
Mikyoung LimKorea Advanced Institute of Science and Technology | KAIST · Department of Mathematical Sciences
Mikyoung Lim
Doctor of Philosophy
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113
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Publications (113)
We focus on designing and solving the neutral inclusion problem via neural networks. The neutral inclusion problem has a long history in the theory of composite materials, and it is exceedingly challenging to identify the precise condition that precipitates a general-shaped inclusion into a neutral inclusion. Physics-informed neural networks (PINNs...
Quantum information processing comprises physical processes, which obey the quantum speed limit (QSL): high speed requires strong driving. Single-qubit gates using Rabi oscillation, which is based on the rotating wave approximation (RWA), satisfy this bound in the form that the gate time $T$ is inversely proportional to the Rabi frequency $\Omega$,...
We investigate the inverse scattering problem for tracking the location and orientation of a moving scatterer using a single incident field. We solve the problem by adopting the optimization approach with the objective function defined by the discrepancy in far-field data. We rigorously derive formulas for the far-field data under translation and r...
We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet condition on the inner boundary sphere. The first eigenfunction admits a Fourier--Gegenbauer series expansion via the bispherical co...
We investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call as the generalized polarization tensors...
We analyze the spectrum of the Neumann-Poincar\'e (NP) operator for a doubly connected domain lying between two level curves defined by a conformal mapping, where the inner boundary of the domain is of general shape. The analysis relies on an infinite-matrix representation of the NP operator involving the Grunsky coefficients of the conformal mappi...
We investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call the generalized polarization tensors (GP...
We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problem...
In this study, we propose a sampling-type algorithm for a real-time identification of a set of short, linear perfectly conducting cracks in a two-dimensional bistatic measurement configuration. The indicator function is defined based on the asymptotic formula of the far-field pattern of the scattered field due to cracks. To clarify the applicabilit...
We present monostatic sampling methods for limited-aperture scattering problems in two dimensions. The direct sampling method (DSM) is well known to provide a robust, stable, and fast numerical scheme for imaging inhomogeneities from multistatic measurements even with only one or two incident fields. However, in practical applications, monostatic m...
This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover a homogeneous inclusion from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements. The key result is an inversion formula for conformal mapping coefficients associated with the inclusion...
A conductivity inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from multistatic measurements. As a modification of GPTs, the Faber polynomial polarizat...
We present monostatic sampling methods for limited-aperture scattering problems in two dimensions. The direct sampling method (DSM) is well known to provide a robust, stable, and fast numerical scheme for imaging inhomogeneities from multistatic measurements even with only one or two incident fields. However, in practical applications, monostatic m...
In this paper, we investigate the first Steklov–Dirichlet eigenvalue on eccentric annuli. The main geometric parameter is the distance t between the centers of the inner and outer boundaries of an annulus. We first show the differentiability of the eigenvalue in t and obtain an integral expression for the derivative value in two and higher dimensio...
The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investi...
We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with a rigid inclusion. Our analysis is based on the boundary integral formulation of the elastostatic problem and geometric function theory. Specifically, we use the coordinate system provided by the exterior conformal mapping of the inclusion to...
According to the Eshelby conjecture, an ellipse or ellipsoid is the only shape that induces an interior uniform strain under a uniform far-field loading. We extend the Eshelby conjecture to domains of general shape for anti-plane elasticity. Specifically, we show that for each positive integer N, an inclusion induces an interior uniform strain unde...
When seeking a solution to the interface problem, the potential theory to represent solution has been widely used. As the solution representation of the interface problem is only well-known for domains with simple inclusion, such as a disk, conformal mapping is often used to transform the arbitrary inclusion to a manageable inclusion to utilize the...
The Neumann-Poincar\'{e} operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the Neumann-Poincar\'{e} operator was developed in two dimensions based on geometric function theory. In this paper...
We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with a rigid isotropic inclusion. Our analysis is based on the boundary integral formulation of the elastostatic problem and geometric function theory. Specifically, we use the coordinate system provided by the exterior conformal mapping of the in...
In this paper, we investigate the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli with respect to the distance, namely $t$, between the centers of the inner and outer boundaries of annulus. We first show the differentiability of the eigenvalue in $t$ and obtain an integral expression for the derivative value in two and hi...
An elastic or electrical inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion, whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from the exterior measurements. As a modification of GPTs, we recently introd...
We consider the decay property of the eigenvalues of the Neumann-Poincaré operator in two dimensions. As is well known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having C 1 , α C^{1,\alpha } boundary with α ∈ ( 0 , 1 ) \alpha \in (0,1) . We show that the eigenvalues λ k \lambda...
In this paper we analyze the gradient blow-up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core-shell geometry. Assuming that the core and shell have circular boundaries that are nearly touching, we derive an asymptotic formula for the solution in terms of the single and double layer p...
We consider a boundary value problem of the anti-plane elasticity in a domain containing an inclusion which is nearly touching to the domain's boundary. We assume that the domain and the inclusion are disks. By using the boundary integral formulation for the interface problem and adopting the bipolar coordinates, we derive the asymptotic formulas w...
We consider the shape reconstruction of a conductivity inclusion in two dimensions. We use the concept of Faber polynomials Polarization Tensors (FPTs) introduced in \cite{choi:2018:GME} to derive an exact shape recovery formula for an inclusion with the extreme conductivity. This shape can be a good initial guess in the shape recovery optimization...
We investigate the decay property of the eigenvalues of the Neumann-Poincar\'{e} operator in two dimensions. As is well-known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having $C^{1,\alpha}$ boundary with $\alpha\in (0,1)$. In this paper, we show that the eigenvalue $\lambda_k$...
We consider the Neumann Poincare operator on domains generated by two touching disks. There can be two types of such domains, each of which has a cusp point at the touching point of two circles. For each domain we define a Hilbert space on which the Neumann Poincare operator is self-adjoint and continuous. Then we computed the complete spectral res...
Neutral coated inclusions do not perturb a uniform electric field when they are inserted into a homogeneous matrix. While the coated inclusions can have a variety of shapes to be neutral only to a single field, they should be either concentric ellipses or confocal ellipsoids to be neutral to all directions of uniform fields. In this paper we constr...
The field perturbation induced by an elastic or electrical inclusion admits a multipole expansion in terms of the outgoing potential functions. In the classical expansion, basis functions are defined independently of the inclusion. In this paper, we introduce the new concept of the geometric multipole expansion for the two-dimensional conductivity...
We consider the elastic field perturbation due to an inclusion for the anti-plane elasticity. The Eshelby conjecture asserts that if an inclusion induces a uniform field on the inclusion for probing by a uniform loading, then the inclusion should be an ellipse. In this paper we extend the Eshelby conjecture to domains of general shape. In particula...
We derive analytically series representation for the Neumann-Poincare operator using the exterior conformal mapping for general shape domains. We derive the formula by using the Faber polynomial basis for arbitrary Lipschitz domains. With the proposed method we can approximate the spectrum of the smooth domain by finding the eigenvalues of matrixes...
In this paper, we analyze the gradient blow-up of the solution to the conductivity equation in two dimensions in the presence of an inclusion with eccentric core-shell geometry. We derive an asymptotic approximation for the solution in terms of the single and double layer potentials with line charges, assuming that the core and shell have circular...
We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coefficients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the dom...
We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coefficients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the dom...
The purpose of this paper is to investigate the spectral nature of the Neumann–Poincaré operator on the intersecting disks, which is a domain with the Lipschitz boundary. The complete spectral resolution of the operator is derived, which shows, in particular, that it admits only the absolutely continuous spectrum; no singularly continuous spectrum...
A robust algorithm is proposed to reconstruct the spatial support and the Lam\'e parameters of multiple inclusions in a homogeneous background elastic material using a few measurements of the displacement field over a finite collection of boundary points. The algorithm does not require any linearization or iterative update of Green's function but s...
We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalu...
We consider the plane elasticity problem for two circular holes. When two holes are close to touching, the stress concentration happens in the narrow gap region. In this paper, we characterize the stress singularity between the two holes by an explicit function. A new method of a singular asymptotic expansion for the Fourier series with slowing dec...
A cylindrical plasmonic structure with a concentric core exhibits an anomalous localized resonance which results in cloaking effects. Here we show that if the structure has an eccentric core, a new kind of shielding effect can happen. In contrast to conventional shielding devices, our proposed structure can block the effect of external electrical s...
A robust algorithm is proposed to reconstruct the spatial support and the Lam\'e parameters of multiple inclusions in a homogeneous background elastic material using a few measurements of the displacement field over a finite collection of boundary points. The algorithm does not require any linearization or iterative update of Green's function but s...
We study spectral properties of the Neumann-Poincar\'e operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenva...
This paper considers an electrical impedance tomography (EIT) problem to reconstruct multiple small anomalies from boundary measurements. The inverse problem of EIT is a severely ill-posed nonlinear inverse problem so that the conventional methods usually require linear approximation or iterative procedure. In this paper, we propose a non-iterative...
The cylindrical plasmonic structure with concentric core exhibits the
anomalous localized resonance and the resulting cloaking effect. It turns out
that the plasmonic structure of eccentric core also has the anomalous resonant
behavior. Differently from the concentric case, the eccentric superlens has the
shielding effect as well as the cloaking ef...
Line-scanning microscopes are often used to overcome the limited scanning speed of conventional point-scanning confocal microscopes, at the cost, however, of spatial resolution. In this paper, we present a dual-beam fluorescence line-scanning microscope that can restore the original confocal resolution. This microscope forms two orthogonal line foc...
We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradien...
We consider the enhancement of electric field in the presence of two
perfectly conducting spheres accepting different radii. When the two spheres
get closer, the electric field have a much larger magnitude compared to the
external field in the small gap region between the two spheres. The enhanced
field can be arbitrary large with the generic blow-...
The purpose of this paper is to propose a non-iterative method for the inverse conductivity problem of recovering multiple small anomalies from the boundary measurements. When small anomalies are buried in a conducting object, the electric potential values inside the object can be expressed by integrals of densities with a common sparse support on...
We present a new systematic method to compute the Riemann mapping from the outside of the unit disc to the outside of a simply connected domain. We derive explicit relations between the coefficients of the Riemann mapping and the generalized polarization tensors associated with the domain. Because the generalized polarization tensors can be compute...
We consider the conductivity problem in the presence of adjacent circular
inclusions having arbitrary constant conductivity. When two inclusions get
closer and their conductivities degenerate to zero or infinity, the gradient of
the solution can be arbitrary large. We characterize the gradient blow-up by
deriving an explicit formula for the singula...
In this chapter we consider the nonlinear optimization problem for reconstructing the shape of an extended target from multistatic data. Because of the nonlinearity of the problem, iterative algorithms have to be introduced.
This chapter aims to reconstruct GPTs from MSR measurements. We consider the effect of the presence of measurement noise in the MSR on the reconstruction of the GPTs of a small conductivity inclusion. Given a signal-to-noise ratio, we determine the statistical stability in the reconstruction of the GPTs, and show that such an inverse problem is exp...
In this chapter we apply the accurate asymptotic formulas derived in Chap. 3 for the purpose of identifying the location and certain properties of the inclusions. We single out simple fundamental localization algorithms.
The aim of this chapter is to introduce the concept of generalized polarization tensors (GPTs). The GPTs are the basic building blocks for the asymptotic expansions of the boundary voltage perturbations due to the presence of small conductivity inclusions inside a conductor. The GPTs contain important geometrical information on the inclusion.
In multistatic wave imaging, waves are emitted by a set of sources and they are recorded by a set of sensors in order to probe an unknown medium. The responses between each pair of source and receiver are collected and assembled in the form of the multi-static response (MSR) matrix. The indices of the MSR matrix are the index of the source and the...
In this chapter we analyze the structure of the MSR matrices, using the multipolar expansions (4.46) and (5.8). We show the linear dependence of the multistatic data with respect to the GPTs or the FDPTs in which geometrical features of the target are encoded in a nonlinear way. As will be shown later, a least-squares approach will allow an accurat...
This chapter introduces the notion of higher-order frequency-dependent polarization tensors (FDPTs). Multipolar asymptotic expansions for wave scattered by a target of characteristic size smaller than the wavelength can be written in terms of high-order derivatives of the fundamental solution to the Helmholtz equation and high-order FDPTs. This key...
The inclusion detection, localization, and reconstruction algorithms described in this book rely on asymptotic expansions of the fields when the medium contains inclusions of small volume. Such asymptotics will be investigated in the cases of the conductivity and the Helmholtz equations. As it will be shown in the subsequent chapters, a remarkable...
In their most general forms imaging problems are severely ill-posed and nonlinear. These are the main obstacles to find non-iterative reconstruction algorithms. In this chapter we use structural information about the profile of the material property in order to determine specific features about them with a satisfactory resolution.
This chapter reviews some mathematical and statistical concepts essential for understanding multistatic imaging principles. We first review commonly used special functions, function spaces, and an integral transform: the Fourier transform. We then collect basic facts about the Moore-Penrose generalized inverse, singular value decomposition, and com...
The problem addressed in this chapter is to detect and localize point reflectors or small inclusions embedded in a medium from MSR measurements. We use random matrix theory tools and the results of Chap. 6 to study these problems in the presence of measurement noise. The measurement noise can be modeled by an additive complex Gaussian matrix with z...
In this chapter we first recall the notion of contracted GPTs. Then we show that the CGPTs have some nice properties, such as simple rotation and translation formulas, simple relation with shape symmetry, etc. More importantly, we derive new invariants for the CGPTs. Based on those invariants, we develop a dictionary matching algorithm. We suppose...
The aim of this chapter is to provide classical techniques for solving inverse extended source problems. From time-domain or broadband measurements, time-reversal techniques yield direct reconstruction of the source. In the frequency domain, from measurements at a single frequency or bandlimited measurements, diffraction tomography can be used to r...
Our aim in this chapter is to reconstruct shape perturbations of an extended inclusion from MSR measurements. As for small volume inclusions, we present direct imaging algorithms and analyze their resolution and stability. Our algorithms are based on an asymptotic expansion for the perturbations in the data due to small shape perturbations. A conce...
The concept of GPTs is used in Chaps. 4 and 11 for resolved imaging, identification, and tracking of small targets. In this chapter, we use the concept of GPTs for constructing cloaking devices.
This chapter provides Matlab codes for the main algorithms described in this book.
This chapter presents numerical illustrations using the codes described in Chap. 17 in order to highlight the performance and show the limitations of our numerical approaches for multistatic imaging.
The aim of this chapter is to review a general method based on the potential theory to study cloaking due to anomalous resonance.
When two perfectly conducting inclusions are located closely to each other,
the electric field concentrates in a narrow region in between two inclusions,
and becomes arbitrarily large as the distance between two inclusions tends to
zero. The purpose of this paper is to derive an asymptotic formula of the
concentration which completely characterizes...
The aim of this paper is to propose a new optimal control formulation for recovering an extended inclusion from boundary measurements. Our approach provides an optimal representation of the shape of the inclusion. It guarantees local Lipschitz stability for the reconstruction problem. Some numerical experiments are performed to demonstrate the vali...
In this paper, we consider near cloaking for the full Maxwell equations. We
extend the recent results, where the quasi-static limit case and the Helmholtz
equation are considered, to electromagnetic scattering problems. We construct
very effective near cloaking structures for the electromagnetic scattering
problem at a fixed frequency. These new st...
With each C 2 -domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain and its material parameter. They generalize the concept of Polarization Tensor (PT), which can be seen as the first-order GPT. It is known...
In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic regimes. We consider both the Born approximation in the non...
With each domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper [9], a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this pa...
The aim of this paper is to extend the method of improving cloaking
structures in the conductivity to scattering problems. We construct very
effective near-cloaking structures for the scattering problem at a fixed
frequency. These new structures are, before using the transformation optics,
layered structures and are designed so that their first sca...
When inclusions with extreme conductivity (insulator or perfect conductor)
are closely located, the gradient of the solution to the conductivity equation
can be arbitrarily large. And computation of the gradient is extremely
challenging due to its nature of blow-up in a narrow region in between
inclusions. In this paper we characterize explicitly t...
The aim of this paper is to provide an original method of constructing very
effective near-cloaking structures for the conductivity problem. These new
structures are such that their first Generalized Polarization Tensors vanish.
We show that this in particular significantly enhances the cloaking effect. We
then present some numerical examples of Ge...
In stiff fiber-reinforced composites, it has been known that the shear stress increases at the rate of as the distance ϵ between adjacent fibers approaches 0. This paper reveals a strong influence of a combination of a triple fiber, as well as the distance between a pair of fibers, on the blow-up so that the stress concentration can be significantl...
The goal of this paper is to illustrate the efficiency and the stability of the near-cloaking structures proposed in [4] and [5]. These new structures are, before using transformation optics, layered structures and are designed so that their first contracted generalized polarization tensors (in the quasi-static limit) or scattering coefficients (in...
We develop a non-iterative method to address the inverse problem of identifying a collection of disjoint internal corrosive parts of small Hausdorff measures in pipelines from exterior ultrasound boundary measurements. The method is based on an asymp- totic expansion of the effect of the corrosion in terms of the size of the corrosive parts. We num...
In this paper we develop an iterative approach for reconstructing fine shape details of an inclusion using higher-order EMTs. Starting from the integral equation formulation, we derive an asymptotic formula for the perturbation in the EMTs that are due to small changes in the interface of the inclusion. Based on this formula, we propose an optimiza...
In order to reconstruct small changes in the interface of an elastic inclusion from modal measurements, we rigorously derive an asymptotic formula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the eigenvalues due to interface changes of the inclusion. Based on this (dual) formula we propos...
In this paper, we propose an original and promising optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmission problem for the Laplacian. Based on a rigorous asymptotic analysis, we derive an asymptotic formula for the perturbations in t...
We provide a rigorous derivation of new complete asymptotic ex- pansions for eigenvalues of the Laplacian in domains with small inclusions. The inclusions, somewhat apart from or nearly touching the boundary, are of arbitrary shape and arbitrary conductivity contrast vis-`a-vis the background domain, with the limiting perfectly conducting inclusion...
With each Lipschitz domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain and its material parameter. They generalize the concept of Polarization Tensor (PT), which can be seen as the first-order GPT. It is...
We derive high-order terms in the asymptotic expansions of the boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with C2-boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for C1-perturbation...
When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution to the conductivity equation blows up in general. In this paper, we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form: one of them has a bounded gradient and the gradient of the...
In stiff fiber-reinforced material, the high shear stress concentration occurs in the narrow region between fibers. With the addition of a small geometric change in cross-section, such as a thin fiber or a overhanging part of fiber, the concentration is significantly increased. This paper presents mathematical analysis to explain the rapidly increa...
we consider the problem of determining the boundary perturbations of an object from far-field electric or acoustic measurements. Assuming that the unknown object boundary is a small perturbation of a circle, we develop a linearized relation between the far-field data that result from fixed Dirichlet boundary conditions, entering as parameters, and...
We consider the problem of determining the boundary perturbations of an object from far-field electric or acoustic measurements. Assuming that the unknown object boundary is a small perturbation of a circle, we develop a linearized relation between the far-field data that result from fixed Dirichlet boundary conditions, entering as parameters, and...
The electric field increases toward infinity in the narrow region between closely adjacent perfect conductors as they approach each other. Much attention has been devoted to the blow-up estimate, especially in two dimensions, for the practical relevance to high stress concentration in fiber-reinforced elastic composites. In this paper, we establish...
We prove that, in three-dimensional case, the electric field does not blow-up the presence of closely adjacent spherical perfect conductors even though the separation distance between the conducting inclusions approaches zero.
The electric field for an infinite array of conducting nanosized objects in two-dimensional space has been calculated. The mirror symmetry for this physical problem has been introduced. By taking into account this symmetry, we transform the original problem into an infinite two-dimensional array of nanosized objects with the same solution. The elec...
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