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Pu-Zhao Kow

Publications and source records attributed to Pu-Zhao Kow.

At least 19 recordsLinked to original sources
arXiv2026

An inverse problem for semilinear elliptic equations with generalized Kerr-type nonlinearities

We study the inverse problem of reconstructing the convex hull of unknown inclusions in semilinear elliptic equations with nonanalytic nonlinearities, by extending Ikehata's enclosure method to accommodate such nonlinear effects. To overcome the lack of the higher-order differentiability required for the Taylor-based construction, we construct an approximate solution based on the linearized equation through a first-iteration approximation. Under suitable structural conditions on the nonlinearity, we establish a reconstruction result for the convex hull of the inclusion. The method applies to a broad class of generalized Kerr-type nonlinearities that need not be analytic, without requiring an explicit formula for the nonlinear term.

math.AP↗
arXiv2026

Bayesian inference for the fractional Calderón problem with a single measurement

This paper investigates the consistency of a posterior distribution in the single-measurement fractional Calderón problem with additive Gaussian noise. We consider a Bayesian framework with rescaled and Gaussian sieve priors, using a collection of noisy, discrete observations taken from a suitable exterior domain. Our main result shows that the posterior distribution concentrates around the true parameter as the number of measurements increases. Furthermore, we establish tight convergence rates for the reconstruction error of the posterior mean. A central technical challenge is to obtain refined stability estimates for both the forward and inverse problems. In particular, the required forward estimates are delicate to obtain because the fractional elliptic problems do not enjoy as strong regularity theory as their classical counterparts.

math.ST↗
arXiv2026

Multiphase quadrature domains (existence and uniqueness)

The primary goal of this paper is to give a precise definition and prove existence and uniqueness of multiphase quadrature domains for subharmonic functions, ensuring that the prescribed measures are supported in the interior of the resulting domains. The approach to prove existence is based on a variational framework, where we minimize an energy functional over so called segregated states. In this respect we refine earlier results in this direction. But we also show that this approach alone is not enough for two reasons. First of all it seems hard to get existence results which ensure that the interior support condition is satisfied. And second it may happen, as we show by an example, that a multiphase quadrature domain exists but is not a minimizer of the energy functional. The main novelty of this work is the study of minimizers, if they exist, of the energy functional over a subset of the segregated states given by a natural constraint with respect to the given measures. From this approach we are able to prove uniqueness, and also give sufficient conditions for existence. We also give an example showing that, unlike the energy minimization and partial balayage approaches which are equivalent in the one-phase case, this equivalence breaks down already in the two-phase setting.

math.AP↗
arXiv2026

Existence and regularity of minimizers for a variational problem of species population density

We study a variational problem motivated by models of species population density in a nonhomogeneous environment. We first analyze local minimizers and the structure of the saturated region (where the population attains its maximal density) from a free boundary perspective. By comparing the original problem with a radially symmetric minimization problem and studying its properties, we then establish the existence and structure of a global solution. Analytic examples of radially symmetric solutions and numerical simulations illustrate the theoretical results and provide insight into spatial saturation patterns in population models. We further highlight an unresolved question regarding the quasiconcavity of minimizers.

math.AP↗
arXiv2025

On scattering behavior of corner domains with anisotropic inhomogeneities: part II

We study the scattering behavior of an anisotropic inhomogeneous Lipschitz medium at a fixed wave number, continuing our previous work [SIAM J. Math. Anal., 56(4):4834-4853, 2024] and using free boundary techniques from [arXiv:2506.22328]. Our main results can be categorized into two distinct cases. In the first case, we show that in two dimensions, piecewise $C^{1}$ or convex penetrable obstacles with corners, and in higher dimensions, obstacles with edge points, always induce nontrivial scattering for any incoming wave. In the second case, we prove that piecewise $C^{1}$ obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles $\notinπ\mathbb{Q}$ always produce nontrivial scattering for any incoming wave.

math.AP↗
arXiv2025

A Speech Enhancement Method Using Fast Fourier Transform and Convolutional Autoencoder

This paper addresses the reconstruction of audio signals from degraded measurements. We propose a lightweight model that combines the discrete Fourier transform with a Convolutional Autoencoder (FFT-ConvAE), which enabled our team to achieve second place in the Helsinki Speech Challenge 2024. Our results, together with those of other teams, demonstrate the potential of neural-network-free approaches for effective speech signal reconstruction.

cs.SD↗
arXiv2025

Unique continuation for the momentum ray transform

The present article focuses on a unique continuation result for certain weighted ray transforms, utilizing the unique continuation property (UCP) of the fractional Laplace operator. Specifically, we demonstrate a conservative property for momentum ray transforms acting on tensors, as well as the antilocality property for both weighted ray and cone transforms acting on functions.

math.AP↗
arXiv2024

Increasing resolution and instability for linear inverse scattering problems

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-Hörmander estimates, and a Schwartz kernel computation based on the coarea formula.

math.AP↗
arXiv2024

Multi-phase $k$-quadrature domains and applications to acoustic waves and magnetic fields

The primary objective of this paper is to explore the multi-phase variant of quadrature domains associated with the Helmholtz equation, commonly referred to as $k$-quadrature domains. Our investigation employs both the minimization problem approach, which delves into the segregation ground state of an energy functional, and the partial balayage procedure, drawing inspiration from the recent work by Gardiner and Sjödin. Furthermore, we present practical applications of these concepts in the realms of acoustic waves and magnetic fields.

math.AP↗
arXiv2024

On scattering behavior of corner domains with anisotropic inhomogeneities

This paper investigates the possible scattering and non-scattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic non-scattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not $C^{1,α}$ boundary scatters every incident wave that satisfies a non-degeneracy condition.

math.AP↗
arXiv2023

On positivity sets for Helmholtz solutions

We address the question of finding global solutions of the Helmholtz equation that are positive in a given set. This question arises in inverse scattering for penetrable obstacles. In particular, we show that there are solutions that are positive on the boundary of a bounded Lipschitz domain.

math.AP↗
arXiv2022

An inverse problem for semilinear equations involving the fractional Laplacian

Our work concerns the study of inverse problems of heat and wave equations involving the fractional Laplacian operator with zeroth order nonlinear perturbations. We recover nonlinear terms in the semilinear equations from the knowledge of the fractional Dirichlet-to-Neumann type map combined with the Runge approximation and the unique continuation property of the fractional Laplacian.

math.AP↗
arXiv2022

Landis-type conjecture for the half-Laplacian

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage's Liouville-type theorem.

math.AP↗
arXiv2021

On Landis Conjecture for the Fractional Schrödinger Equation

In this paper, we study a Landis-type conjecture for the general fractional Schrödinger equation $((-P)^{s}+q)u=0$. As a byproduct, we also proved the additivity and boundedness of the linear operator $(-P)^{s}$ for non-smooth coefficents. For differentiable potentials $q$, if a solution decays at a rate $\exp(-|x|^{1+})$, then the solution vanishes identically. For non-differentiable potentials $q$, if a solution decays at a rate $\exp(-|x|^{\frac{4s}{4s-1}+})$, then the solution must again be trivial. The proof relies on delicate Carleman estimates. This study is an extension of the work by Rüland-Wang (2019).

math.AP↗
arXiv2021

Optimality of increasing stability for an inverse boundary value problem

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for Schrödinger equation. The rigorous justification of increasing stability for the IBVP for Schrödinger equation were established by Isakov \cite{Isa11} and by Isakov, Nagayasu, Uhlmann, Wang of the paper \cite{INUW14}. In \cite{Isa11}, \cite{INUW14}, the authors showed that the stability of this IBVP increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a Hölder type. In this work, we prove that the instability changes from an exponential type to a Hölder type when the frequency increases. This result verifies that results in \cite{Isa11}, \cite{INUW14} are optimal.

math.AP↗