Ping Liu

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Liu

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Ping

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0000-0002-7857-7040

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Publications 1 - 10 of 25

Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
Ammari, Habib; Barandun, Silvio Alberto; De Bruijn, Yannick; et al. (2024)
SAM Research Report
We establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials, proving the observation and conjecture in [H. Ammari et al., arXiv:2307.13551]. We also numerically verify our theory for these systems.
Tunable localisation in parity-time-symmetric resonator arrays with imaginary gauge potentials
Ammari , Habib; Barandun, Silvio Alberto; Liu, Ping; et al. (2025)
Forum of Mathematics, Sigma
The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in the deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity - an imaginary gauge potential and on-site gain and loss - we prove that all but two eigenmodes of the system pass through exceptional points and decouple. By tuning the gain-to-loss ratio, the system changes from a phase with unbroken parity-time symmetry to a phase with broken parity-time symmetry. At the macroscopic level, this is observed as a transition from symmetrical eigenmodes to condensated eigenmodes at one edge of the structure. Mathematically, it arises from a topological state change. The results of this paper open the door to the justification of a variety of phenomena arising from the interplay between non-Hermitian reciprocal and nonreciprocal mechanisms not only in subwavelength wave physics but also in quantum mechanics, where the tight-binding model coupled with the nearest neighbour approximation can be analysed with the same tools as those developed here.
Improved Resolution Estimate for the Two-Dimensional Super-Resolution and a New Algorithm for Direction of Arrival Estimation with Uniform Rectangular Array
Liu, Ping; Ammari, Habib (2024)
Foundations of Computational Mathematics
In this paper, we develop a new technique to obtain improved estimates for the computational resolution limits in two-dimensional super-resolution problems and present a new idea for developing two-dimensional super-resolution algorithms. To be more specific, our main contributions are fourfold: (1) Our work improves the resolution estimates for number detection and location recovery in two-dimensional super-resolution problems; (2) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems [or direction of arrival Problems (DOA)]. The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (3) Inspired by the new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (4) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys some advantages compared to the conventional DOA algorithms.
A mathematical theory of super-resolution and two-point resolution
Liu, Ping; Ammari, Habib (2024)
Forum of Mathematics, Sigma
This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_{0}$ minimization algorithm in the super-resolution problem.The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as $$\begin{align*}\mathscr R = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)<^>{\frac{1}{2}} \right)}{\Omega} \end{align*}$$ for ${\frac {\sigma }{m_{\min }}}{\leqslant }{\frac {1}{2}}$ , where ${\frac {\sigma }{m_{\min }}}$ represents the inverse of the signal-to-noise ratio ( ${\mathrm {SNR}}$ ) and $\Omega $ is the cutoff frequency. It also demonstrates that for resolving two point sources, the resolution can exceed the Rayleigh limit ${\frac {\pi }{\Omega }}$ when the signal-to-noise ratio (SNR) exceeds $2$ . Moreover, we find a tractable algorithm that achieves the resolution ${\mathscr {R}}$ when distinguishing two sources.
Stability of the Non-Hermitian Skin Effect in One Dimension
Ammari, Habib; Barandun, Silvio Alberto; Davies, Bryn; et al. (2024)
SIAM Journal on Applied Mathematics
This paper shows both analytically and numerically that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of a directional damping term (motivated by an imaginary gauge potential), which leads to a skin effect that is manifested by the system's eigenmo des accumulating at one edge of the structure. We elucidate the topological protection of the associated (real) eigenfrequencies and illustrate numerically the competition between the two different localization effects present when the system is randomly perturbed: the non-Hermitian skin effect and the disorder-induced Anderson localization. We show numerically that, as the strength of the disorder increases, more and more eigenmo des become localized in the bulk. Our results are based on an asymptotic matrix model for subwavelength physics and can be generalized also to tight-binding models in condensed matter theory.
Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
Ammari, Habib; Barandun, Silvio Alberto; De Bruijn, Yannick; et al. (2025)
Journal of Physics A: Mathematical and Theoretical
We establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials. We also numerically verify our theory for these polymer systems.
Generalised Brillouin Zone for Non-Reciprocal Systems
Ammari, Habib; Barandun, Silvio Alberto; Liu, Ping; et al. (2024)
SAM Research Report
Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the k-Toeplitz matrix cases.
A mathematical theory of resolution limits for super-resolution of positive sources
Liu, Ping; He, Yanchen; Ammari, Habib (2023)
SAM Research Report
The superresolving capacity for number and location recoveries in the super-resolution of positive sources is analyzed in this work. Specifically, we introduce the computational resolution limit for respectively the number detection and location recovery in the one-dimensional super-resolution problem and quantitatively characterize their dependency on the cutoff frequency, signal-to-noise ratio, and the sparsity of the sources. As a direct consequence, we show that targeting at the sparest positive solution in the super-resolution already provides the optimal resolution order. These results are generalized to multi-dimensional spaces. Our estimates indicate that there exist phase transitions in the corresponding reconstructions, which are confirmed by numerical experiments. Our theory fills in an important puzzle towards fully understanding the super-resolution of positive sources.
Mathematical Foundation of Sparsity-Based Multi-Illumination Super-Resolution
Liu, Ping; Yu, Sanghyeon; Sabet, Ola; et al. (2022)
SSRN
It is well-known that the resolution of traditional optical imaging system is limited by the so-called Rayleigh resolution or diffraction limit, which is of several hundreds of nanometers. By employing fluorescence techniques, modern microscopic methods can resolve point scatterers separated by a distance much lower than the Rayleigh resolution limit. Localization-based fluorescence subwavelength imaging techniques such as PALM and STORM can achieve spatial resolution of several tens of nanometers. However, these techniques have limited temporal resolution as they require tens of thousands of exposures. Employing sparsity-based models and recovery algorithms is a natural way to reduce the number of exposures, and hence obtain high temporal resolution. Nevertheless, to date fluorescence techniques suffer from the trade-off between spatial and temporal resolutions.In [34], a new multi-illumination imaging technique called Brownian Excitation Amplitude Modulation microscopy (BEAM) is introduced. BEAM achieves a threefold resolution improvement by applying a compressive sensing recovery algorithm over only few frames. Motivated by BEAM, our aim in this paper is to pioneer the mathematical foundation for sparsity-based multi-illumination super-resolution. More precisely, we consider several diffraction-limited images from sample exposed to different illumination patterns and recover the source by considering the sparsest solution. We estimate the minimum separation distance between point scatterers so that they could be stably recovered. By this estimation of the resolution of the sparsity recovery, we reveal the dependence of the resolution on the cut-off frequency of the imaging system, the signal-to-noise ratio, the sparsity of point scatterers, and the incoherence of illumination patterns. Our theory particularly highlights the importance of the high incoherence of illumination patterns in enhancing the resolution. It also demonstrates that super-resolution can be achieved using sparsity-based multi-illumination imaging with very few frames, whereby the spatio-temporal super-resolution becomes possible. BEAM can be viewed as the ﬁrst experimental realization of our theory, which is demonstrated to hold in both the one- and two-dimensional cases.
Generalized Brillouin zone for non-reciprocal systems
Ammari, Habib; Barandun, Silvio Alberto; Liu, Ping; et al. (2025)
Royal Society of London. Proceedings A
Non-reciprocal physical systems, such as waveguides with imaginary gauge potentials or mass-spring chains with active elements, present new mathematical challenges beyond those of ordinary Hermitian structures. In a phenomenon known as the non-Hermitian skin effect, eigenmodes condensate at one edge of the structure and decay exponentially in space. Traditional Floquet-Bloch theory, relying on real-valued quasi-periodicities, fails to capture this decay and thus gives incomplete predictions of spectral behaviour. In this paper, we develop a rigorous theory to address this shortcoming. By extending the Brillouin zone into the complex plane, we introduce the notion of generalized Brillouin zone. We show that this complex formulation reflects the unidirectional decay inherent in non-reciprocal systems and provides an accurate framework for spectral analysis. Our main results are proven in the general context of k-Toeplitz matrices and operators, which model a wide variety of both finite and semi-infinite or infinite non-Hermitian periodic structures. We demonstrate that our generalized Floquet-Bloch formalism correctly identifies spectra and restores spectral convergences for large finite lattices.

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