arXiv subjects

Qin Li

Publications and source records attributed to Qin Li.

At least 181 records · Page 10Linked to original sources

arXiv2021

Semi-classical limit of an inverse problem for the Schrödinger equation

It is a classical derivation that the Wigner equation, derived from the Schrödinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant $ε\to0$. Since the latter presents the Newton's second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse problem for the Schrödinger equation. More specifically, we show that using the initial condition and final state of the Schrödinger equation to reconstruct the potential term, in the classical regime with $ε\to0$, becomes using the initial and final state to reconstruct the potential term in the Liouville equation. This formally bridges an inverse problem in quantum mechanics with an inverse problem in classical mechanics.

math-ph↗

arXiv2021

Improving speech recognition models with small samples for air traffic control systems

In the domain of air traffic control (ATC) systems, efforts to train a practical automatic speech recognition (ASR) model always faces the problem of small training samples since the collection and annotation of speech samples are expert- and domain-dependent task. In this work, a novel training approach based on pretraining and transfer learning is proposed to address this issue, and an improved end-to-end deep learning model is developed to address the specific challenges of ASR in the ATC domain. An unsupervised pretraining strategy is first proposed to learn speech representations from unlabeled samples for a certain dataset. Specifically, a masking strategy is applied to improve the diversity of the sample without losing their general patterns. Subsequently, transfer learning is applied to fine-tune a pretrained or other optimized baseline models to finally achieves the supervised ASR task. By virtue of the common terminology used in the ATC domain, the transfer learning task can be regarded as a sub-domain adaption task, in which the transferred model is optimized using a joint corpus consisting of baseline samples and new transcribed samples from the target dataset. This joint corpus construction strategy enriches the size and diversity of the training samples, which is important for addressing the issue of the small transcribed corpus. In addition, speed perturbation is applied to augment the new transcribed samples to further improve the quality of the speech corpus. Three real ATC datasets are used to validate the proposed ASR model and training strategies. The experimental results demonstrate that the ASR performance is significantly improved on all three datasets, with an absolute character error rate only one-third of that achieved through the supervised training. The applicability of the proposed strategies to other ASR approaches is also validated.

cs.SD↗

arXiv2021

On developing piecewise rational mapping with fine regulation capability for WENO schemes

On the idea of mapped WENO-JS scheme, properties of mapping methods are analyzed, uncertainties in mapping development are investigated, and new rational mappings are proposed. Based on our former understandings, i.e. mapping at endpoints {0, 1} tending to identity mapping, an integrated Cm,n condition is summarized for function development. Uncertainties, i.e., whether the mapping at endpoints would make mapped scheme behave like WENO or ENO, whether piecewise implementation would entail numerical instability, and whether WENO3-JS could preserve the third-order at first-order critical points by mapping, are analyzed and clarified. A new piecewise rational mapping with sufficient regulation capability is developed afterwards, where the flatness of mapping around the linear weights and its endpoint convergence toward identity mapping can be coordinated explicitly and simultaneously. Hence, the increase of resolution and preservation of stability can be balanced. Especially, concrete mappings are determined for WENO3,5,7-JS. Numerical cases are tested for the new mapped WENO-JS, which regards numerical stability including that in long time computation, resolution and robustness. In purpose of comparison, some recent mappings such as IM by [App. Math. Comput. 232, 2014:453-468], RM by [J. Sci. Comput. 67, 2016:540-580] and AIM by [J. Comput. Phys. 381, 2019:162-188] are chosen; in addition, some recent WENO-Z type scheme are selected also. Proposed new schemes can preserve optimal orders at corresponding critical points, achieve numerical stability and indicate overall comparative advantages regarding accuracy, resolution and robustness.

math.NA↗

arXiv2021

Classical limit for the varying-mass Schrödinger equation with random inhomogeneities

The varying-mass Schrödinger equation (VMSE) has been successfully applied to model electronic properties of semiconductor hetero-structures, for example, quantum dots and quantum wells. In this paper, we consider VMSE with small random heterogeneities, and derive a radiative transfer equation as its asymptotic limit. The main tool is to systematically apply the Wigner transform in the classical regime when the rescaled Planck constant $ε\ll 1$, and expand the Wigner equation to proper orders of $ε$. As a proof of concept, we numerically compute both VMSE and its limiting radiative transfer equation, and show that their solutions agree well in the classical regime.

math.NA↗

arXiv2021

Cryptanalysis and improvement of a semi-quantum private comparison protocol based on Bell states

Semi-quantum private comparison (SQPC) allows two participants with limited quantum ability to securely compare the equality of their secrets with the help of a semi-dishonest third party (TP). Recently, Jiang proposed a SQPC protocol based on Bell states (Quantum Inf Process 19(6): 180, 2020) and claimed it is secure. In this paper, we present two types of attack on Jiang's SQPC protocol. In the first type of attack, an outside eavesdropper will make participants accept a wrong result. In the second type of attack, a malicious participant will not only make the other participant accept a wrong result, but also learn the secret of the honest participant. Neither type of attack will be detected. In addition, we propose an improved SQPC protocol that can resist these two types of attack.

quant-ph↗

arXiv2020

Geometric quantization and quantum moment maps on coadjoint orbits and Kähler-Einstein manifolds

Deformation quantization and geometric quantization on Kähler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a Kähler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and Kähler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).

math.DG↗

arXiv2020

Variance reduction for Random Coordinate Descent-Langevin Monte Carlo

Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions. A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbation stochastic approximation (SPSA). We show by a counter-example that blindly applying RCD does not achieve the goal in the most general setting. The high variance induced by the randomness means a larger number of iterations are needed, and this balances out the saving in each iteration. We then introduce a new variance reduction approach, termed Randomized Coordinates Averaging Descent (RCAD), and incorporate it with both overdamped and underdamped LMC. The methods are termed RCAD-O-LMC and RCAD-U-LMC respectively. The methods still sit in the random gradient approximation framework, and thus the computational cost in each iteration is low. However, by employing RCAD, the variance is reduced, so the methods converge within the same number of iterations as the classical overdamped and underdamped LMC. This leads to a computational saving overall.

stat.ML↗

arXiv2020

Random Coordinate Underdamped Langevin Monte Carlo

The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.

stat.ML↗

arXiv2020

A Survey on Machine Reading Comprehension: Tasks, Evaluation Metrics and Benchmark Datasets

Machine Reading Comprehension (MRC) is a challenging Natural Language Processing(NLP) research field with wide real-world applications. The great progress of this field in recent years is mainly due to the emergence of large-scale datasets and deep learning. At present, a lot of MRC models have already surpassed human performance on various benchmark datasets despite the obvious giant gap between existing MRC models and genuine human-level reading comprehension. This shows the need for improving existing datasets, evaluation metrics, and models to move current MRC models toward "real" understanding. To address the current lack of comprehensive survey of existing MRC tasks, evaluation metrics, and datasets, herein, (1) we analyze 57 MRC tasks and datasets and propose a more precise classification method of MRC tasks with 4 different attributes; (2) we summarized 9 evaluation metrics of MRC tasks, 7 attributes and 10 characteristics of MRC datasets; (3) We also discuss key open issues in MRC research and highlighted future research directions. In addition, we have collected, organized, and published our data on the companion website(https://mrc-datasets.github.io/) where MRC researchers could directly access each MRC dataset, papers, baseline projects, and the leaderboard.

cs.CL↗

arXiv2020

Reconstruction of the emission coefficient in the nonlinear radiative transfer equation

In this paper, we investigate an inverse problem for the radiative transfer equation that is coupled with a heat equation in a nonscattering medium in $\mathbb{R}^n$, $n\geq 2$. The two equations are coupled through a nonlinear blackbody emission term that is proportional to the fourth power of the temperature. By measuring the radiation intensity on the surface of the blackbody, we prove that the emission property of the system can be uniquely reconstructed. In particular, we design a reconstruction procedure that uses merely one set of experiment setup to fully recover the emission parameter.

math.AP↗

arXiv2020

Random Coordinate Langevin Monte Carlo

Langevin Monte Carlo (LMC) is a popular Markov chain Monte Carlo sampling method. One drawback is that it requires the computation of the full gradient at each iteration, an expensive operation if the dimension of the problem is high. We propose a new sampling method: Random Coordinate LMC (RC-LMC). At each iteration, a single coordinate is randomly selected to be updated by a multiple of the partial derivative along this direction plus noise, and all other coordinates remain untouched. We investigate the total complexity of RC-LMC and compare it with the classical LMC for log-concave probability distributions. When the gradient of the log-density is Lipschitz, RC-LMC is less expensive than the classical LMC if the log-density is highly skewed for high dimensional problems, and when both the gradient and the Hessian of the log-density are Lipschitz, RC-LMC is always cheaper than the classical LMC, by a factor proportional to the square root of the problem dimension. In the latter case, our estimate of complexity is sharp with respect to the dimension.

stat.ML↗

arXiv2020

Sensitivity analysis of Burgers' equation with shocks

Generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but this is what hyperbolic type problems cannot provide. We provide a counter-argument in this paper, and show that even though the solution profile develops singularities in the random space, which destroys the spectral accuracy of gPC, the physical quantities (such as the shock emergence time, the shock location, and the shock strength) are all smooth functions of the uncertainties coming from both initial data and the wave speed: with proper shifting, the solution's polynomial interpolation approximates the real solution accurately, and the error decays as the order of the polynomial increases. Therefore this work provides a new perspective to "quantify uncertainties" and significantly improves the accuracy of the gPC method with a slight reformulation. We use the Burgers' equation as an example for the thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.

math.NA↗

arXiv2020

Ensemble Kalman Sampler: mean-field limit and convergence analysis

Ensemble Kalman Sampler (EKS) is a method to find approximately $i.i.d.$ samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. The continuous version of the algorithm is a set of coupled stochastic differential equations (SDEs). In this paper, we prove the wellposedness of the SDE system, justify its mean-field limit is a Fokker-Planck equation, whose long time equilibrium is the target distribution. We further demonstrate that the convergence rate is near-optimal ($J^{-1/2}$, with $J$ being the number of particles). These results, combined with the in-time convergence of the Fokker-Planck equation to its equilibrium, justify the validity of EKS, and provide the convergence rate as a sampling method.

math.NA↗

arXiv2020

Ensemble Kalman Inversion: mean-field limit and convergence analysis

Ensemble Kalman Inversion (EKI) has been a very popular algorithm used in Bayesian inverse problems. It samples particles from a prior distribution, and introduces a motion to move the particles around in pseudo-time. As the pseudo-time goes to infinity, the method finds the minimizer of the objective function, and when the pseudo-time stops at $1$, the ensemble distribution of the particles resembles, in some sense, the posterior distribution in the linear setting. The ideas trace back further to Ensemble Kalman Filter and the associated analysis, but to today, when viewed as a sampling method, why EKI works, and in what sense with what rate the method converges is still largely unknown. In this paper, we analyze the continuous version of EKI, a coupled SDE system, and prove the mean field limit of this SDE system. In particular, we will show that 1. as the number of particles goes to infinity, the empirical measure of particles following SDE converges to the solution to a Fokker-Planck equation in Wasserstein 2-distance with an optimal rate, for both linear and weakly nonlinear case; 2. the solution to the Fokker-Planck equation reconstructs the target distribution in finite time in the linear case.

math.NA↗

arXiv2020

Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds

Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear. Important Sampling (IS), on the other hand, obtains the unbiased sampling at the expense of large variance of weights, leading to slow convergence of high moments. We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.

math.NA↗

arXiv2020

PAC Model Checking of Black-Box Continuous-Time Dynamical Systems

In this paper we present a novel model checking approach to finite-time safety verification of black-box continuous-time dynamical systems within the framework of probably approximately correct (PAC) learning. The black-box dynamical systems are the ones, for which no model is given but whose states changing continuously through time within a finite time interval can be observed at some discrete time instants for a given input. The new model checking approach is termed as PAC model checking due to incorporation of learned models with correctness guarantees expressed using the terms error probability and confidence. Based on the error probability and confidence level, our approach provides statistically formal guarantees that the time-evolving trajectories of the black-box dynamical system over finite time horizons fall within the range of the learned model plus a bounded interval, contributing to insights on the reachability of the black-box system and thus on the satisfiability of its safety requirements. The learned model together with the bounded interval is obtained by scenario optimization, which boils down to a linear programming problem. Three examples demonstrate the performance of our approach.

eess.SY↗

arXiv2020

A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient

Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an accelerated Schwarz method for radiative transfer equations that makes use of approximate local solution maps constructed offline via a random sampling strategy. Numerical examples demonstrate the accuracy, robustness, and efficiency of the proposed approach.

math.NA↗

arXiv2020

Machine Learning in Heliophysics and Space Weather Forecasting: A White Paper of Findings and Recommendations

The authors of this white paper met on 16-17 January 2020 at the New Jersey Institute of Technology, Newark, NJ, for a 2-day workshop that brought together a group of heliophysicists, data providers, expert modelers, and computer/data scientists. Their objective was to discuss critical developments and prospects of the application of machine and/or deep learning techniques for data analysis, modeling and forecasting in Heliophysics, and to shape a strategy for further developments in the field. The workshop combined a set of plenary sessions featuring invited introductory talks interleaved with a set of open discussion sessions. The outcome of the discussion is encapsulated in this white paper that also features a top-level list of recommendations agreed by participants.

astro-ph.SR↗