparse recovery problem, and the Constrained Split Augmented Lagrangian Shrinkage Algorithm (C-SALSA) for the constrained optimization problem.