arXiv subjects

Catharine W. K. Lo

Publications and source records attributed to Catharine W. K. Lo.

At least 19 recordsLinked to original sources

arXiv2026

Determining a parabolic-elliptic-elliptic system by boundary observation of its non-negative solutions under chemotaxis background

This paper addresses a profoundly challenging inverse problem that has remained largely unexplored due to its mathematical complexity: the unique identification of all unknown coefficients in a coupled nonlinear system of mixed parabolic-elliptic-elliptic type using only boundary measurements. The system models attraction-repulsion chemotaxis -- an advanced mathematical biology framework for studying sophisticated cellular processes -- yet despite its significant practical importance, the corresponding inverse problem has never been investigated, representing a true frontier in the field. The mixed-type nature of this system introduces significant theoretical difficulties that render conventional methodologies inadequate, demanding fundamental extensions beyond existing techniques developed for simpler, purely parabolic models. Technically, the problem presents formidable obstacles: the coupling between parabolic and elliptic components creates inherent analytical complications, while the nonlinear structure resists standard approaches. From an applied perspective, the biological relevance adds another layer of complexity, as solutions must maintain physical interpretability through non-negativity constraints. Our work provides a complete theoretical framework for this challenging problem, establishing rigorous unique identifiability results that create a one-to-one correspondence between boundary data and the model's parameters. We demonstrate the power of our general theory through a central biological application: the full parameter recovery for an attraction-repulsion chemotaxis model with logistic growth, thus opening new avenues for quantitative analysis in mathematical biology.

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arXiv2026

Parameter Identification in Reaction-Diffusion-Chemotaxis Systems from Single Turing Pattern Amplitudes

Turing patterns encode key information about biological mechanisms, yet traditional inverse problems rely on non-biological data like boundary measurements, neglecting the patterns themselves. We introduce a new direction that directly uses the amplitudes of Turing patterns for parameter identification. Motivated by stripe-like biological patterns, we study two 1D models: one with density-dependent chemotaxis ($χ(n,c)=χ_0 n$) and one with ratio-dependent chemotaxis ($χ(n,c)=χ_0 n/c$). We present a framework that uses the spatial amplitude profile of a stationary pattern to recover system parameters, including wavelength, diffusion constants, and chemotactic and kinetic coefficients---offering a biologically grounded paradigm for reverse-engineering pattern formation. The core contribution is a proof-of-concept showing that, under a Fourier truncation at mode $M=3$, the amplitude data can reconstruct the parameter combinations $\{d_n k^2,\, d_c k^2,\, χ_0 k^2,\, r\}$ for both models. These uniquely determine the ratios $d_c/d_n$ and $χ_0/d_n$, while individual parameters are recovered up to an overall scale. This is a purely theoretical work; numerical validation and stability analysis are left for future research. The framework is extensible to other models with suitable modifications.

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arXiv2026

Identifying unknown time delay and spatially varying coefficients in a reaction-diffusion equation from boundary measurements

This work investigates an inverse problem for a general class of linear reaction-diffusion systems incorporating multiple delayed contributions, namely retarded diffusion, retarded time derivatives, and retarded source terms. The objective is to simultaneously recover the unknown time lag $τ>0$ and spatially heterogeneous coefficients from boundary flux measurements alone. The recovery strategy exploits the singular temporal behavior generated by an incompatibility between the prescribed initial history and the boundary data. In contrast to prior inverse problems for delay equations, which assume $τ$ known and are confined to ODE or abstract settings, our approach operates in a parabolic PDE framework with spatially varying coefficients. Once $τ$ is identified, we establish a Lipschitz stability estimate via a Carleman inequality, in the case of time-independent coefficients $p=p(x)$, $q=q(x)$. This is the first result to simultaneously recover an unknown delay and spatially dependent coefficients in a delayed parabolic PDE from boundary data.

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arXiv2026

Determining evolutionary equations from a single passive boundary observation

We study inverse boundary problems for evolutionary PDEs using only a single passive boundary observation, where data from an unknown internal source propagate through an unknown medium without active inputs. The goal is the simultaneous recovery of coupled unknowns (sources and coefficients) from severely limited data. Unlike active methods with rich, structured inputs, passive observation poses two core challenges: minimal information and intrinsic coupling of multiple unknowns. Consequently, such problems remain largely open and unsystematically studied. We develop a unified framework based on integral identities, harmonic and microlocal analysis, and low-/high-frequency asymptotics. This approach yields the first systematic resolution for second-order hyperbolic, parabolic, and Schrödinger equations under a single coherent method. The key condition requires the measurement dataset's cardinality to exceed the unknowns' by at least one dimension, providing room to decouple unknowns and linearize the nonlinear inverse problem. Our unique identifiability results subsume all existing literature and cover more general configurations of practical interest. This framework complements classical theories and opens a promising new direction for future development.

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arXiv2025

Determining habitat anomalies in cross-diffusion predator-prey chemotaxis models

This paper addresses an open inverse problem at the interface of mathematical analysis and spatial ecology: the unique identification of unknown spatial anomalies -- interpreted as zones of habitat degradation -- and their associated ecological parameters in multi-species predator-prey systems with multiple chemical signals, using only boundary measurements. We formulate the problem as the simultaneous recovery of an unknown interior subdomain and discontinuous ecological interaction rules across its boundary. A unified theooretical framework is developed that unique determines both the anomaly's geometry and discontinuous coefficients characterizing the altered interactions within the degraded region. Our results cover smooth anomalies in time-dependent systems and are extended to non-smooth polyhedral inclusions in stationary regimes. This work bridges a gap between ecological sensing and the quantitative inference of internal habitat heterogeneity, offering a mathamtical basis for detecting and characterizing habitat degradation from limited external data.

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arXiv2025

Inverse Problems for Mean Field Games

In this book, we present a curated collection of existing results on inverse problems for Mean Field Games (MFGs), a cutting-edge and rapidly evolving field of research. Our aim is to provide fresh insights, novel perspectives, and a comprehensive foundation for future investigations into this fascinating area. MFGs, a class of differential games involving a continuum of non-atomic players, offer a powerful framework for analyzing the collective behavior of large populations of symmetric agents as the number of agents approaches infinity. This framework has proven to be an invaluable tool for quantitatively modeling the macroscopic dynamics of agents striving to minimize specific costs in complex systems, such as crowd dynamics, financial markets, traffic flows, and social networks. The study of MFGs has traditionally focused on forward problems, where the goal is to determine the equilibrium behavior of agents given a set of model parameters, such as cost functions, interaction mechanisms, and initial conditions. However, the inverse problems for MFGs -- which seek to infer these underlying parameters from observed data -- have received comparatively less attention in the literature. This book seeks to address this gap by delving into the fundamental aspects of MFG inverse problems, with a particular emphasis on issues of unique identifiability, stability, and reconstruction of unknown parameters. These problems are not only mathematically challenging but also of immense practical significance, as they enable the calibration and validation of MFG models using real-world data. This book is intended to serve as a valuable resource for researchers interested in the theory and applications of MFGs, particularly in inverse problems. Through this book, we hope to inspire further exploration and innovation in this dynamic and interdisciplinary area of study.

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arXiv2025

Simultaneously decoding the unknown stationary state and function parameters for mean field games

Mean field games (MFGs) offer a versatile framework for modeling large-scale interactive systems across multiple domains. This paper builds upon a previous work, by developing a state-of-the-art unified approach to decode or design the unknown stationary state of MFGs, in addition to the underlying parameter functions governing their behavior. This result is novel, even in the general realm of inverse problems for nonlinear PDEs. By enabling agents to distill crucial insights from observed data and unveil intricate hidden structures and unknown states within MFG systems, our approach surmounts a significant obstacle, enhancing the applicability of MFGs in real-world scenarios. This advancement not only enriches our understanding of MFG dynamics but also broadens the scope for their practical deployment in various contexts.

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arXiv2024

Decoding a mean field game by the Cauchy data around its unknown stationary states

In recent years, mean field games (MFGs) have garnered considerable attention and emerged as a dynamic and actively researched field across various domains, including economics, social sciences, finance, and transportation. The inverse design and decoding of MFGs offer valuable means to extract information from observed data and gain insights into the intricate underlying dynamics and strategies of these complex physical systems. This paper presents a novel approach to the study of inverse problems in MFGs by analyzing the Cauchy data around their unknown stationary states. This study distinguishes itself from existing inverse problem investigations in three key significant aspects: Firstly, we consider MFG problems in a highly general form. Secondly, we address the technical challenge of the probability measure constraint by utilizing Cauchy data in our inverse problem study. Thirdly, we enhance existing high order linearization methods by introducing a novel approach that involves conducting linearization around non-trivial stationary states of the MFG system, which are not a-priori known. These contributions provide new insights and offer promising avenues for studying inverse problems for MFGs. By unraveling the hidden structure of MFGs, researchers and practitioners can make informed decisions, optimize system performance, and address real-world challenges more effectively.

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arXiv2024

Inverse problems for coupled nonlocal nonlinear systems arising in mathematical biology

In this paper, we propose and study several inverse problems of determining unknown parameters in nonlocal nonlinear coupled PDE systems, including the potentials, nonlinear interaction functions and time-fractional orders. In these coupled systems, we enforce non-negativity of the solutions, aligning with realistic scenarios in biology and ecology. There are several salient features of our inverse problem study: the drastic reduction in measurement/observation data due to averaging effects, the nonlinear coupling between multiple equations, and the nonlocality arising from fractional-type derivatives. These factors present significant challenges to our inverse problem, and such inverse problems have never been explored in previous literature. To address these challenges, we develop new and effective schemes. Our approach involves properly controlling the injection of different source terms to obtain multiple sets of mean flux data. This allows us to achieve unique identifiability results and accurately determine the unknown parameters. Finally, we establish a connection between our study and practical applications in biology, further highlighting the relevance of our work in real-world contexts.

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arXiv2024

Determining state space anomalies in mean field games

In this paper, we are concerned with the inverse problem of determining anomalies in the state space associated with the stationary mean field game (MFG) system. We establish novel unique identifiability results for the intrinsic structure of these anomalies in mean field games systems, including their topological structure and parameter configurations, in several general scenarios of practical interest, including traffic flow, market economics and epidemics. To the best of our knowledge, this is the first work that considers anomalies in the state space for the nonlinear coupled MFG system.

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arXiv2024

On the Obstacle Problem in Fractional Generalised Orlicz Spaces

We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $g$-Laplacian $\mathcal{L}_g^s$, with $0<s<1$. We prove the strict T-monotonicity of $\mathcal{L}_g^s$ and we obtain the Lewy-Stampacchia inequalities. We consider the approximation of the solutions through semilinear problems, for which we prove a global $L^\infty$-estimate, and we extend the local Hölder regularity to the solutions of the obstacle problems in the case of the fractional $p(x,y)$-Laplacian operator. We make further remarks on a few elementary properties of related capacities in the fractional generalised Orlicz framework, with a special reference to the Hilbertian nonlinear case in fractional Sobolev spaces.

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arXiv2024

On inverse problems in multi-population aggregation models

This paper focuses on inverse problems arising in studying multi-population aggregations. The goal is to reconstruct the diffusion coefficient, advection coefficient, and interaction kernels of the aggregation system, which characterize the dynamics of different populations. In the theoretical analysis of the physical setup, it is crucial to ensure non-negativity of solutions. To address this, we employ the high-order variation method and introduce modifications to the systems. Additionally, we propose a novel approach called transformative asymptotic technique that enables the recovery of the diffusion coefficient preceding the Laplace operator, presenting a pioneering method for this type of problems. Through these techniques, we offer comprehensive insights into the unique identifiability aspect of inverse problems associated with multi-population aggregation models.

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arXiv2024

On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-Δ_p^s$ operator, when the fractional parameter $s\toσ$ for $0<σ\leq1$, converge to the solution of the corresponding obstacle problem for $-Δ_p^σ$, being $σ=1$ the classical obstacle problem for the local $p$-Laplacian. We discuss the weak stability of the quasi-characteristic functions of coincidence sets of the solution with the obstacle, which is a strong convergence of their characteristic functions when $s\nearrow 1$ under a nondegeneracy condition. This stability can be shown also in terms of the convergence of the free boundaries, as well as of the coincidence sets, in Hausdorff distance when $s\nearrow 1$, under non-degeneracy local assumptions on the external force and a local topological property of the coincidence set of the limit classical obstacle problem for the local $p$-Laplacian, essentially when the limit coincidence set is the closure of its interior.

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arXiv2023

On inverse problems in predator-prey models

In this paper, we consider the inverse problem of determining the coefficients of interaction terms within some Lotka-Volterra models, with support from boundary observation of its non-negative solutions. In the physical background, the solutions to the predator-prey model stand for the population densities for predator and prey and are non-negative, which is a critical challenge in our inverse problem study. We mainly focus on the unique identifiability issue and tackle it with the high-order variation method, a relatively new technique introduced by the second author and his collaborators. This method can ensure the positivity of solutions and has broader applicability in other physical models with non-negativity requirements. Our study improves this method by choosing a more general solution $(u_0,v_0)$ to expand around, achieving recovery for all interaction terms. By this means, we improve on the previous results and apply this to physical models to recover coefficients concerning compression, prey attack, crowding, carrying capacity, and many other interaction factors in the system. Finally, we apply our results to study three specific cases: the hydra-effects model, the Holling-Tanner model and the classic Lotka-Volterra model.

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arXiv2023

Determining Sources in the Bioluminescence Tomography Problem

In this paper, we revisit the bioluminescence tomography (BLT) problem, where one seeks to reconstruct bioluminescence signals (an internal light source) from external measurements of the Cauchy data. As one kind of optical imaging, the BLT has many merits such as high signal-to-noise ratio, non-destructivity and cost-effectiveness etc., and has potential applications such as cancer diagnosis, drug discovery and development as well as gene therapies and so on. In the literature, BLT is extensively studied based on diffusion approximation (DA) equation, where the distribution of peak sources is to be reconstructed and no solution uniqueness is guaranteed without adequate a priori information. Motivated by the solution uniqueness issue, several theoretical results are explored. The major contributions in this work that are new to the literature are two-fold: first, we show the theoretical uniqueness of the BLT problem where the light sources are in the shape of $C^2$ domains or polyhedral- or corona-shaped; second, we support our results with plenty of problem-orientated numerical experiments.

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arXiv2023

Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems

In this work, we are concerned with inverse problems involving poly-fractional operators, where the poly-fractional operator is of the form \[P( (-Δ_g)^s)u := \sum_{i=1}^M α_i(-Δ_{g_i})^{s_i}u\] for $s=(s_1,\dots,s_M)$, $0<s_1<\cdots<s_M<\infty$, $s_M\in\mathbb{R}_+\backslash\mathbb{Z}$, $g=(g_1,\dots,g_M)$. There are three major contributions in this work that are new to the literature. First, we propose equations involving such poly-fractional operators $P$, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. Secondly, we give novel results for the unique continuation properties for fractional polyharmonic $u$, in the sense that $u$ satisfies $\tilde{P}((-Δ_{\tilde{g}})^{\tilde{s}})=0$ in a bounded Lipschitz domain $Ω$ for some $\tilde{P}$. With these results in hand, we consider the inverse problems for $P$, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator.

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arXiv2023

Determining a parabolic system by boundary observation of its non-negative solutions with biological applications

In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in PDE analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.

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arXiv2022

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $Ω\subset\mathbb{R}^d$ with time-dependent Dirichlet boundary condition for the temperature $\vartheta=\vartheta(x,t)$, $\vartheta=g$ on $Ω^c\times]0,T[$, and initial condition $η_0$ for the enthalpy $η=η(x,t)$, given in $Ω\times]0,T[$ by \[\frac{\partial η}{\partial t} +\mathcal{L}_A^s \vartheta= f\quad\text{ with }η\in β(\vartheta),\] where $\mathcal{L}_A^s$ is an anisotropic fractional operator defined in the distributional sense by \[\langle\mathcal{L}_A^su,v\rangle=\int_{\mathbb{R}^d}AD^su\cdot D^sv\,dx,\] $β$ is a maximal monotone graph, $A(x)$ is a symmetric, strictly elliptic and uniformly bounded matrix, and $D^s$ is the distributional Riesz fractional gradient for $0<s<1$. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $s\nearrow 1$ towards the classical local problem, the asymptotic behaviour as $t\to\infty$, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $β$.

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