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Yi-Hsuan Lin

Publications and source records attributed to Yi-Hsuan Lin.

At least 37 records · Page 2Linked to original sources
arXiv2025

Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities

The main purpose of this article is to establish the Runge-type approximation in $L^2(0,T;\widetilde{H}^s(Ω))$ for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calderón problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calderón problem of nonlocal wave equations with polyhomogeneous nonlinearities.

math.AP↗
arXiv2024

Electrostatics of Salt-Dependent Reentrant Phase Behaviors Highlights Diverse Roles of ATP in Biomolecular Condensates

Liquid-liquid phase separation (LLPS) involving intrinsically disordered protein regions (IDRs) is a major physical mechanism for biological membraneless compartmentalization. The multifaceted electrostatic effects in these biomolecular condensates are exemplified here by experimental and theoretical investigations of the different salt- and ATP-dependent LLPSs of an IDR of messenger RNA-regulating protein Caprin1 and its phosphorylated variant pY-Caprin1, exhibiting, e.g., reentrant behaviors in some instances but not others. Experimental data are rationalized by physical modeling using analytical theory, molecular dynamics, and polymer field-theoretic simulations, indicating that interchain ion bridges enhance LLPS of polyelectrolytes such as Caprin1 and the high valency of ATP-magnesium is a significant factor for its colocalization with the condensed phases, as similar trends are observed for other IDRs. The electrostatic nature of these features complements ATP's involvement in $π$-related interactions and as an amphiphilic hydrotrope, underscoring a general role of biomolecular condensates in modulating ion concentrations and its functional ramifications.

q-bio.BM↗
arXiv2024

The Calderón problem for the logarithmic Schrödinger equation

We study the Calderón problem for a logarithmic Schrödinger type operator of the form $L_Δ +q$, where $L_Δ$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-Δ)^s$ of the family of fractional Laplacian operators. This operator enjoys remarkable nonlocal properties, such as the unique continuation and Runge approximation. Based on these tools, we can uniquely determine bounded potentials using the Dirichlet-to-Neumann map. Additionally, we can build a constructive uniqueness result by utilizing the monotonicity method. Our results hold for any space dimension.

math.AP↗
arXiv2024

Entanglement principle for the fractional Laplacian with applications to inverse problems

We prove an entanglement principle for fractional Laplace operators on $\mathbb R^n$ for $n\geq 2$ as follows; if different fractional powers of the Laplace operator acting on several distinct functions on $\mathbb R^n$, which vanish on some nonempty open set $O$, are known to be linearly dependent on $O$, then all the functions must be globally zero. This remarkable principle was recently discovered to be true for smooth functions on compact Riemannian manifolds without boundary \cite{FKU24}. Our main result extends the principle to the noncompact Euclidean space stated for tempered distributions under suitable decay conditions at infinity. We also present applications of this principle to solve new inverse problems for recovering anisotropic principal terms as well as zeroth order coefficients in fractional polyharmonic equations. Our proof of the entanglement principle uses the heat semigroup formulation of fractional Laplacian to establish connections between the principle and the study of several topics including interpolation properties for holomorphic functions under certain growth conditions at infinity, meromorphic extensions of holomorphic functions from a subdomain, as well as support theorems for spherical mean transforms on $\mathbb R^n$ that are defined as averages of functions over spheres.

math.AP↗
arXiv2024

The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds

We consider the fractional anisotropic Calderón problem for the nonlocal parabolic equation $(\partial_t -Δ_g)^s u=f$ ($0<s<1$) on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold $(M,g)$ up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calderón problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.

math.AP↗
arXiv2024

The Calderón problem for the Schrödinger equation in transversally anisotropic geometries with partial data

We study the partial data Calderón problem for the anisotropic Schrödinger equation \begin{equation} \label{eq: a1} (-Δ_{\widetilde{g}}+V)u=0\text{ in }Ω\times (0,\infty), \end{equation} where $Ω\subset\mathbb{R}^n$ is a bounded smooth domain, $\widetilde{g}=g_{ij}(x)dx^{i}\otimes dx^j+dy\otimes dy$ and $V$ is translationally invariant in the $y$ direction. Our goal is to recover both the metric $g$ and the potential $V$ from the (partial) Neumann-to-Dirichlet (ND) map on $Γ\times \{0\}$ with $Γ\Subset Ω$. Our approach can be divided into three steps: Step 1. Boundary determination. We establish a novel boundary determination to identify $(g,V)$ on $Γ$ with help of suitable approximate solutions for the Schrödinger equation with inhomogeneous Neumann boundary condition. Step 2. Relation to a nonlocal elliptic inverse problem. We relate inverse problems for the Schrödinger equation with the nonlocal elliptic equation \begin{equation} \label{eq: a2} (-Δ_g+V)^{1/2}v=f\text{ in }Ω, \end{equation} via the Caffarelli--Silvestre type extension, where the measurements are encoded in the source-to-solution map. The nonlocality of this inverse problem allows us to recover the associated heat kernel. Step 3. Reduction to an inverse problem for a wave equation. Combining the knowledge of the heat kernel with the Kannai type transmutation formula, we transfer the inverse problem for the nonlocal equation to an inverse problem for the wave equation \begin{equation} \label{eq: a3} (\partial_t^2-Δ_g+V)w=F\text{ in }Ω\times (0,\infty), \end{equation} where the measurement operator is also the source-to-solution map. We can finally determine $(g,V)$ on $Ω\setminusΓ$ by solving the inverse problem for the wave equation.

math.AP↗
arXiv2024

Approximation and uniqueness results for the nonlocal diffuse optical tomography problem

We investigate the inverse problem of recovering the diffusion and absorption coefficients $(σ,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( σ\nabla))^s u+q u =0 \text{ in }Ω$ from the nonlocal Dirichlet-to-Neumann (DN) map $Λ^s_{σ,q}$. The purpose of this article is to establish the following approximation and uniqueness results. - Approximation: We show that solutions to the conductivity equation $ \text{div}( σ\nabla v)=0 \text{ in }Ω$ can be approximated in $H^1(Ω)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $Λ_σ$ related to conductivity equation can be approximated by the nonlocal DN map $Λ_{σ,q}^s$. - Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partialΩ$ provided $σ$ is already known in $\mathcal{N}$. - Global uniqueness: Under the same assumptions as for the local uniqueness result, and if one of the potentials vanishes in $Ω$, then one can turn with the help of \ref{item 1 abstract} the local determination into a global uniqueness result. It is worth mentioning that the approximation result relies on the Caffarelli--Silvestre type extension technique and the geometric form of the Hahn--Banach theorem.

math.AP↗
arXiv2024

Enhancing Taiwanese Hokkien Dual Translation by Exploring and Standardizing of Four Writing Systems

Machine translation focuses mainly on high-resource languages (HRLs), while low-resource languages (LRLs) like Taiwanese Hokkien are relatively under-explored. The study aims to address this gap by developing a dual translation model between Taiwanese Hokkien and both Traditional Mandarin Chinese and English. We employ a pre-trained LLaMA 2-7B model specialized in Traditional Mandarin Chinese to leverage the orthographic similarities between Taiwanese Hokkien Han and Traditional Mandarin Chinese. Our comprehensive experiments involve translation tasks across various writing systems of Taiwanese Hokkien as well as between Taiwanese Hokkien and other HRLs. We find that the use of a limited monolingual corpus still further improves the model's Taiwanese Hokkien capabilities. We then utilize our translation model to standardize all Taiwanese Hokkien writing systems into Hokkien Han, resulting in further performance improvements. Additionally, we introduce an evaluation method incorporating back-translation and GPT-4 to ensure reliable translation quality assessment even for LRLs. The study contributes to narrowing the resource gap for Taiwanese Hokkien and empirically investigates the advantages and limitations of pre-training and fine-tuning based on LLaMA 2.

cs.CL↗
arXiv2024

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form $f(x,u)$ under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension $n\in \N$.

math.AP↗
arXiv2023

Determining both leading coefficient and source in a nonlocal elliptic equation

In this short note, we investigate an inverse source problem associated with a nonlocal elliptic equation $\left( -\nabla \cdot σ\nabla \right)^s u =F$ that is given in a bounded open set $Ω\subset \mathbb{R}^n$, for $n\geq 3$ and $0<s<1$. We demonstrate both $σ$ and $F$ can be determined uniquely by using the exterior Dirichlet-to-Neumann (DN) map in $Ω_e:=\mathbb{R}^n\setminus \overlineΩ$. The result is intriguing in that analogous theory cannot be true for the local case generally, that is, $s=1$. The key ingredients to prove the uniqueness is based on the unique continuation principle for nonlocal elliptic operators and the reduction from the nonlocal to the local via the Stinga-Torrea extension problem.

math.AP↗
arXiv2023

Unique determination of coefficients and kernel in nonlocal porous medium equations with absorption term

The main purpose of this article is the study of an inverse problem for nonlocal porous medium equations (NPMEs) with a linear absorption term. More concretely, we show that under certain assumptions on the time-independent coefficients $ρ,q$ and the time-independent kernel $K$ of the nonlocal operator $L_K$, the (partial) Dirichlet-to-Neumann map uniquely determines the three quantities $(ρ,K,q)$ in the nonlocal porous medium equation $ρ\partial_tu+L_K(u^m)+qu=0$, where $m>1$. In the first part of this work we adapt the Galerkin method to prove existence and uniqueness of nonnegative, bounded solutions to the homogenoeus NPME with regular initial and exterior conditions. Additionally, a comparison principle for solutions of the NPME is proved, whenever they can be approximated by sufficiently regular functions like the one constructed for the homogeneous NPME. These results are then used in the second part to prove the unique determination of the coefficients $(ρ,K,q)$ in the inverse problem. Finally, we show that the assumptions on the nonlocal operator $L_K$ in our main theorem are satisfied by the fractional conductivity operator $\mathcal{L}_γ$, whose kernel is $γ^{1/2}(x)γ^{1/2}(y)/|x-y|^{n+2s}$ up to a normalization constant.

math.AP↗
arXiv2023

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on $\mathbb{R}^n$. Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.

math.AP↗
arXiv2023

The Calderón problem for nonlocal parabolic operators

We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, respectively.

math.AP↗
arXiv2023

Determining a nonlinear hyperbolic system with unknown sources and nonlinearity

We investigate inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. We establish in several generic scenarios that one can uniquely determine the nonlinearity or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and a Runge approximation with higher order linearization for the semilinear hyperbolic equation.

math.AP↗
arXiv2023

Uniqueness results and gauge breaking for inverse source problems of semilinear elliptic equations

We study inverse source problems associated to semilinear elliptic equations of the form \[ Δu(x)+a(x,u)=F(x), \] on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 2$. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities $a(x,u)$. This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential nonlinearities. For these nonlinearities, we determine both $a(x,u)$ and $F(x)$ uniquely from the associated DN map. Moreover, for general nonlinearities $a(x,u)$, we show that we can recover the derivatives $\partial_u^ka(x,u)$ and the source $F(x)$ up to a gauge. Especially, we recover general polynomial nonlinearities up to a gauge and generalize results of [FO20,LLLS20] by removing the assumption that $u\equiv 0$ is a solution.

math.AP↗
arXiv2022

Simultaneous recovery of piecewise analytic coefficients in a semilinear elliptic equation

In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.

math.AP↗
arXiv2022

Determining coefficients for a fractional $p$-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional $p\,$-Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficents. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

math.AP↗