arXiv subjects

Qin Li

Publications and source records attributed to Qin Li.

At least 199 records · Page 11Linked to original sources

arXiv2020

An efficient and provably secure arbitrated quantum signature scheme

In this paper, an efficient arbitrated quantum signature scheme is proposed by combining quantum cryptographic techniques and some ideas in classical cryptography. In the presented scheme, the signatory and the receiver can share a long-term secret key with the arbitrator by utilizing the key together with a random number. While in previous quantum signature schemes, the key shared between the signatory and the arbitrator or between the receiver and the arbitrator could be used only once, and thus each time when a signatory needs to sign, the signatory and the receiver have to obtain a new key shared with the arbitrator through a quantum key distribution protocol. Detailed theoretical analysis shows that the proposed scheme is efficient and provably secure.

quant-ph↗

arXiv2020

Structured random sketching for PDE inverse problems

For an overdetermined system $\mathsf{A}\mathsf{x} \approx \mathsf{b}$ with $\mathsf{A}$ and $\mathsf{b}$ given, the least-square (LS) formulation $\min_x \, \|\mathsf{A}\mathsf{x}-\mathsf{b}\|_2$ is often used to find an acceptable solution $\mathsf{x}$. The cost of solving this problem depends on the dimensions of $\mathsf{A}$, which are large in many practical instances. This cost can be reduced by the use of random sketching, in which we choose a matrix $\mathsf{S}$ with fewer rows than $\mathsf{A}$ and $\mathsf{b}$, and solve the sketched LS problem $\min_x \, \|\mathsf{S}(\mathsf{A} \mathsf{x}-\mathsf{b})\|_2$ to obtain an approximate solution to the original LS problem. Significant theoretical and practical progress has been made in the last decade in designing the appropriate structure and distribution for the sketching matrix $\mathsf{S}$. When $\mathsf{A}$ and $\mathsf{b}$ arise from discretizations of a PDE-based inverse problem, tensor structure is often present in $\mathsf{A}$ and $\mathsf{b}$. For reasons of practical efficiency, $\mathsf{S}$ should be designed to have a structure consistent with that of $\mathsf{A}$. Can we claim similar approximation properties for the solution of the sketched LS problem with structured $\mathsf{S}$ as for fully-random $\mathsf{S}$? We give estimates that relate the quality of the solution of the sketched LS problem to the size of the structured sketching matrices, for two different structures. Our results are among the first known for random sketching matrices whose structure is suitable for use in PDE inverse problems.

math.NA↗

arXiv2020

Security Improvements of Several Basic Quantum Private Query Protocols with O(log N) Communication Complexity

New quantum private database (with N elements) query protocols are presented and analyzed. Protocols preserve O(logN) communication complexity of known protocols for the same task, but achieve several significant improvements in security, especially concerning user privacy. For example, the randomized form of our protocol has a cheat-sensitive property - it allows the user to detect a dishonest database with a nonzero probability, while the phase-encoded private query protocols for the same task do not have such a property. Moreover, when the database performs the computational basis measurement, a particular projective measurement which can cause a significant loss of user privacy in the previous private query protocols with O(logN) communication complexity, at most half of the user privacy could leak to such a database in our protocol, while in the QPQ protocol, the entire user privacy could leak out. In addition, it is proved here that for large N, the user could detect a cheating via the computational basis measurement, with a probability close to 1/2 using O(\sqrt{N}) special queries. Finally, it is shown here, for both forms of our protocol, basic and randomized, how a dishonest database has to act in case it could not learn user's queries.

cs.CR↗

arXiv2020

Non-recurrent Traffic Congestion Detection with a Coupled Scalable Bayesian Robust Tensor Factorization Model

Non-recurrent traffic congestion (NRTC) usually brings unexpected delays to commuters. Hence, it is critical to accurately detect and recognize the NRTC in a real-time manner. The advancement of road traffic detectors and loop detectors provides researchers with a large-scale multivariable temporal-spatial traffic data, which allows the deep research on NRTC to be conducted. However, it remains a challenging task to construct an analytical framework through which the natural spatial-temporal structural properties of multivariable traffic information can be effectively represented and exploited to better understand and detect NRTC. In this paper, we present a novel analytical training-free framework based on coupled scalable Bayesian robust tensor factorization (Coupled SBRTF). The framework can couple multivariable traffic data including traffic flow, road speed, and occupancy through sharing a similar or the same sparse structure. And, it naturally captures the high-dimensional spatial-temporal structural properties of traffic data by tensor factorization. With its entries revealing the distribution and magnitude of NRTC, the shared sparse structure of the framework compasses sufficiently abundant information about NRTC. While the low-rank part of the framework, expresses the distribution of general expected traffic condition as an auxiliary product. Experimental results on real-world traffic data show that the proposed method outperforms coupled Bayesian robust principal component analysis (coupled BRPCA), the rank sparsity tensor decomposition (RSTD), and standard normal deviates (SND) in detecting NRTC. The proposed method performs even better when only traffic data in weekdays are utilized, and hence can provide more precise estimation of NRTC for daily commuters.

physics.soc-ph↗

arXiv2020

On diffusive scaling in acousto-optic imaging

Acousto-optic imaging (AOI) is a hybrid imaging process. By perturbing the to-be-reconstructed tissues with acoustic waves, one introduces the interaction between the acoustic and optical waves, leading to a more stable reconstruction of the optical properties. The mathematical model was described in [25], with the radiative transfer equation serving as the forward model for the optical transport. In this paper we investigate the stability of the reconstruction. In particular, we are interested in how the stability depends on the Knudsen number, Kn, a quantity that measures the intensity of the scattering effect of photon particles in a media. Our analysis shows that as Kn decreases to zero, photons scatter more frequently, and since information is lost, the reconstruction becomes harder. To counter this effect, devices need to be constructed so that laser beam is highly concentrated. We will give a quantitative error bound, and explicitly show that such concentration has an exponential dependence on Kn. Numerical evidence will be provided to verify the proof.

math.AP↗

arXiv2020

Random Sampling and Efficient Algorithms for Multiscale PDEs

We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on detailed analytical understanding of specific multiscale PDEs, in particular, their asymptotic limits. We present the application of the framework on two examples --- a linear kinetic equation and an elliptic equation with rough media. On these two examples, this framework achieves the asymptotic preserving property for the kinetic equations and numerical homogenization for the elliptic equations.

math.NA↗

arXiv2019

Error Lower Bounds of Constant Step-size Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) plays a central role in modern machine learning. While there is extensive work on providing error upper bound for SGD, not much is known about SGD error lower bound. In this paper, we study the convergence of constant step-size SGD. We provide error lower bound of SGD for potentially non-convex objective functions with Lipschitz gradients. To our knowledge, this is the first analysis for SGD error lower bound without the strong convexity assumption. We use experiments to illustrate our theoretical results.

math.OC↗

arXiv2019

Schwarz iteration method for elliptic equation with rough media based on random sampling

We propose a computationally efficient Schwarz method for elliptic equations with rough media. A random sampling strategy is used to find low-rank approximations of all local solution maps in the offline stage; these maps are used to perform fast Schwarz iterations in the online stage. Numerical examples demonstrate accuracy and robustness of our approach.

math.NA↗

arXiv2019

Applications of Kinetic Tools to Inverse Transport Problems

We show that the inverse problems for a class of kinetic equations can be solved by classical tools in PDE analysis including energy estimates and the celebrated averaging lemma. Using these tools, we give a unified framework for the reconstruction of the absorption coefficient for transport equations in the subcritical and critical regimes. Moreover, we apply this framework to obtain, to the best of our knowledge, the first result in a nonlinear setting. We also extend the result of recovering the scattering coefficient in [14] from 3D to 2D convex domains.

math.AP↗

arXiv2019

A Numerical method for coupling the BGK model and Euler equation through the linearized Knudsen layer

The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is small. In practice, however, Knudsen layers emerge at the physical boundary, or at the interfaces between the two regimes. We model the Knudsen layer using a half-space kinetic equation, and apply a half-space numerical solver [ESAIM: M2AN 51 (2017) 1583-1615] [Math. Comp. 86 (2017), 1269-1301] to quantify the transition between the kinetic to the fluid regime. A full domain numerical solver is developed with a domain-decomposition approach, where we apply the Euler solver and kinetic solver on the appropriate subdomains and connect them via the half-space solver. In the nonlinear case, linearization is performed upon local Maxwellian. Despite the lack of analytical support, the numerical evidence nevertheless demonstrates that the linearization approach is promising.

math.NA↗

arXiv2019

Parameter Reconstruction for general transport equation

We consider the inverse problem for the general transport equation with external field, source term and absorption coefficient. We show that the source and the absorption coefficients can be uniquely reconstructed from the boundary measurement, in a Lipschitz stable manner. Specifically, the uniqueness and stability are obtained by using the Carleman estimate in which a special weight function is designed to pick up information on the desired parameter.

math.AP↗

arXiv2019

Generalized multiscale finite element method for the steady state linear Boltzmann equation

The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this article, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime, and study the numerical behavior of the Generalized Multi-scale Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a general approach to numerically treat equations with multi-scale structures. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the wellposedness of the method on the Boltzmann equation, and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number.

math.NA↗

arXiv2019

Hybridizing WENO implementations of interpolation and reconstruction-wise operation for upwind-biased schemes with free-stream preservation

Cases have shown that WENO schemes usually behave robustly on problems containing shocks with high pressure ratios when uniformed or smooth grids are present, while nonlinear schemes based on WENO interpolations might relatively be liable to numerical instability. In the meanwhile, the latter have manifested their advantages in computations on grids of bad quality, because the free-stream preservation is easily realized there, and what is more flux-splitting schemes with low dissipations can be engaged inherently as well. Targeting at above dissatisfactions, a method by hybridizing WENO implementations of interpolation and reconstruction-wise operation for upwind-biased schemes with flux splitting employed is proposed and corresponding third-, fifth- and seventh-order upwind-biased schemes are proposed. Based on the understandings of [Q. Li, et al. Commun. Comput. Phys. 22 (2017) 64-94], the free-stream preservation of proposed schemes is achieved with incorporation of frozen grid metrics in WENO reconstructions-wise operations on split fluxes. In proposed schemes, flux-splitting schemes with low dissipation can also be applied for the flux on a cell edge. As a byproduct, an implementation of WENO scheme with free-stream preservation is obtained. Numerical examples are provided as following with the third- and fifth-order schemes being tested. In tests of free-stream preservation, the property is achieved as expected (including two implementations of WENO). The computation of 1-D Sod problem shows the capability of proposed schemes on solving ordinary shock discontinuity. 2-D vortex preservation and double Mach reflection are tested on uniformed and randomized grids. The accomplishment by proposed schemes manifests their capability and robustness on solving problems under rigorous circumstances.

physics.comp-ph↗

arXiv2019

Diffusive optical tomography in the Bayesian framework

Many naturally-occuring models in the sciences are well-approximated by simplified models, using multiscale techniques. In such settings it is natural to ask about the relationship between inverse problems defined by the original problem and by the multiscale approximation. We develop an approach to this problem and exemplify it in the context of optical tomographic imaging. Optical tomographic imaging is a technique for infering the properties of biological tissue via measurements of the incoming and outgoing light intensity; it may be used as a medical imaging methodology. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering and the absorption coefficients in the RTE from boundary measurements. We study this problem in the Bayesian framework, focussing on the strong scattering regime. In this regime the forward RTE is close to the diffusion equation (DE). We study the RTE in the asymptotic regime where the forward problem approaches the DE, and prove convergence of the inverse RTE to the inverse DE in both nonlinear and linear settings. Convergence is proved by studying the distance between the two posterior distributions using the Hellinger metric, and using Kullback-Leibler divergence.

math.NA↗

arXiv2019

WENO interpolation-based and upwind-biased schemes with free-stream preservation

Based on the understandings regarding linear upwind schemes with flux splitting to achieve free-stream preservation (Q. Li, etc. Commun. Comput. Phys., 22 (2017) 64-94), a series of WENO interpolation-based and upwind-biased nonlinear schemes are proposed in this study. By means of engagement of fluxes on midpoints, the nonlinearity of schemes is introduced through WENO interpolations, and upwind-biased features are acquired through the choice of dependent grid stencil. Regarding the third- and fifth-order versions, schemes with one and two midpoints are devised and carefully tested. With the integration of the piecewise-polynomial mapping function methods (Q. Li, etc. Commun. Comput. Phys. 18 (2015) 1417-1444), the proposed schemes are found to achieve the designed orders and free-stream preservation property. In 1-D Sod and Shu-Osher problems, all schemes succeed in yielding well predictions. In 2-D cases, the vortex preservation, supersonic inviscid flow around cylinder at M=4, Riemann problem and Shock-vortex interaction problems are tested. In each problem, two types of grids are employed, i.e. the uniformed/smooth grids and the randomized/partially-randomized grids. On the latter, the shock wave and complex flow structures are located/partially located. All schemes fulfill computations in uniformed/smooth grids with satisfactory results. On randomized grids, all schemes accomplish computations and yield reasonable results except the third-order one with two midpoints engaged fails in Riemann problem and shock-vortex interaction problem. Overall speaking, the proposed schemes manifest the capability to solve problems on grids with bad quality, and therefore indicate their potential in engineering applications.

physics.comp-ph↗

arXiv2018

Randomized sampling for basis functions construction in generalized finite element methods

In the framework of generalized finite element methods for elliptic equations with rough coefficients, efficiency and accuracy of the numerical method depend critically on the use of appropriate basis functions. This work explores several random sampling strategies that construct approximations to the optimal set of basis functions of a given dimension, and proposes a quantitative criterion to analyze and compare these sampling strategies. Numerical evidence shows that the best results are achieved by two strategies, Random Gaussian and Smooth boundary sampling.

math.OC↗

arXiv2018

Analysis of Stellar Spectra from LAMOST DR5 with Generative Spectrum Networks

In this study, the fundamental stellar atmospheric parameters (Teff, log g, [Fe/H] and [α/Fe]) were derived for low-resolution spectroscopy from LAMOST DR5 with Generative Spectrum Networks (GSN). This follows the same scheme as a normal artificial neural network with stellar parameters as the input and spectra as the output. The GSN model was effective in producing synthetic spectra after training on the PHOENIX theoretical spectra. In combination with Bayes framework, the application for analysis of LAMOST observed spectra exhibited improved efficiency on the distributed computing platform, Spark. In addition, the results were examined and validated by a comparison with reference parameters from high-resolution surveys and asteroseismic results. Our results show good consistency with the results from other survey and catalogs. Our proposed method is reliable with a precision of 80 K for Teff, 0.14 dex for log g, 0.07 dex for [Fe/H] and 0.168 dex for [α/Fe], for spectra with a signal-to-noise in g bands (SNRg) higher than 50. The parameters estimated as a part of this work are available at http://paperdata.china-vo.org/GSN_parameters/GSN_parameters.csv.

astro-ph.IM↗

arXiv2018

Inverse problems for the stationary transport equation in the diffusion scaling

We consider the inverse problem of reconstructing the optical parameters of the radiative transfer equation (RTE) from boundary measurements in the diffusion limit. In the diffusive regime (the Knudsen number $\mathsf{Kn}\ll 1$), the forward problem for the stationary RTE is well approximated by an elliptic equation. However, the connection between the inverse problem for the RTE and the inverse problem for the elliptic equation has not been fully developed. This problem is particularly interesting because the former one is mildly ill-posed , with a Lipschitz type stability estimate, while the latter is well known to be severely ill-posed with a logarithmic type stability estimate. In this paper, we derive stability estimates for the inverse problem for RTE and examine its dependence on $\mathsf{Kn}$. We show that the stability is Lipschitz in all regimes, but the coefficient deteriorates as $e^{\frac{1}{\mathsf{Kn}}}$, making the inverse problem of RTE severely ill-posed when $\mathsf{Kn}$ is small. In this way we connect the two inverse problems. Numerical results agree with the analysis of worsening stability as the Knudsen number gets smaller.

math.AP↗