arXiv subjects

Zhiliang Deng

Publications and source records attributed to Zhiliang Deng.

At least 19 recordsLinked to original sources

arXiv2026

Hankel-Christoffel-Nevai Screening of Posterior Relevance in Bayesian Inverse Problems

We introduce a Hankel--Christoffel--Nevai framework for screening posterior-relevant candidates in Bayesian inverse problems. A likelihood-weighted moment matrix records how Bayesian updating changes the geometry of the prior, and Christoffel and Nevai constructions convert this information into inexpensive relevance scores. The Christoffel ratio captures relative local moment mass, whereas the Nevai score provides a stable polynomially localized approximation of the likelihood. The moment matrix can be estimated from deterministic likelihood values, bounded unbiased marks, posterior samples, or binary unbiased likelihood observations. The construction extends to function-valued unknowns through nested feature maps, where the induced conditional likelihood has an exact Bayesian interpretation. We establish consistency and error bounds that separate feature resolution, polynomial localization, and pilot estimation. The same learned geometry can also be used to reorder exact stochastic likelihood blocks, reducing expected work without altering the posterior target. Numerical experiments on nonlinear, PDE-based, and function-space inverse problems demonstrate effective posterior-mass screening and computational savings.

stat.CO↗

arXiv2026

Likelihood-Coin Poisson Sampling for Bayesian Inverse Problems: Exact Sampling and Sharp Complexity

We develop an exact direct-sampling framework for Bayesian inverse problems whose bounded forward observables can be queried through Bernoulli events rather than numerically evaluated. A Bernstein--Poisson construction converts these events into scaled Gaussian likelihood coins. Thinning a prior-based Poisson point process with these coins yields a posterior point process whose locations, conditional on its cardinality, are independent posterior draws; the cardinality also provides an unbiased estimator of the reduced evidence. The main theoretical contribution is a sharp analysis of the actual early-stopped computational work. In the small-noise regime, the work is governed by the local prior-predictive mass near the exact-fit set, revealing a predictive-dimension effect distinct from the nominal dimension of the unknown. A multivariate extension accommodates correlated Gaussian observation errors and gives a precision-weighted allocation of factory effort. For a bounded class of elliptic resolvent problems, Feynman--Kac representations, Poisson killing, and lazy random-series evaluation provide exact continuum Bernoulli oracles without introducing a spatial discretization or a fixed parameter truncation in the posterior target. Numerical experiments illustrate posterior thinning in parameter space, confirm the sharp work regimes, and validate the function-valued PDE construction.

math.ST↗

arXiv2026

Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems

Bayesian targets may converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this probability--sensitivity mismatch and its consequences for Metropolized Hamiltonian proposals. A vanishing-amplitude wiggly-energy model first gives the basic analytic obstruction: the potential perturbation tends to zero while its classical derivative is of order $r_\varepsilon/\varepsilon$. We then show that the same scaling arises naturally in a periodic elliptic inverse problem, where homogenization makes the forward map and Gaussian likelihood converge while differentiation with respect to a microscopic scale parameter retains an $O(1)$ oscillatory contribution. This provides a PDE origin for the single-scale wiggly mechanism. The main construction concerns a more demanding nested Weierstrass hierarchy, interpreted as an analytically tractable prototype for repeated corrector contributions across geometrically separated scales. There all previously resolved scales persist, adjacent classical-force increments grow geometrically like $(ab)^N$, and the limiting rough component may fail to possess a classical derivative. In this self-similar setting the matched Jackson quotient is structurally adapted to the refinement through dilation covariance and exact finite closure. Uniform negative-log-likelihood approximation yields explicit total-variation, Hellinger, and bounded quantity-of-interest bounds. Measurable kick--drift--kick maps remain exact after Metropolis correction, local field convergence propagates to fixed-length proposals and kernels, and the first classical HMC half-kick can have no fixed-step refinement limit. Numerical experiments on scale-structured inverse problems test the resulting resolution-stability mechanism across one- and two-dimensional inverse problems.

math.NA↗

arXiv2026

Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra

We study sparse Mellin spectral recovery from geometric samples. Dilation covariance is shown to generate the Hankel structure directly, rather than only after reduction to a classical exponential-sum model. For complex exponents, incommensurate sampling scales remove the logarithmic aliasing that persists at a single scale. A second recovery channel is obtained from the minimal Hankel pencil: contour winding counts spectral nodes regionally, while Rouché-type bounds certify isolated nodes and unresolved clusters under noise. The associated contour margin has an explicit degeneration rate as the sampling ratio approaches one. Numerical experiments show that the auxiliary scale improves both identifiability and resolution, and that certified contour counts can remain reliable beyond the regime of accurate pointwise recovery.

math.NA↗

arXiv2026

Fourier--Hankel Moment Recovery in Acoustic Scattering: Multichannel Stabilization and Radial Interface Resolution

We study direct recovery of visible phase centers and concentric radial interfaces from full-aperture acoustic far-field data under the Born approximation. Two complementary moment structures are extracted from the two-angle Fourier matrix. At fixed positive total Fourier order, the nonnegative-order channels share the same leading phase-center moment, and a generalized least-squares combination yields an exact variance gain over a single Fourier row. The corresponding Hankel rank and shifted pencil recover distinct phase centers. Signed moments reveal off-center cavities but become degenerate when material and cavity centers coincide. Zero-total-order coefficients retain complementary radial Bessel moments. For a piecewise-constant radial average, low-frequency extrapolation produces a second finite exponential sequence whose nodes are the squared interface radii. We establish rank and perturbation results for both reductions. Numerical experiments verify multichannel stabilization, concentric-cavity resolution, and recovery of multiple radial interfaces. A final full-wave Helmholtz experiment, generated without the Born substitution, assesses the Born-derived reconstruction under model mismatch and shows how nonlinear scattering eventually appears as an additional Hankel tail.

math.NA↗

arXiv2026

Dual-Filtration Topology Recovery from Qualitative Acoustic Scattering Indicators

Qualitative inverse scattering methods produce gray-scale indicator fields, from which the topology of the scatterer must be inferred without a prescribed threshold. We propose a dual-filtration method that estimates the component and cavity structure directly from the indicator. Superlevel sets are used to recover the exterior components, while sublevel persistent homology identifies interior cavities associated with the reconstructed envelope. The two types of information are combined into a binary reconstruction, allowing different topological features to be resolved at different intensity levels. We establish geometric and stability results that characterize when the topology can be reliably detected. Full-wave Lippmann--Schwinger experiments with factorization-method indicators show accurate recovery for well-resolved configurations and clear resolution transitions as component separation and cavity size vary.

math.NA↗

arXiv2026

Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes

Identification of dominant polynomial-chaos modes is usually formulated as a sparse-regression problem on a sampled multivariate polynomial dictionary. We develop coded Hankel polynomial chaos (CH-PC), a complementary spectral formulation for dominant-mode identification. A finite generating transform converts PCE coefficients into a coefficient-generating polynomial, and evaluation along a geometric phase orbit produces a finite exponential sum. Its model order and spectral nodes are encoded by low-rank Hankel matrices, while coordinate phase shifts attach root-of-unity labels from which the full polynomial multi-indices are recovered. Coordinate-shifted probes are combined as common-node snapshots, and independent phase encodings provide redundant representations when a single spectral encoding is poorly conditioned. For finite observations, population, finite-data, and observed probes are kept distinct: sampling or quadrature error and observation error enter as separate Hankel perturbations, which are then connected to spectral stability, discrete decoding, and phase voting. For tensor-product candidate sets, the generating kernel factorizes into one-dimensional sums and can be evaluated without assembling the full multivariate PCE design matrix. Numerical experiments on sparse Legendre benchmarks and a stochastic Darcy problem illustrate exact recovery, noise stabilization, unknown-order identification by phase persistence, and dominant-mode recovery for a PDE-generated quantity of interest.

stat.ML↗

arXiv2026

Contour Hankel dynamics and indicator fields for the Riemann $Ξ$-function

We develop a moving-contour Hankel framework for encoding local zero configurations of the Riemann $Ξ$-function. Weighted contour integrals of the logarithmic derivative $Ξ'/Ξ$, expressed in a holomorphic coordinate associated with the contour, are identified with the power moments of a finite atomic measure supported at the coordinate images of the enclosed zeros. This representation yields exact zero-free and rank criteria and, for conjugation-compatible contour-coordinate pairs, an inertia formula: once the matrix order is at least the number of distinct coordinate nodes, the negative index equals the number of distinct nonreal conjugate pairs. Consequently, the Riemann hypothesis admits a local finite-dimensional Hankel-positivity formulation, although establishing this positivity independently of the zero set remains unresolved. As the contour moves, the Hankel matrix evolves by a continuous congruence flow between zero crossings and undergoes finite-rank jumps at crossing events. An isolated zero produces a signed rank-one jump, whereas a nonreal conjugate pair produces a rank-two indefinite event in a real-axis circular scan that meets the pair. Removing the continuous coordinate drift yields a piecewise-constant matrix process from which crossing coordinates, multiplicities, and zero locations can be recovered. Numerical experiments validate the contour quadrature, indicator fields, continuous flow, crossing signatures, and recovery procedure.

math.NA↗

arXiv2026

Representation--Symbol Correspondence for $q$-Hamiltonian Mechanics on the Quantum Plane

We develop a representation--symbol correspondence for $q$-Hamiltonian mechanics on the quantum plane. The coordinate algebra and its covariant differential calculus are realized simultaneously on a smooth commutative function space by multiplication, dilation, and Jackson operators. Normal ordering identifies the coordinate algebra with a polynomial symbol space and transports operator composition to an explicit associative star product. Under this correspondence, the induced covariant $q$-derivatives intertwine exactly with the Jackson operators, and the ordered Hamiltonian action descends to an exact star-Jackson action. For the coordinate observables, the relevant Jackson derivatives are constants, so the star factors reduce to the unit and the Jackson coordinate equations are exact symbol images of the formal quantum-plane equations. For nonlinear observables, pointwise multiplication produces a genuine commutativization error. We quantify this distinction through a first-order expansion of the operator quantization map, study the derivation, divergence, and energy defects of the induced pointwise Jackson dynamics, and prove convergence of the represented Hamiltonian, the symbol actions, the Jackson vector field, and its finite-time trajectories to their classical counterparts as $q\to1$. The resulting framework connects covariant quantum-plane differential calculus, deformation products, and Hamiltonian dynamics while keeping exact algebraic statements separate from commutative and classical approximations.

math-ph↗

arXiv2026

Contour-count indicator fields for visible pole clusters in meromorphic continuation

We develop a contour-count indicator method for visible pole clusters in outward meromorphic continuation from circular boundary data. The method starts from determinant characteristics built from positive Fourier coefficients. In the pure finite-pole model, the correct determinant characteristic factors into a polynomial whose zeros are the reciprocals of the exterior poles. In the presence of a holomorphic background, finite sampling, and noise, roots of individual determinants are unstable and are used only as local evidence. We aggregate this evidence into a scalar indicator field on the reciprocal pole plane: at each sampling point, the field records the fraction of determinant orders and shifts for which a small contour centered at that point encloses exactly one empirical determinant zero. The resulting field plays the role of a sampling-type imaging functional for pole visibility. Fixed superlevel sets give visible-pole clusters, while zero-dimensional persistent homology is used only as a threshold-robust post-processing step. We prove deterministic results linking pure-pole contour counts, Rouché stability, indicator-field contrast, fixed-threshold component recovery, and persistence-gap stability. These results explain why isolated poles with sufficient residue and separation generate stable high-value components, whereas weak, close, boundary-near, or noise-dominated poles may give low, short-lived, or merged components. The framework is a cluster-certification and imaging method, not an unconditional all-pole recovery procedure.

math.NA↗

arXiv2026

Determinant Characteristics and Argument-Principle Certification for Visible Poles in Meromorphic Continuation

We study outward meromorphic continuation from circular boundary data in the unit disk. The unknown function is holomorphic in the unit disk and admits a meromorphic continuation to a larger disk, where finitely many exterior simple poles are superposed on an unknown holomorphic background. The positive Fourier coefficients of the boundary trace are Taylor coefficients at the origin, and exterior poles generate a finite exponential-sum component in these coefficients. We introduce shifted determinant characteristics and prove that, in the pure finite-pole model, the determinant for the correct order factors exactly into a nonzero constant times the polynomial whose zeros are the reciprocals of the exterior poles. The same zero set is obtained for noiseless equispaced discrete Fourier coefficients; sampling changes only the amplitudes through an aliasing factor. For data containing a holomorphic background, discretization effects, and noise, roots of a single empirical determinant are only candidate reciprocal poles. We therefore propose a root-propose and contour-certify procedure: determinant roots generate candidate regions, while local argument-principle counts, contour moments, empirical margins, and persistence over determinant orders and shifts certify visible poles. A Rouché-type perturbation analysis gives sufficient conditions for stable local zero counts and explains how residues, pole separation, distance to the target annulus boundary, shifts, and noise affect visibility. Numerical experiments verify the pure-pole identity, demonstrate certification under background and noise, and show that high noise, weak residues, boundary-near poles, and close poles naturally lead to partial recovery of contour-certified visible poles.

math.NA↗

arXiv2026

Hamiltonian Monte Carlo from $q$-deformed phase-space mechanics

Hamiltonian Monte Carlo (HMC) generates efficient Markov transitions by combining Hamiltonian dynamics with a Metropolis correction. This paper develops a geometric $q$-analogue of HMC by replacing classical Hamiltonian dynamics with a $q$-deformed Hamiltonian system arising from $q$-calculus. Starting from a Lagrangian formulation, we derive the corresponding $q$-Hamiltonian equations and prove the formal invariance of the associated $q$-symplectic form within the $q$-deformed differential calculus. To obtain a computable sampler, we introduce a Jackson-derivative realization and construct a Metropolis-corrected $q$-HMC algorithm. The proposal reduces to classical HMC as $q\to1$, while for $q\neq1$ it replaces ordinary derivatives by $q$-Jackson finite differences. We establish detailed balance, which ensures that the resulting Markov transition preserves the target distribution. Numerical experiments examine the computational behavior of the proposed method. For positive-scale black-box targets, the $q$-Jackson force has a scale-consistent interpretation: multiplicative perturbations of $s>0$ correspond to centered finite differences in $y=\log s$. In such examples, $q$-HMC closely tracks log-coordinate finite-difference HMC and the exact-gradient benchmark, whereas raw additive finite differences may produce large force and Hamiltonian errors. These results suggest that the proposed $q$-analogue provides a valid HMC-type sampling framework with a visible advantage for positive and multiplicative black-box targets.

math.NA↗

arXiv2026

A persistent-homology-Gaussian prior for solving infinite-dimensional Bayesian inverse scattering problems

Bayesian inference methods have been developed to address inverse problems in function spaces where the unknown parameters are of infinite dimension. However, conventional Gaussian priors remain inadequate for reconstructing discontinuous or sharply varying target functions encountered in practical applications like obstacle reconstruction. Although hybrid priors have emerged as a promising solution, significant challenges remain in developing theoretically rigorous and computationally tractable frameworks in engineering applications. To address these issues, we propose a persistent-homology-Gaussian (PHG) prior for solving the acoustic obstacle scattering inverse problem in the infinite-dimensional Bayesian setting, which combines a weighted persistence-based regularization term with a periodic Gaussian reference measure through a Gibbs tilt. Then, the complex boundary is represented by a log-radial function on the unit circle, so that the reconstruction from far-field data is formulated as a function-space inverse problem. The well-posedness of the resulting posterior measure is established in the Hellinger, total variation, and Wasserstein-$p$ metrics. Furthermore, the convergence of finite-dimensional posterior approximations is obtained, and posterior sampling is performed by a preconditioned Crank--Nicolson (pCN) method. Numerical experiments show that the proposed PHG prior yields accurate and stable reconstructions under more extensive noisy conditions, providing explicit control of multiscale topological features and better performance compared to other conventional priors.

math.NA↗

arXiv2026

A Moment--Hankel Rank Method for Identifying the Number of Point Sources in the Heat Equation

We develop a low-frequency moment--Hankel rank method for identifying the number of time-independent point sources in the heat equation from boundary flux data. The method is formulated in the unit disk, where the Laplace-transformed and normalized boundary flux admits an explicit Fourier moment representation. By taking the low-frequency limit, we obtain a finite exponential-sum moment sequence in which the nodes encode the source locations, the weights encode the source strengths, and the number of terms equals the number of point sources. The associated Hankel matrix admits a Vandermonde factorization, and its rank is exactly the source number under the natural assumptions that the source locations are distinct and the source strengths are nonzero. We also analyze the effect of discrete and noisy boundary data. A uniform moment perturbation bound is propagated to the empirical Hankel matrix, and Weyl's singular-value perturbation inequality yields a sufficient condition for stable numerical rank recovery in terms of the smallest nonzero singular value of the ideal Hankel matrix. Numerical experiments confirm the exact rank pattern in the noiseless case, validate the stability threshold under moment noise, and illustrate the loss of resolution for close or weak sources. After the source number is identified, the same moment sequence can be used for location and strength recovery through an annihilating-polynomial and Vandermonde reconstruction procedure.

math.NA↗

arXiv2026

A Hankel determinant zero-order principle for source counting in an inverse heat point-source problem

This paper studies the identification of an unknown number of stationary point sources in a two-dimensional heat equation from boundary flux measurements. Unlike many reconstruction approaches that assume the number of sources to be known in advance, we develop a determinant-based counting method that extracts this number directly from the measured data. In the unit disk, the Laplace-transformed and normalized boundary flux admits a Fourier moment representation whose low-frequency limit has a finite exponential-sum structure. This structure leads to a family of Hankel matrices and associated determinant characteristics. We prove that, under a generic determinant lifting condition, the vanishing order of the Hankel determinant at the zero Laplace frequency changes exactly when the Hankel order exceeds the true number of sources. Consequently, the source number is characterized by the first nonzero contour count of the determinant characteristic through the argument principle. We further establish a Rouché-type stability result showing that the determinant zero count is preserved under sufficiently small perturbations induced by measurement noise, boundary discretization, and time truncation. After the source number is identified, the source locations and strengths are recovered from the low-frequency moment sequence by an annihilating-polynomial and Vandermonde reconstruction procedure. Numerical experiments confirm the predicted count pattern, demonstrate robustness with respect to contour selection, illustrate the role of the Rouché margin under noise and near-degenerate configurations, and validate the subsequent recovery of source locations and strengths.

math.NA↗

arXiv2026

A Bayesian approach with persistent homology prior for Robin coefficient identification in a parabolic problem

The reconstruction of time-dependent Robin coefficients is a challenging inverse heat transfer problem due to its inherent ill-posedness. This paper introduces a hierarchical Bayesian approach integrated with a persistent homology (PH) prior for robust coefficient estimation. By quantifying the birth and death of topological features, the PH-based prior provides a global structural constraint that transcends local derivative based penalties. Numerical experiments show that this topological perspective allows for the preservation of complex temporal profiles without the typical staircase distortions of total variation (TV) priors or the excessive blurring of Gaussian models. A key feature of our framework is the hierarchical implementation, which yields an automated, data-driven selection of hyperparameters. The results demonstrate that while PH-based inference yields competitive accuracy compared to TV regularization, it offers superior performance in preserving the multiscale characteristics of the Robin coefficient, providing a robust alternative for convective heat transfer diagnostics

stat.CO↗

arXiv2025

A novel viewpoint for Bayesian inversion based on the Poisson point process

We present a novel Bayesian framework for inverse problems in which the pos terior distribution is interpreted as the intensity measure of a Poisson point process (PPP). The posterior density is approximated using kernel density estimation, and the superposition property of PPPs is then exploited to enable efficient sampling from each kernel component. This methodology offers a new means of exploring the posterior distribution and facilitates the generation of independent and identically distributed samples, thereby enhancing the analysis of inverse problem solutions.

math.NA↗

arXiv2025

A persistent-homology-based Bayesian prior for potential coefficient reconstruction in an elliptic PDE

We address the reconstruction of a potential coefficient in an elliptic partial differential equation from distributed observations within the Bayesian framework. The choice of prior distribution is crucial in such inverse problems, particularly when the target function exhibits sharp discontinuities that conventional Gaussian priors fail to capture effectively. To overcome this limitation, we introduce a novel prior based on persistent homology (PH), which quantifies and encodes the topological features of candidate functions through their persistent pairs. To ensure a well-defined distribution in infinite-dimensional spaces, the prior is constructed with respect to a Gaussian reference measure. A significant advantage over classical approaches is that the PH prior only requires the unknown functions to belong to a suitable topological space, which substantially enhances its applicability. Numerical results demonstrate that the proposed PH prior outperforms the Gaussian prior and achieves a modest yet consistent improvement over the classical total variation (TV) prior.

math.NA↗