arXiv subjects

Chiu-Yen Kao

Publications and source records attributed to Chiu-Yen Kao.

17 recordsLinked to original sources

arXiv2026

Extremal Steklov-Neumann Eigenvalues

Let $Ω$ be a bounded open planar domain with smooth connected boundary, $Γ$, that has been partitioned into two disjoint components, $Γ= Γ_S \sqcup Γ_N$. We consider the Steklov-Neumann eigenproblem on $Ω$, where a harmonic function is sought that satisfies the Steklov boundary condition on $Γ_S$ and the Neumann boundary condition on $Γ_N$. We pose the extremal eigenvalue problems (EEPs) of minimizing/maximizing the $k$-th non-trivial Steklov-Neumann eigenvalue among boundary partitions of prescribed measure. We formulate a relaxation of these EEPs in terms of weighted Steklov eigenvalues where an $L^\infty(Γ)$ density replaces the boundary partition. For these relaxed EEPs, we establish existence and prove optimality conditions. We also prove a homogenization result that allows us to use solutions to the relaxed EEPs to infer properties of solutions to the original EEPs. For a disk, we provide numerical and asymptotic evidence that the minimizing arrangement of $Γ_S\sqcup Γ_N$ for the $k$-th eigenvalue consists of $k+1$ connected components that are symmetrically arranged on the boundary. For a disk, for $k = 1$, the constant density is a maximizer for the relaxed problem; we also provide numerical and asymptotic evidence that for $k\ge 2$, the maximizing density for the relaxed problem is a non-trivial function; a sequence of rapidly oscillating Steklov/Neumann boundary conditions approach the supremum value.

math.OC↗

arXiv2026

An accelerated rearrangement method for two-phase composite optimization

We propose and analyze an Accelerated Rearrangement Method (ARM) for solving a class of nonconvex optimization problems involving two-phase composites. These problems include maximizing the (work) energy of a membrane governed by the Poisson equation and minimizing the principal eigenvalue of a weighted Dirichlet-Laplacian, both subject to material distribution constraints. Building on the classical rearrangement method, we introduce momentum-like acceleration by extrapolating the Fréchet derivative, leading to a provably convergent algorithm. We also introduce a restarted variant that guarantees monotonic improvement of the objective. In one dimension, we derive asymptotic convergence rates for ARM and prove that they improve upon the classical rearrangement method. Numerical experiments in both two and three dimensions confirm the accelerated convergence and demonstrate practical efficiency.

math.OC↗

arXiv2025

Optimal Control For Anti-Abeta Treatment in Alzheimer's Disease using a Reaction-Diffusion Model

Alzheimer's disease (AD) is a progressive neurodegenerative disorder that severely impairs survival and quality of life. While anti-amyloid beta (Abeta) therapies can slow disease progression, their efficacy depends on personalized dosing that maximizes benefits and minimizes risks such as amyloid related imaging abnormalities (ARIA). Mathematical modeling offers a powerful tool for understanding AD dynamics and optimizing treatment, yet most models focus solely on temporal behavior, overlooking spatial heterogeneity within the brain. In this study, we propose a spatially explicit reaction-diffusion model to describe Abeta plaque dynamics. We formulate an optimal control problem to minimize plaque concentration while balancing therapeutic efficacy and treatment risk. Under reasonable assumptions, we establish well-posedness and uniqueness of the optimal solution. A Finite Element Method (FEM) based numerical framework is developed to compute personalized treatment strategies. Our model is calibrated using longitudinal Abeta positron emission tomography (PET) data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), enabling estimation of patient-specific parameters such as growth rate and effective diffusivity. Results show that optimized treatment strategies consistently outperform constant dosing regimens across patient groups, achieving substantial reductions in cumulative amyloid burden while minimizing side effects. This integrated, data-driven framework advances personalized, spatially informed therapeutic optimization for Alzheimer's disease.

math.OC↗

arXiv2024

A Semi-definite Optimization Method for Maximizing the Shared Band Gap of Topological Photonic Crystals

Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.

math.OC↗

arXiv2023

Steklov Eigenvalue Problems on Nearly Spherical and Nearly Annular Domains

We consider Steklov eigenvalues on nearly spherical and nearly annular domains in $d$ dimensions. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain perturbation parameter can be determined by the eigenvalues of a matrix involving the integral of the product of three spherical harmonic functions. By using the addition theorem for spherical harmonic functions, we determine conditions when the trace of this matrix becomes zero. These conditions can then be used to determine when spherical and annular regions are critical points while we optimize Steklov eigenvalues subject to a volume constraint. In addition, we develop numerical approaches based on particular solutions and show that numerical results in two and three dimensions are in agreement with our analytic results.

math.SP↗

arXiv2023

Harmonic functions on finitely-connected tori

In this paper, we prove a Logarithmic Conjugation Theorem on finitely-connected tori. The theorem states that a harmonic function can be written as the real part of a function whose derivative is analytic and a finite sum of terms involving the logarithm of the modulus of a modified Weierstrass sigma function. We implement the method using arbitrary precision and use the result to find approximate solutions to the Laplace problem and Steklov eigenvalue problem. Using a posteriori estimation, we show that the solution of the Laplace problem on a torus with a few circular holes has error less than $10^{-100}$ using a few hundred degrees of freedom and the Steklov eigenvalues have similar error.

math.NA↗

arXiv2022

Flat tori with large Laplacian eigenvalues in dimensions up to eight

We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $λ_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d \leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k \to \infty$.

math.SP↗

arXiv2020

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, $Σ$, with genus $γ$ and $b$ boundary components, we maximize $σ_j(Σ,g) \ L(\partial Σ, g)$ over a class of smooth metrics, $g$, where $σ_j(Σ,g)$ is the $j$-th nonzero Steklov eigenvalue and $L(\partial Σ, g)$ is the length of $\partial Σ$. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus $γ=0$ and $b=2,\dots, 9, 12, 15, 20$ boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with $b=2$ boundary components and for the third and fifth eigenvalues with $b=3$ boundary components.

math.SP↗

arXiv2018

Extremal Spectral Gaps for Periodic Schrödinger Operators

The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m-th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.

math.OC↗

arXiv2018

A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping

In this paper, a spectral method based on conformal mappings is proposed to solve Steklov eigenvalue problems and their related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes $k-$th Steklov eigenvalue with a fixed area for a given $k$. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different $k$

math.NA↗

arXiv2016

Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $λ_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto λ_k(M,g)$. We propose a computational method for finding the conformal spectrum $Λ^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $λ_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $γ$. This is known as the topological spectrum for genus $γ$ and denoted by $Λ^t_k(γ)$. Our computations support a conjecture of N. Nadirashvili (2002) that $Λ^t_k(0) = 8 πk$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that $Λ^t_k(1) = \frac{8π^2}{\sqrt{3}} + 8π(k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$-th Laplace-Beltrami eigenvalue has a local maximum with value $λ_k = 4π^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

math.DG↗

arXiv2016

A New Algorithm to Simulate the First Exit Times of a Vector of Brownian Motions, with an Application to Finance

We provide a new methodology to simulate the first exit times of a vector of Brownian motions from an orthant. This new approach can be used to simulate the first exit times of dimension higher than two. When at least one Brownian motion has non-zero drift, the joint density function of the first exit times in N dimensions needs to be known, or approximated. However, when the drifts are all zero, a simpler simulation algorithm is obtained without using the joint density function.

math.PR↗

arXiv2016

Computational Methods For Extremal Steklov Problems

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the $p$-th Steklov eigenvalue is unique (up to dilations and rigid transformations), has p-fold symmetry, and an axis of symmetry. The $p$-th Steklov eigenvalue has multiplicity 2 if $p$ is even and multiplicity 3 if $p\geq3$ is odd.

math.SP↗

arXiv2013

Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations

This paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations (PDEs) in 1-D. The goal of the paper is to extend the successful framework of monotone, consistent, and stable finite difference methods for first order fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs such as Monge-Ampère and Bellman type equations. New concepts of consistency, generalized monotonicity, and stability are introduced; among them, the generalized monotonicity and consistency, which are easier to verify in practice, are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear Hamilton-Jacobi equations. The main component of the proposed framework is the concept of "numerical operator", and the main idea used to design consistent, monotone and stable finite difference methods is the concept of "numerical moment". These two new concepts play the same roles as the "numerical Hamiltonian" and the "numerical viscosity" play in the finite difference framework for first order fully nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also are proved to be stable and the second class contains Godunov-like methods. Numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper.

math.NA↗

arXiv2011

Bounded domain problem for the modified Buckley-Leverett equation

The focus of the present study is the modified Buckley-Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley-Leverett (BL) equation by including a balanced diffusive-dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that the solution of the finite interval [0,L] boundary value problem converges to that of the half-line [0,+\infty) boundary value problem for the MBL equation as L-> +\infty. This result provides a justification for the use of the finite interval boundary value problem in numerical studies for the half line problem. Furthermore, we extend the classical central schemes for the hyperbolic conservation laws to solve the MBL equation which is of pseudo-parabolic type. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.

math.NA↗

arXiv2010

Numerical study of the KP equation for non-periodic waves

The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions in this paper. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. In this paper, we study the initial value problem of the KP equation with V- and X-shape initial waves consisting of two distinct line-solitons by means of the direct numerical simulation. We then show that the solution converges asymptotically to some of those exact soliton solutions. The convergence is in a locally defined $L^2$-sense. The initial wave patterns considered in this paper are related to the rogue waves generated by nonlinear wave interactions in shallow water wave problem.

nlin.PS↗

arXiv2009

A moving boundary model motivated by electric breakdown: II. Initial value problem

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a 'kinetic undercooling' boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for $t\to\infty$, which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.

nlin.PS↗