The Almost Global and Global Existence for Quasi-linear Wave Equations with Multiple-Propagation Speeds in High Dimensions

In this paper, we consider the Cauchy problem for systems of quasi-linear wave equations with multiple propagation speeds in spatial dimensions n ≥ 4. The problem when the nonlinearities depend on both the unknown function and their derivatives is studied. Based on some Klainerman- Sideris type weighted estimates and space-time L2 estimates, the results that the almost global existence for space dimensions n = 4 and global existence for n≥ 5 of small amplitude solutions are presented.

Recent advances in statistical methodologies in evaluating program for high-dimensional data

The era of big data brings opportunities and challenges to developing new statistical methods and models to evaluate social programs or economic policies or interventions. This paper provides a comprehensive review on some recent advances in statistical methodologies and models to evaluate programs with high-dimensional data. In particular, four kinds of methods for making valid statistical inferences for treatment effects in high dimensions are addressed. The first one is the so-called doubly robust type estimation, which models the outcome regression and propensity score functions simultaneously. The second one is the covariate balance method to construct the treatment effect estimators. The third one is the sufficient dimension reduction approach for causal inferences. The last one is the machine learning procedure directly or indirectly to make statistical inferences to treatment effect. In such a way, some of these methods and models are closely related to the de-biased Lasso type methods for the regression model with high dimensions in the statistical literature. Finally, some future research topics are also discussed.

High-dimension multiparty quantum secret sharing scheme with Einstein-Podolsky-Rosen pairs

In this paper a high-dimension multiparty quantum secret sharing scheme is proposed by using Einstein-Podolsky-Rosen pairs and local unitary operators. This scheme has the advantage of not only having higher capacity, but also saving storage space. The security analysis is also given.

Boundedness in a fully parabolic quasilinear repulsion chemotaxis model of higher dimension

We deal with the boundedness of solutions to a class of fully parabolic quasilinear repulsion chemotaxis systems{ut=∇・(ϕ(u)∇u)+∇・(ψ(u)∇v),(x,t)∈Ω×(0,T),vt=Δv−v+u,(x,t)∈Ω×(0,T),under homogeneous Neumann boundary conditions in a smooth bounded domainΩ⊂R^N(N≥3),where 0<ψ(u)≤K(u+1)^a,K1(s+1)^m≤ϕ(s)≤K2(s+1)^m withα,K,K1,K2>0 and m∈R.It is shown that ifα−m<4/N+2,then for any sufficiently smooth initial data,the classical solutions to the system are uniformly-in-time bounded.This extends the known result for the corresponding model with linear diffusion.

Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

Demonstration of controlled high-dimensional quantum teleportation

Controlled quantum teleportation(CQT), which is regarded as the prelude and backbone for a genuine quantum internet, reveals the cooperation, supervision, and control relationship among the sender, receiver, and controller in the quantum network within the simplest unit. Compared with low-dimensional counterparts, high-dimensional CQT can exhibit larger information transmission capacity and higher superiority of the controller's authority. In this article, we report a proof-of-principle experimental realization of three-dimensional(3D) CQT with a fidelity of 97.4% ± 0.2%. To reduce the complexity of the circuit, we simulate a standard 4-qutrit CQT protocol in a 9×9-dimensional two-photon system with high-quality operations. The corresponding control powers are 48.1% ± 0.2% for teleporting a qutrit and 52.8% ± 0.3% for teleporting a qubit in the experiment, which are both higher than the theoretical value of control power in 2-dimensional CQT protocol(33%). The results fully demonstrate the advantages of high-dimensional multi-partite entangled networks and provide new avenues for constructing complex quantum networks.

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control.We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians.Additionally,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of dimensionality.Using existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial costs.We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demonstrates the promising performance boosts FPGAs can achieve.Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

Feature Selection via Analysis of Relevance and Redundancy

Feature selection is an important problem in pattern classification systems. High dimension fisher criterion（HDF） is a good indicator of class separability. However, calculating the high dimension fisher ratio is difficult. A new feature selection method, called fisher-and-correlation （FC）, is proposed. The proposed method is combining fisher criterion and correlation criterion based on the analysis of feature relevance and redundancy. The proposed methodology is tested in five different classification applications. The presented resuits confirm that FC performs as well as HDF does at much lower computational complexity.

Parallel Web Mining System Based on Cloud Platform

Traditional machine-learning algorithms are struggling to handle the exceedingly large amount of data being generated by the internet. In real-world applications, there is an urgent need for machine-learning algorithms to be able to handle large-scale, high-dimensional text data. Cloud computing involves the delivery of computing and storage as a service to a heterogeneous community of recipients, Recently, it has aroused much interest in industry and academia. Most previous works on cloud platforms only focus on the parallel algorithms for structured data. In this paper, we focus on the parallel implementation of web-mining algorithms and develop a parallel web-mining system that includes parallel web crawler; parallel text extract, transform and load （ETL） and modeling; and parallel text mining and application subsystems. The complete system enables variable real-world web-mining applications for mass data.

Inconsistency of Classical Penalized Likelihood Approaches under Endogeneity

With the high speed development of information technology, contemporary data from a variety of fields becomes extremely large. The number of features in many datasets is well above the sample size and is called high dimensional data. In statistics, variable selection approaches are required to extract the efficacious information from high dimensional data. The most popular approach is to add a penalty function coupled with a tuning parameter to the log likelihood function, which is called penalized likelihood method. However, almost all of penalized likelihood approaches only consider noise accumulation and supurious correlation whereas ignoring the endogeneity which also appeared frequently in high dimensional space. In this paper, we explore the cause of endogeneity and its influence on penalized likelihood approaches. Simulations based on five classical pe-nalized approaches are provided to vindicate their inconsistency under endogeneity. The results show that the positive selection rate of all five approaches increased gradually but the false selection rate does not consistently decrease when endogenous variables exist, that is, they do not satisfy the selection consistency.

Automated Dynamic Cellular Analysis in Time-Lapse Microscopy

Analysis of cellular behavior is significant for studying cell cycle and detecting anti-cancer drugs. It is a very difficult task for image processing to isolate individual cells in confocal microscopic images of non-stained live cell cultures. Because these images do not have adequate textural variations. Manual cell segmentation requires massive labor and is a time consuming process. This paper describes an automated cell segmentation method for localizing the cells of Chinese hamster ovary cell culture. Several kinds of high-dimensional feature descriptors, K-means clustering method and Chan-Vese model-based level set are used to extract the cellular regions. The region extracted are used to classify phases in cell cycle. The segmentation results were experimentally assessed. As a result, the proposed method proved to be significant for cell isolation. In the evaluation experiments, we constructed a database of Chinese Hamster Ovary Cell’s microscopic images which includes various photographing environments under the guidance of a biologist.

Distribution/correlation-free test for two-sample means in high-dimensional functional data with eigenvalue decay relaxed

We propose a methodology for testing two-sample means in high-dimensional functional data that requires no decaying pattern on eigenvalues of the functional data.To the best of our knowledge,we are the first to consider and address such a problem.To be specific,we devise a confidence region for the mean curve difference between two samples,which directly establishes a rigorous inferential procedure based on the multiplier bootstrap.In addition,the proposed test permits the functional observations in each sample to have mutually different distributions and arbitrary correlation structures,which is regarded as the desired property of distribution/correlation-free,leading to a more challenging scenario for theoretical development.Other desired properties include the allowance for highly unequal sample sizes,exponentially growing data dimension in sample sizes and consistent power behavior under fairly general alternatives.The proposed test is shown uniformly convergent to the prescribed significance,and its finite sample performance is evaluated via the simulation study and an application to electroencephalography data.

High-dimensional frequency conversion in a hot atomic system

One of the major difficulties in realizing a high-dimensional frequency converter for conventional optical vortex(COV)modes stems from the difference in ring diameter of the COV modes with different topological charge numbers l.Here,we implement a high-dimensional frequency converter for perfect optical vortex(POV)modes with invariant sizes by way of the four-wave mixing(FWM)process using Bessel–Gaussian beams instead of Laguerre–Gaussian beams.The measured conversion efficiency from 1530 to 795 nm is independent of l at least in subspace l∈{-6,………,6},and the achieved conversion fidelities for two-dimensional(2D)superposed POV states exceed 97%.We further realize the frequency conversion of 3D,5D,and 7D superposition states with fidelities as high as 96.70%,89.16%,and 88.68%,respectively.The proposed scheme is implemented in hot atomic vapor.It is also compatible with the cold atomic system and may find applications in high-capacity and long-distance quantum communication.

Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

We study a new algorithm for solvingparabolic partial differential equations(PDEs)and backward stochastic differential equations(BSDEs)in high dimension,which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function,and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE.The policy function is then approximated by a neural network,as is done in deep reinforcement learning.Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation,the Hamilton–Jacobi–Bellman equation,and a nonlinear pricing model for financial derivatives.

Variance Variation Criterion and Consistency in Estimating the Number of Significant Signals of High-dimensional PCA

In this paper,we propose a criterion based on the variance variation of the sample eigenvalues to correctly estimate the number of significant components in high-dimensional principal component analysis(PCA),and it corresponds to the number of significant eigenvalues of the covariance matrix for p-dimensional variables.Using the random matrix theory,we derive that the consistent properties of the proposed criterion for the situations that the significant eigenvalues tend to infinity,as well as that the bounded significant population eigenvalues.Numerical simulation shows that the probability of estimator is correct by our variance variation criterion converges to 1 is faster than that by criterion of Passemier and Yao[Estimation of the number of spikes,possibly equal,in the high-dimensional case.J.Multivariate Anal.,(2014)](PYC),AIC and BIC under the finite fourth moment condition as the dominant population eigenvalues tend to infinity.Moreover,in the case of the maximum eigenvalue bounded,once the gap condition is satisfied,the rate of convergence to 1 is faster than that of PYC and AIC,especially the effect is better than AIC when the sample size is small.It is worth noting that the variance variation criterion significantly improves the accuracy of model selection compared with PYC and AIC when the random variable is a heavy-tailed distribution or finite fourth moment not exists.

High-dimensional proportionality test of two covariance matrices and its application to gene expression data

With the development of modern science and technology, more and more high-dimensionaldata appear in the application fields. Since the high dimension can potentially increase the com-plexity of the covariance structure, comparing the covariance matrices among populations isstrongly motivated in high-dimensional data analysis. In this article, we consider the proportion-ality test of two high-dimensional covariance matrices, where the data dimension is potentiallymuch larger than the sample sizes, or even larger than the squares of the sample sizes. We devisea novel high-dimensional spatial rank test that has much-improved power than many exist-ing popular tests, especially for the data generated from some heavy-tailed distributions. Theasymptotic normality of the proposed test statistics is established under the family of ellipticallysymmetric distributions, which is a more general distribution family than the normal distribu-tion family, including numerous commonly used heavy-tailed distributions. Extensive numericalexperiments demonstrate the superiority of the proposed test in terms of both empirical sizeand power. Then, a real data analysis demonstrates the practicability of the proposed test forhigh-dimensional gene expression data.

A Generalized Stationary Algorithm for Resonant Tunneling:Multi-Mode Approximation and High Dimension

The multi-mode approximation is presented to compute the interior wave function of Schr¨odinger equation.This idea is necessary to handle the multi-barrier and high dimensional resonant tunneling problems where multiple eigenvalues are considered.The accuracy and efficiency of this algorithm is demonstrated via several numerical examples.

Convergence of the deep BSDE method for coupled FBSDEs

The recently proposed numerical algorithm,deep BSDE method,has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations(FBSDEs)and parabolic partial differential equations(PDEs).This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs.In particular,a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks.Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.

Outlier Detection via a Block Diagonal Product Estimator

Outlier detection is a fundamental topic in robust statistics.Traditional outlier detection methods try to find a clean subset of given size,which is used to estimate the location vector and scatter matrix,and the outliers can be flagged by the Mahalanobis distance.However,methods such as the minimum covariance determinant approach cannot be applied directly to high-dimensional data,especially when the dimension of the sample is greater than the sample size.A novel fast detection procedure based on a block diagonal partition is proposed,and the asymptotic distribution of the modified Mahalanobis distance is obtained.The authors verify the specificity and sensitivity of this procedure by simulation and real data analysis in high-dimensional settings.

Posterior contraction rate of sparse latent feature models with application to proteomics

The Indian buffet process(IBP)and phylogenetic Indian buffet process(pIBP)can be used as prior models to infer latent features in a data set.The theoretical properties of these models are under-explored,however,especially in high dimensional settings.In this paper,we show that under mild sparsity condition,the posterior distribution of the latent feature matrix,generated via IBP or pIBP priors,converges to the true latent feature matrix asymptotically.We derive the posterior convergence rate,referred to as the contraction rate.We show that the convergence results remain valid even when the dimensionality of the latent feature matrix increases with the sample size,therefore making the posterior inference valid in high dimensional settings.We demonstrate the theoretical results using computer simulation,in which the parallel-tempering Markov chain Monte Carlo method is applied to overcome computational hurdles.The practical utility of the derived properties is demonstrated by inferring the latent features in a reverse phase protein arrays(RPPA)dataset under the IBP prior model.