arXiv subjects

Yat Tin Chow

Publications and source records attributed to Yat Tin Chow.

At least 19 recordsLinked to original sources

arXiv2026

Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media

We develop the first rigorous topological framework for an auxiliary boundary-integral model motivated by physical scaling identities for wave scattering from self-similar or fractal media. The model is periodic in logarithmic scale, and a multiplicative Bloch-Floquet-Zak transform fiberizes its interscale coupling. For sufficiently small, well-separated components, equilibrium densities define a computable finite-dimensional projected matrix, while Riesz projections select the corresponding invariant spectral subspaces of the full boundary symbol. Under suitable spectral-isolation and point-gap conditions, and after recentering the two families at their respective same-scale reference values, we prove that the determinant winding of the exact Riesz-reduced family agrees with that of the projected matrix family. For regular-simplex configurations, we derive explicit nonzero winding formulas and track the resulting local winding data across finite prefractal levels and along a geometric sequence of wavenumbers. We also formulate conditional winding data for multiple dilation centers and show that the Zak phase of the chiral Hermitianization equals $π$ times the point-gap winding modulo $2π$. Thus, the scale-periodic boundary model admits a rigorous topological reduction.

math.SP↗

arXiv2026

Truncated Differentiation Through Primal-Dual Solvers for Inverse Potential Mean-Field Games

We study inverse potential mean-field games (MFGs), in which an unknown spatial inverse-cost (mobility) map is inferred from observed population densities. We solve the forward MFG with a preconditioned primal-dual hybrid gradient (PDHG) method and develop Jacobian-free backpropagation (JFB-$r$), which records only the final $r$ iterations from a detached warm start while retaining the full forward solve. To analyze this truncated differentiation method, we show that the exact-proximal dual-extrapolated PDHG map is a metric resolvent of the maximal monotone KKT operator. This resolvent view shows that JFB-$r$ exactly differentiates a finite-trajectory surrogate and, under a locally fixed active set at an exact equilibrium detach point, converges to the implicit gradient as the tracked depth increases. Across several inverse-MFG settings, numerical experiments show that JFB-r at moderate tracked depths can achieve recovery accuracy comparable to full unrolling while reducing memory and runtime.

math.OC↗

arXiv2025

A notion of fractality for a class of states and noncommutative relative distance zeta functional

In this work, we first recall the definition of the relative distance zeta function in [42, 43, 44, 46, 47] and slightly generalize this notion from sets to probability measures, and then move on to propose a novel definition a relative distance (and tube) zeta functional for a class of states over a C* algebra. With such an extension, we look into the chance to define relative Minkowski dimensions in this context, and explore the notion of fractality for this class of states. Relative complex dimensions as poles of this newly proposed relative distance zeta functional, as well as its geometric and transformation properties, decomposition rules and properties that respects tensor products are discussed. We then explore some examples that possess fractal properties with this new zeta functional and propose functional equations similar to [11,35,36,42].

math-ph↗

arXiv2025

Koopman-Hopf Hamilton-Jacobi Reachability and Control

The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional differential games, producing the set of initial states and corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable method, the Hopf formula avoids the curse of dimensionality that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf solution with Koopman theory for global linearization. By first lifting a nonlinear system to a linear space and then solving the Hopf formula, approximate reachable sets can be efficiently computed that are much more accurate than local linearizations. Furthermore, we construct a Koopman-Hopf disturbance-rejecting controller, and test its ability to drive a 10-dimensional nonlinear glycolysis model. We find that it significantly out-competes expectation-minimizing and game-theoretic model predictive controllers with the same Koopman linearization in the presence of bounded stochastic disturbance. In summary, we demonstrate a dimension-robust method to approximately solve HJR, allowing novel application to analyze and control high-dimensional, nonlinear systems with disturbance. An open-source toolbox in Julia is introduced for both Hopf and Koopman-Hopf reachability and control.

eess.SY↗

arXiv2025

Conservative Linear Envelopes for Nonlinear, High-Dimensional, Hamilton-Jacobi Reachability

Hamilton-Jacobi reachability (HJR) provides a value function that encodes the set of states from which a system with bounded control inputs can reach or avoid a target despite any bounded disturbance, and the corresponding robust, optimal control policy. Though powerful, traditional methods for HJR rely on dynamic programming (DP) and suffer from exponential computation growth with respect to state dimension. The recently favored Hopf formula mitigates this ``curse of dimensionality'' by providing an efficient and space-parallelizable approach for solving the reachability problem. However, the Hopf formula can only be applied to linear time-varying systems. To overcome this limitation, we show that the error between a nonlinear system and a linear model can be transformed into an adversarial bounded artificial disturbance. One may then solve the dimension-robust generalized Hopf formula for a linear game with this ``antagonistic error" to perform guaranteed conservative reachability analysis and control synthesis of nonlinear systems; this can be done for problem formulations in which no other HJR method is both computationally feasible and guaranteed. In addition, we offer several technical methods for reducing conservativeness in the analysis. We demonstrate the effectiveness of our results through one illustrative example (the controlled Van der Pol system) that can be compared to standard DP, and one higher-dimensional 15D example (a 5-agent pursuit-evasion game with Dubins cars).

eess.SY↗

arXiv2025

State-Augmented Linear Games with Antagonistic Error for High-Dimensional, Nonlinear Hamilton-Jacobi Reachability

Hamilton-Jacobi Reachability (HJR) is a popular method for analyzing the liveness and safety of a dynamical system with bounded control and disturbance. The corresponding HJ value function offers a robust controller and characterizes the reachable sets, but is traditionally solved with Dynamic Programming (DP) and limited to systems of dimension less than six. Recently, the space-parallelizeable, generalized Hopf formula has been shown to also solve the HJ value with a nearly three-log increase in dimension limit, but is limited to linear systems. To extend this potential, we demonstrate how state-augmented (SA) spaces, which are well-known for their improved linearization accuracy, may be used to solve tighter, conservative approximations of the value function with any linear model in this SA space. Namely, we show that with a representation of the true dynamics in the SA space, a series of inequalities confirms that the value of a SA linear game with antagonistic error is a conservative envelope of the true value function. It follows that if the optimal controller for the HJ SA linear game with error may succeed, it will also succeed in the true system. Unlike previous methods, this result offers the ability to safely approximate reachable sets and their corresponding controllers with the Hopf formula in a non-convex manner. Finally, we demonstrate this in the slow manifold system for clarity, and in the controlled Van der Pol system with different lifting functions.

eess.SY↗

arXiv2025

Stochastic Moving Anchor Algorithms and a Popov's Scheme with Moving Anchor

Since their introduction, anchoring methods in extragradient-type saddlepoint problems have inspired a flurry of research due to their ability to provide order-optimal rates of accelerated convergence in very general problem settings. Such guarantees are especially important as researchers consider problems in artificial intelligence (AI) and machine learning (ML), where large problem sizes demand immense computational power. Much of the more recent works explore theoretical aspects of this new acceleration framework, connecting it to existing methods and order-optimal convergence rates from the literature. However, in practice introducing stochastic oracles allows for more computational efficiency given the size of many modern optimization problems. To this end, this work provides the moving anchor variants [1] of the original anchoring algorithms [36] with stochastic implementations and robust analyses to bridge the gap from deterministic to stochastic algorithm settings. In particular, we demonstrate that an accelerated convergence rate theory for stochastic oracles also exists for our moving anchor scheme, itself a generalization of the original fixed anchor algorithms, and provide numerical results that validate our theoretical findings. We also develop a tentative moving anchor Popov scheme based on the work in [33], with promising numerical results pointing towards an as-of-yet uncovered general convergence theory for such methods.

math.OC↗

arXiv2025

Inexact Proximal Point Algorithms for Zeroth-Order Global Optimization

This work concerns the zeroth-order global minimization of continuous nonconvex functions with a unique global minimizer and possibly multiple local minimizers. We formulate a theoretical framework for inexact proximal point (IPP) methods for global optimization, establishing convergence guarantees under mild assumptions when either deterministic or stochastic estimates of proximal operators are used. The quadratic regularization in the proximal operator and the scaling effect of a parameter $δ>0$ create a concentrated landscape of an associated Gibbs measure that is practically effective for sampling. The convergence of the expectation under the Gibbs measure as $δ\to 0^+$ is established, and the convergence rate of $\mathcal O(δ)$ is derived under additional assumptions. These results provide a theoretical foundation for evaluating proximal operators inexactly using sampling-based methods such as Monte Carlo (MC) integration. In addition, we propose a new approach based on tensor train (TT) approximation. This approach employs a randomized TT cross algorithm to efficiently construct a low-rank TT approximation of a discretized function using a small number of function evaluations, and we provide an error analysis for the TT-based estimation. We then propose two practical IPP algorithms, TT-IPP and MC-IPP. The TT-IPP algorithm leverages TT estimates of the proximal operators, while the MC-IPP algorithm employs MC integration to estimate the proximal operators. Both algorithms are designed to adaptively balance efficiency and accuracy in inexact evaluations of proximal operators. The effectiveness of the two algorithms is demonstrated through experiments on diverse benchmark functions and various applications.

math.OC↗

arXiv2025

Moving Anchor Extragradient Methods For Smooth Structured Minimax Problems

This work introduces a moving anchor acceleration technique to extragradient algorithms for smooth structured minimax problems. The moving anchor is introduced as a generalization of the original algorithmic anchoring framework, i.e. the EAG method introduced in [32], in hope of further acceleration. We show that the optimal order of convergence in terms of worst-case complexity on the squared gradient, O(1/k2), is achieved by our new method (where k is the number of iterations). We have also extended our algorithm to a more general nonconvex-nonconcave class of saddle point problems using the framework of [14], which slightly generalizes [32]. We obtain similar order-optimal complexity results in this extended case. In both problem settings, numerical results illustrate the efficacy of our moving anchor algorithm variants, in particular by attaining the theoretical optimal convergence rate for first order methods, as well as suggesting a better optimized constant in the big O notation which surpasses the traditional fixed anchor methods in many cases. A proximal-point preconditioned version of our algorithms is also introduced and analyzed to match optimal theoretical convergence rates.

math.OC↗

arXiv2024

Monotone inclusion methods for a class of second-order non-potential mean-field games

We propose a monotone splitting algorithm for solving a class of second-order non-potential mean-field games. Following [Achdou, Capuzzo-Dolcetta, "Mean Field Games: Numerical Methods," SINUM (2010)], we introduce a finite-difference scheme and observe that the scheme represents first-order optimality conditions for a primal-dual pair of monotone inclusions. Based on this observation, we prove that the finite-difference system obtains a solution that can be provably recovered by an extension of the celebrated primal-dual hybrid gradient (PDHG) algorithm.

math.OC↗

arXiv2023

Surface concentration of transmission eigenfunctions

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in [10] on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based on generalized Weyl's law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions.

math.AP↗

arXiv2022

Quantum integrable systems and concentration of plasmon resonance

We are concerned with the quantitative mathematical understanding of surface plasmon resonance (SPR) when $d \geq 3$. SPR is the resonant oscillation of conducting electrons at the interface between negative and positive permittivity materials and forms the fundamental basis of many cutting-edge applications of metamaterials. It is recently found that the SPR concentrates due to curvature effect. In this paper, we derive sharper and more explicit characterisations of the SPR concentration at high-curvature places in both the static and quasi-static regimes. The study can be boiled down to analyzing the geometries of the so-called Neumann-Poincaré (NP) operators, which are certain pseudo-differential operators sitting on the interfacial boundary. We propose to study the joint Hamiltonian flow of an integral system given by the moment map defined by the NP operator. Via considering the Heisenberg picture and lifting the joint flow to a joint wave propagator, we establish a more general version of quantum ergodicity on each leaf of the foliation of this integrable system, which can then be used to establish the desired SPR concentration results. The mathematical framework developed in this paper leverages the Heisenberg picture of quantization and extends some results of quantum integrable system via generalising the concept of quantum ergodicity, which can be of independent interest to the spectral theory and the potential theory.

math.AP↗

arXiv2022

Continuous data assimilation for two-phase flow: analysis and simulations

We propose, analyze, and test a novel continuous data assimilation two-phase flow algorithm for reservoir simulation. We show that the solutions of the algorithm, constructed using coarse mesh observations, converge at an exponential rate in time to the corresponding exact reference solution of the two-phase model. More precisely, we obtain a stability estimate which illustrates an exponential decay of the residual error between the reference and approximate solution, until the error hits a threshold depending on the order of data resolution. Numerical computations are included to demonstrate the effectiveness of this approach, as well as variants with data on sub-domains. In particular, we demonstrate numerically that synchronization is achieved for data collected from a small fraction of the domain.

math.NA↗

arXiv2022

A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem

In this work, we consider a novel inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs. Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a \emph{finite number of boundary measurements} of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.

math.NA↗

arXiv2021

Localized sensitivity analysis at high-curvature boundary points of reconstructing inclusions in transmission problems

In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a high magnitude of mean curvature can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarisation tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via the microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one steps depend on the global geometry, and one particular step depends on the mean curvature at a given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave super-resolution imaging.

math.AP↗

arXiv2021

Recovery of a Time-Dependent Bottom Topography Function from the Shallow Water Equations via an Adjoint Approach

We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly recover the bottom topography using only noisy boundary data from one measurement event and the inclusion of two regularization terms in the iterative update scheme. The adjoint scheme is determined from a linearization of the forward system and is used to compute the gradient of a cost function. The bottom topography function is recovered through an iterative process given by a three-operator splitting method which allows the feasibility to include two regularization terms. Numerous numerical tests demonstrate the robustness of the method regardless of the choice of initial guess and in the presence of discontinuities in the solution of the forward problem.

math.OC↗

arXiv2020

Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes

We present the discovery of a novel and intriguing global geometric structure of the (interior) transmission eigenfunctions associated with the Helmholtz system. It is shown in generic scenarios that there always exists a sequence of transmission eigenfunctions with the corresponding eigenvalues going to infinity such that those eigenfunctions are localized around the boundary of the domain. We provide a comprehensive and rigorous justification in the case within the radial geometry, whereas for the non-radial case, we conduct extensive numerical experiments to quantitatively verify the localizing behaviours. The discovery provides a new perspective on wave localization. As significant applications, we develop a novel inverse scattering scheme that can produce super-resolution imaging effects and propose a method of generating the so-called pseudo surface plasmon resonant (PSPR) modes with a potential sensing application.

math.AP↗

arXiv2020

A Direct Sampling Method for the Inversion of the Radon Transform

We propose a novel direct sampling method (DSM) for the effective and stable inversion of the Radon transform. The DSM is based on a generalization of the important almost orthogonality property in classical DSMs to fractional order Sobolev duality products and to a new family of probing functions. The fractional order duality product proves to be able to greatly enhance the robustness of the reconstructions in some practically important but severely ill-posed inverse problems associated with the Radon transform. We present a detailed analysis to better understand the performance of the new probing and index functions, which are crucial to stable and effective numerical reconstructions. The DSM can be computed in a very fast and highly parallel manner. Numerical experiments are carried out to compare the DSM with a popular existing method, and to illustrate the efficiency, stability, and accuracy of the DSM.

math.NA↗