An Asymptotically Tighter Bound on Sampling for Frequent Itemsets Mining

In this paper we present a new error bound on sampling algorithms for frequent itemsets mining. We show that the new bound is asymptotically tighter than the state-of-art bounds, i.e., given the chosen samples, for small enough error probability, the new error bound is roughly half of the existing bounds. Based on the new bound, we give a new approximation algorithm, which is much simpler compared to the existing approximation algorithms, but can also guarantee the worst approximation error with precomputed sample size. We also give an algorithm which can approximate the top-$k$ frequent itemsets with high accuracy and efficiency.Comment: 13 pages, 2 figures, 2 table

Performance Enhancement for High-order Gas-kinetic Scheme Based on WENO-adaptive-order Reconstruction

High-order gas-kinetic scheme (HGKS) has been well-developed in the past years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness. However, there are still rooms for its further improvement. Firstly, the reconstruction in the previous scheme mainly achieves a third-order accuracy for the initial non-equilibrium states. At the same time, the equilibrium state in space and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with adaptive order (WENO-AO) method is implemented for reconstruction.A whole polynomial inside each cell is provided in WENO-AO reconstruction. The HGKS becomes simpler than the previous one with the direct implementation of cell interface values and their slopes from WENO-AO. The additional reconstruction of equilibrium state at the beginning of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the above existing problems in previous HGKS. The accuracy of the scheme from 1D to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO reconstruction.In the two- and three-dimensional simulations, the new HGKS shows better robustness and efficiency than the previous scheme in all test cases

Adaptive Modular Exponentiation Methods v.s. Python's Power Function

In this paper we use Python to implement two efficient modular exponentiation methods: the adaptive m-ary method and the adaptive sliding-window method of window size k, where both m's are adaptively chosen based on the length of exponent. We also conduct the benchmark for both methods. Evaluation results show that compared to the industry-standard efficient implementations of modular power function in CPython and Pypy, our algorithms can reduce 1-5% computing time for exponents with more than 3072 bits.Comment: 4 page

Quark Fragmentation Functions in Low-Energy Chiral Theory

We examine the physics content of fragmentation functions for inclusive hadron production in a quark jet and argue that it can be calculated in low energy effective theories. As an example, we present a calculation of $u$-quark fragmentation to $\pi^+$ and $\pi^-$ mesons in the lowest order in the chiral quark model. The comparison between our result and experimental data is encouraging.Comment: 4 (tightly packed) pages in ReVTeX and 2 PostScript figures, MIT-CTP-225

Filter-design perspective applying to dynamical decoupling of a multi-qubit system

We employ the filter-design perspective and derive the filter functions according to nested Uhrig dynamical decoupling (NUDD) and Symmetric dynamical decoupling (SDD) in the pure-dephasing spin-boson model with N qubits. The performances of NUDD and SDD are discussed in detail for a two-qubit system. The analysis shows that (i) SDD outperforms NUDD for the bath with a soft cutoff while NUDD approaches SDD as the cutoff becomes harder; (ii) if the qubits are coupled to a common reservoir, SDD helps to protect the decoherence-free subspace while NUDD destroys it; (iii) when the the imperfect control pulses with finite width are considered, NUDD is affected in both the high-fidelity regime and coherence time regime while SDD is affected in the coherence time regime only.Comment: 20 pages,5 figures, submitted to Journal of Physics B: Atomic, Molecular and Optics Physic

A HWENO Reconstruction Based High-order Compact Gas-kinetic Scheme on Unstructured Meshes

As an extension of previous fourth-order compact gas kinetic scheme (GKS) on structured meshes (Ji et al. 2018), this work is about the development of a third-order compact GKS on unstructured meshes for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution at a cell interface, the time dependent gas distribution function in GKS provides not only the flux function and its time derivative at a cell interface, but also the time accurate flow variables there at next time level. As a result, besides updating the conservative flow variables inside each control volume through the interface fluxes, the cell averaged first-order spatial derivatives of flow variables in the cell can be also obtained using the updated flow variables at the cell interfaces around that cell through the divergence theorem. Therefore, with the flow variables and their first-order spatial derivatives inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of a new time step. Following the reconstruction method in (Zhu et al. 2018), a new HWENO reconstruction on triangular meshes is designed in the current scheme. Combined with a two-stage temporal discretization and second-order gas-kinetic flux function, a third-order spatial and temporal accuracy in the current compact scheme can be achieved. Accurate solutions can be obtained for both inviscid and viscous flows without sensitive dependence on the quality of triangular meshes. The robustness of the scheme has been validated as well through the cases with strong shocks in the hypersonic viscous flow simulations

On The Degrees of Freedom of Reduced-rank Estimators in Multivariate Regression

In this paper we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of Stein's unbiased risk estimation (SURE). We derive a finite-sample exact unbiased estimator that admits a closed-form expression in terms of the singular values or thresholded singular values of the least squares solution and hence readily computable. The results continue to hold in the high-dimensional scenario when both the predictor and response dimensions are allowed to be larger than the sample size. The derived analytical form facilitates the investigation of its theoretical properties and provides new insights into the empirical behaviors of the degrees of freedom. In particular, we examine the differences and connections between the proposed estimator and a commonly-used naive estimator, i.e., the number of free parameters. The use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection, as demonstrated by simulation studies and a data example.Comment: 29 pages, 3 figure

Compact Higher-order Gas-kinetic Schemes with Spectral-like Resolution for Compressible Flow Simulations

In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral resolution will be presented. Based on the high-order gas evolution model in GKS, both the interface flux function and conservative flow variables can be evaluated explicitly from the time-accurate gas distribution function. As a result, inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step. The flow variable update and slope update are coming from the same physical solution at the cell interface. Different from many other approaches, such as HWENO, there are no additional governing equations in GKS for the slopes or equivalent degrees of freedom independently inside each cell. For nonlinear gas dynamic evolution, the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction. In GKS, the time evolution solution of the gas distribution function at a cell interface covers a physical process from an initial non-equilibrium state to a final equilibrium one. The GKS evolution models unifies the nonlinear and linear reconstructions in gas evolution process for the determination of a time-dependent gas distribution function. This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions. Compared with discontinuous Galerkin (DG) scheme, the current compact GKS uses the same local and compact stencil, achieves the 6th-order and 8th-order accuracy, uses a much larger time step with CFL number $\geq 0.3$, and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process. At the same time, the current scheme solves the Navier-Stokes equations.Comment: 38 pages, 48 figure

A Family of High-order Gas-kinetic Schemes and Its Comparison with Riemann Solver Based High-order Methods

Most high order computational fluid dynamics (CFD) methods for compressible flows are based on Riemann solver for the flux evaluation and Runge-Kutta (RK) time stepping technique for temporal accuracy. The main advantage of this kind of approach is the easy implementation and stability enhancement by introducing more middle stages. However, the nth-order time accuracy needs no less than n stages for the RK method, which is very time and memory consuming for a high order method. On the other hand, the multi-stage multi-derivative (MSMD) method can be used to achieve the same order of time accuracy using less middle stages, once the time derivatives of the flux function is used.The gas kinetic scheme (GKS) provides such a time accurate evolution model. By combining the second-order or third-order GKS flux functions with the MSMD technique, a family of high order gas kinetic methods can be constructed. As an extension of the previous 2-stage 4th-order GKS, the 5th-order schemes with 2 and 3 stages will be developed in this paper. Based on the same 5th-order WENO reconstruction, the performance of gas kinetic schemes from the 2nd- to the 5th-order time accurate methods will be evaluated. The results show that the 5th-order scheme can achieve the theoretical order of accuracy for the Euler equations, and present accurate Navier-Stokes solutions as well due to the coupling of inviscid and viscous terms in the GKS formulation. In comparison with Riemann solver based 5th-order RK method, the high order GKS has advantages in terms of efficiency, accuracy, and robustness, for all test cases. It provides a promising direction for the development of high-order CFD methods for the computation of complex flows, such as turbulence and acoustics with shock interactions.Comment: 33 pages, 13 figure

A Compact Fourth-order Gas-kinetic Scheme for the Euler and Navier-Stokes Solutions

In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier-Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate evolution solution at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the GKS can get the cell averaged slopes inside each control volume as well through the differences of flow variables at the cell interfaces. So, with the updated flow variables and their slopes inside each cell, the HWENO reconstruction can be naturally implemented for the compact high-order reconstruction at the beginning of next step. Therefore, a compact higher-order GKS, such as the two-stages fourth-order compact scheme can be constructed. This scheme is as robust as second-order one, but more accurate solution can be obtained. In comparison with compact fourth-order DG method, the current scheme has only two stages instead of four within each time step for the fourth-order temporal accuracy, and the CFL number used here can be on the order of $0.5$ instead of $0.11$ for the DG method. Through this research, it concludes that the use of high-order time evolution model rather than the first order Riemann solution is extremely important for the design of robust, accurate, and efficient higher-order schemes for the compressible flows