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Yi-Hsuan Lin

Publications and source records attributed to Yi-Hsuan Lin.

At least 19 recordsLinked to original sources
arXiv2026

The Calder\'on problem for the infinity Laplacian with large affine boundary data

We study the Calder\'on problem for the equation $-\Delta_\infty u+q(x)u=0$ in a bounded convex domain with $C^2$ boundary. We prove that a positive potential $q$ is uniquely determined and explicitly reconstructible from the measurements $\Lambda_q\bigl(t(e\cdot x+b)|_{\partial\Omega}\bigr)$, where $e\in\mathbb S^{n-1}$, the offset $b>\sup_{\overline\Omega}|x|$ is fixed, and $t\to\infty$. The first potential-dependent term in the large amplitude asymptotics determines weighted integrals of $q$ over the chords parallel to $e$. Combining the measurements in the directions $e$ and $-e$ gives the X-ray transform of the zero extension of $q$, and the Fourier slice identity yields reconstruction and uniqueness.

math.AP↗
arXiv2026

On the (Non-)Uniqueness of Random Non-Expected Utility

In random expected utility (Gul and Pesendorfer, 2006), the distribution of preferences is uniquely identified from random choice. This paper investigates whether such identification extends beyond expected utility. We first show that when risk preferences conform to the disappointment aversion model of Gul (1991), the distribution of preferences remains uniquely identified. To assess the scope of this result, we then examine other models of non-expected utility. Within the broader class of betweenness preferences (Dekel, 1986), random utility can be unidentifiable. If preferences are confined to the weighted expected utility class (Chew, 1983), a more nuanced picture emerges: unique identification holds in a three-prize setting but fails with four or more prizes. These findings show that the uniqueness property of random expected utility may persist beyond expected utility, but its persistence critically depends on the class of risk preferences under consideration.

econ.TH↗
arXiv2026

Revealed Attentional Interference

We study the impact of external stimuli on attention in the Attentional Interference Model, capturing two opposing forces in consideration-set formation: proactive and retroactive interference. Proactive interference limits the permeation of external information, while retroactive interference displaces internally generated considerations. We model these forces using parameters governing permeation and displacement. In a general setting, we characterize the tight range of these parameters and show that, across several specifications, only an upper bound on permeation is revealed. Imposing monotonicity on internal attention in one case both tightens this upper bound and yields a lower bound. We illustrate our results through simulation.

econ.TH↗
arXiv2026

Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation

We prove a uniqueness result for an inverse boundary problem associated with the prescribed Gaussian curvature equation \[ \det D^2u=K(x)(1+|\nabla u|^2)^2 \] for graphs over a planar domain. The comparison is made on a common open class of smooth boundary values for which both admissible Dirichlet problems are well posed. We show that, if the corresponding nonlinear Dirichlet-to-Neumann maps agree on this class and the two prescribed curvatures have the same first boundary jet, then the curvatures agree in the whole domain. The main difficulty comes from a gauge obstruction already present at the first linearization. In logarithmic form, the linearized equation is a two-dimensional non-divergence form elliptic equation with drift. Its boundary data determine the coefficients only up to a boundary-fixing change of variables and a scalar gauge factor. For the prescribed Gaussian curvature equation this gauge is not an artifact of the method: the gradient dependence leaves a residual gauge which cannot be removed by the first variation. The proof uses the nonlinear structure to remove this remaining gauge. We derive a second-linearized interaction identity in the first-linearized gauge. In this identity, the covariant Hessian of the gauge map appears in the leading part. Testing the identity with two complementary CGO families gives a pair of residual equations for the gauge variables. Combined with the drift and conductivity identities from the first linearization, these equations form a closed system for the residual gauge. A boundary unique continuation argument for this system gives the trivial gauge and hence determines the prescribed curvature.

math.AP↗
arXiv2026

Affine section tomography for inverse source problems in $k$-Hessian equations with restricted large boundary data

We consider an inverse source problem for the $k$-Hessian equation \begin{equation*} \sigma_k(D^2u)=f(x) \end{equation*} on the $k$-admissible branch in a smooth uniformly convex domain in $\mathbb R^n$, where $2\le k\le n$. We prove that the nonlinear Dirichlet-to-Neumann map determines the positive smooth source from its values on the restricted large-data rays $t\phi_E|_{\partial\Omega}$, where $E\in Gr(k-1,n)$, $\phi_E(x)=|P_Ex|^2/2$, and $t$ is sufficiently large. The Hessian of each boundary profile has exactly $k-1$ large directions and is flat on $V=E^\perp$. Thus, the leading profile lies on a rank $k-1$ face of the $k$-Hessian structure, and the first source-dependent correction is governed by the missing directions. More precisely, this correction solves fiberwise Poisson equations on the affine sections $\Omega\cap(y+V)$ of dimension $q=n-k+1$. We prove that these sectionwise solutions patch smoothly through glancing points where the sections collapse, and we obtain boundary normal derivative asymptotics by local barriers. The boundary flux of the correction gives the section integrals $\int_{\Omega\cap(y+V)}f\,d\mathcal H^q$. Varying $E$ yields the affine $q$-plane Radon transform of the zero extension of $f$. We give an explicit reconstruction formula through the Fourier slice identity, and the injectivity of the affine Radon transform gives uniqueness. The endpoint $k=n$ recovers the Monge--Amp\`ere chord/X-ray geometry, while the range $n\ge3$ and $2\le k<n$ gives inverse source results for genuinely non-determinant Hessian equations.

math.AP↗
arXiv2026

Recovering stable kernels from exterior measurements

We study an inverse problem for translation-invariant symmetric stable operators of the form \begin{equation*} L_a u(x)=\mathrm{P.V.}\int_{\mathbb R^n}(u(x)-u(y))\frac{a((x-y)/|x-y|)}{|x-y|^{n+2s}}\,dy, \quad 0<s<1, \end{equation*} where the unknown is the even angular density $a$ on $\mathbb Sn$. For a bounded open set $\Omega\subset\mathbb R^n$, with $\Omega_e=\mathbb R^n\setminus\overline\Omega$, we consider restricted exterior Dirichlet-to-Neumann maps $\Lambda_a^{W_1,W_2}$, where exterior data are supported in $W_1\Subset\Omega_e$ and the nonlocal Neumann data are observed on $W_2\Subset\Omega_e$. We prove three recovery results for the leading angular density. In the overlapping regime $W_1\cap W_2\ne\emptyset$, the exterior diagonal singularity determines every smooth elliptic angular density. In the separated regime $\overline W_1\cap\overline W_2=\emptyset$, where this singularity is absent, we prove uniqueness in the finite harmonic angular class by an exact factorization of the stable symbol. We also prove separated-data uniqueness for real-analytic angular densities when the source and observation sets lie in the unbounded exterior component, using analytic continuation of the off-diagonal Dirichlet-to-Neumann kernel and a far-field asymptotic argument.

math.AP↗
arXiv2026

An inverse source problem for a fully nonlinear elliptic equation

We study an inverse source problem for fully nonlinear elliptic equations of the form \[ F(D^2u)=f \quad \text{in } \Omega. \] The question is whether the source term can be recovered from the Dirichlet-to-Neumann map. In two dimensions, the first linearization does not immediately give uniqueness: it leaves a natural conformal ambiguity in the linearized coefficients. For homogeneous nonlinearities $F$ with injective differential $DF$, we show that this ambiguity has a precise meaning at the level of the equation itself, namely that the source is determined up to an explicit scalar factor. The main point of the paper is to show how this remaining factor can be removed. We use the second linearization to extract information which is invisible at first order, and combine it with an algebraic nondegeneracy condition on the nonlinearity. Under this condition, the residual ambiguity is forced to be trivial, and the Dirichlet-to-Neumann map uniquely determines the source. The result applies, in particular, to homogeneous admissible Hessian equations of Monge--Amp\`ere type and related examples.

math.AP↗
arXiv2026

Entanglement principle for fractional Laplacian on hyperbolic spaces and applications to inverse problem

We establish an entanglement principle for fractional powers of the Laplace-Beltrami operator on hyperbolic space $\mathbb H^n$, $n\ge 2$. More precisely, we prove that if finitely many distinct noninteger powers of $-\Delta_{\mathbb H^n}$, acting on functions that vanish on a common nonempty open set, satisfy a linear dependence relation on that set, then each of these functions must vanish identically on $\mathbb H^n$. This extends the recently developed entanglement principle for the fractional Laplacian on $\mathbb R^n$ to the negatively curved setting of hyperbolic space. As an application, we derive global uniqueness results for inverse problems associated with fractional polyharmonic equations on $\mathbb H^n$, including a fractional Calder\'on problem. The proof relies on the heat semigroup representation of fractional powers together with sharp global heat kernel estimates on hyperbolic space.

math.AP↗
arXiv2026

Finite element error analysis for elliptic parameter identification with power-type nonlinearity

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on piecewise linear finite elements. As one of our main novelties, we establish conditional stability estimates at the continuous level, which form the theoretical foundation of the present finite element analysis. Our stability analysis relies on tailored analytical tools, including Hardy-type inequalities, fractional Gagliardo-Nirenberg inequalities, and weighted spaces with singular distance weights. By invoking the achieved conditional stability together with the Carstensen quasi-interpolation operator and associated estimates in negative Sobolev spaces, we derive a priori error estimates for the proposed finite element approximation in terms of the mesh size, the regularization parameter, the noise level, and the nonlinearity exponent. Our results extend the recent stability and error estimates for the linear case by Jin et al. \cite{jin2022convergence} and sharpen their error estimates and convergence order under weaker regularity assumptions.

math.NA↗
arXiv2025

Entanglement principle and fractional Calder\'on problem for nonlocal parabolic operators

We examine inverse problems for the variable-coefficient nonlocal parabolic operator $(\partial_t - \Delta_g)^s$, where $0 < s < 1$. This article makes two primary contributions. First, we introduce a novel entanglement principle for these operators under suitable smoothness conditions. Second, we prove that lower-order perturbations can be uniquely determined from the associated Dirichlet-to-Neumann map using this principle. However, due to insufficient solution regularity, direct application of the entanglement principle to the inverse problem is not feasible. To address this, we derive a modified entanglement principle, enabling the effective resolution of related inverse problems.

math.AP↗
arXiv2025

An inverse problem for the Monge-Amp\`ere equation

We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Amp\`ere equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the plane, the associated Dirichlet-to-Neumann (DN) map uniquely determines a positive source function $F$. The proof relies on recovering the Hessian of a solution to the equation, which is interpreted as a Riemannian metric $g$. Interestingly, although the equation is posed on a Euclidean domain, the inverse problem becomes anisotropic since the metric $g$ appears as a coefficient matrix in the linearized equation. As an intermediate step, we prove that the DN map of the non-divergence form equation \[ g^{ab} \partial_{ab} v = 0 \] uniquely determines the conformal class of the metric $g$ on a simply connected planar domain, without the usual diffeomorphism invariance. To address the challenges of full nonlinearity, we develop asymptotic expansions for complex geometric optics solutions in the planar setting and solve a resulting nonlocal $\overline{\partial}$-equation by proving a unique continuation principle for it. These techniques are expected to be applicable to a wide range of inverse problems for nonlinear equations.

math.AP↗
arXiv2025

Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot \sigma \nabla)^s u = 0 \quad \text{in } \Omega \subset \mathbb{R}^n, \] where $0 < s < 1$, $n \geq 3$, and $\Omega$ is a bounded open set. We establish a monotonicity relation between the leading coefficient $\sigma$ and the (partial) exterior Dirichlet-to-Neumann (DN) map. Our main result shows that a monotonicity ordering of the coefficients implies a corresponding ordering of the DN maps. Furthermore, we construct localized potentials for the nonlocal equation, which yield a local uniqueness result for the fractional inverse problem.

math.AP↗
arXiv2024

The Calder\'on problem for the logarithmic Schr\"odinger equation

We study the Calder\'on problem for a logarithmic Schr\"odinger type operator of the form $L_{\Delta} +q$, where $L_{\Delta}$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-\Delta)^s$ 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 Calder\'{o}n problem for a nonlocal parabolic equation on closed Riemannian manifolds

We consider the fractional anisotropic Calder\'on problem for the nonlocal parabolic equation $(\partial_t -\Delta_g)^s u=f$ ($0<s<1$) 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 Calder\'on 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

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 $L^2(0,T;\widetilde{H}^s(\Omega))$ 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 Calder\'on problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calder\'on problem of nonlocal wave equations with polyhomogeneous nonlinearities.

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↗