Narrow Results

In this paper, we first present an O(log n) time sorting algorithm on 3-D mesh-connected computers with multiple broadcasting (abbreviated to MCCMB) using n^1/2×n^1/2×n^1/2 processors. Our algorithm is derived from rotate sort. Further, we also show that the result can be extended to k-dimensional MCCMB of size to sort n data items in O(7^k−3 log n) time, for k≥3. The algorithm proposed is optimal speed-up while k is any constant. The contribution of this paper is to show that the proposed algorithm can be run in a higher dimensional MCCMB and using fewer processors but keeps the same time complexity as O(log n).

We consider the multi-particle tight-binding Anderson model and prove that its lower spectral edge is non-random under some mild assumptions on the inter-particle interaction and the random external potential. We also adapt to the low energy regime the multi-particle multi-scale analysis initially developed by Chulaevsky and Suhov in the high disorder limit, if the marginal probability distribution of the i.i.d. random variables is log-Hölder continuous and we obtain the spectral exponential and strong dynamical localization near the bottom of the spectrum.

Site and bond percolation thresholds are calculated for the face centered cubic, body centered cubic and diamond lattices in four, five and six dimensions. The results are used to study the behavior of percolation thresholds as a functions of dimension. It is shown that the predictions from a recently proposed invariant for percolation thresholds are not satisfactory for these lattices.

We analyze the bifurcations of a family of time-delayed Hénon maps of increasing dimension and determine the regions where the motion is attracted to different dynamical states. As a function of parameters that govern nonlinearity and dissipation, boundaries that confine asymptotic periodic motion are determined analytically, and we examine their dependence on the dimension d. For large d these boundaries converge. In low dimensions both the period-doubling and quasiperiodic routes to chaos coexist in the parameter space, but for high dimensions the latter predominates and prior to the onset of chaos, the systems exhibit multistability. When the nonlinearity parameter is varied, the dimension of chaotic attractors in the systems changes smoothly with increasing number of non-negative Lyapunov exponents.

In this paper, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.

This paper analyzes a new regularized learning scheme for high-dimensional partially linear support vector machine (SVM). The proposed approach consists of an empirical risk and the Lasso-type penalty for linear part, as well as the standard functional norm for nonlinear part. Here, the linear kernel is used for model interpretation and feature selection, while the nonlinear kernel is adopted to enhance algorithmic flexibility. In this paper, we develop a new technical analysis on the weighted empirical process, and establish the sharp learning rates for the semi-parametric estimator under the regularized conditions. Specially, our derived learning rates for semi-parametric SVM depend on not only the sample size and the functional complexity, but also the sparsity and the margin parameters.

This paper develops an outlier detection procedure for multinomial data when the number of categories tends to infinity. Most of the outlier detection methods are based on the assumption that the observations follow multivariate normal distribution, while in many modern applications, the observations either are measured on a discrete scale or naturally have some categorical structures. For such multinomial observations, there are rather limited approaches for outlier detection. To overcome the main obstacle, the least trimmed distances estimator for multinomial data and a fast algorithm to identify the clean subset are introduced in this work. Also, a threshold rule is considered through the asymptotic distribution of measure distance to identify outliers. Furthermore, a one-step reweighting scheme is proposed to improve the efficiency of the procedure. Finally, the finite sample performance of our method is evaluated through simulations and is compared with that of available outlier detection methods.

In this paper, we propose some new tests for high-dimensional covariance matrices that are applicable to generally distributed populations with finite fourth moments. The proposed test statistics are the maximum of the likelihood ratio test statistic and the statistic based on the Frobenius norm. The advantage of the new tests is the good performance in terms of power for both the traditional case, in which the dimension is much smaller than the sample size, and the high-dimensional case, in which the dimension is large compared to the sample size. In the one-sample case, the new test is proposed for testing the hypothesis that the high-dimensional covariance matrix equals an identity matrix. In the two-sample case, the new test is developed for testing the equality of two high-dimensional covariance matrices. By using the random matrix theory, the asymptotic distributions of the proposed new tests are derived under the assumption that the dimension and the sample size proportionally tend toward infinity. Finally, numerical studies are conducted to investigate the finite sample performance of the proposed new tests.

Suppose that $y_{\alpha }^{(i)},i=1,\dots ,k,\alpha =1,\dots ,m$ are i.i.d. copies of random vector $y={(y_1,y_2,\dots ,y_p)}^T$. Let

We consider the coupling of two-dimensional (2D) and one-dimensional (1D) models to form a single hybrid 2D–1D model for time-dependent linear wave problems. The 1D model is used to represent a 2D computational domain where the solution behaves approximately in a 1D way. This hybrid model, if designed properly, is a more efficient way to solve the full 2D model over the entire problem. Two important issues related to such hybrid 2D–1D models are (a) the design of the hybrid model and its validation (with respect to the original problem) and (b) the way the 2D–1D coupling is done, and the coupling error generated. This paper focuses on the second issue. The method used in this paper to couple the 1D and 2D models is the one proposed by Panasenko. This method has been used for mixed-dimensional coupling in many steady-state problems, and here it is being used for the first time for time-dependent problems. The hybrid formulation is derived, and the numerical accuracy and efficiency of the method are explored for a couple of basic problems.

Understanding how proteins fold is essential to our quest in discovering how life works at the molecular level. Current computation power enables researchers to produce a huge amount of folding simulation data. Hence there is a pressing need to be able to interpret and identify novel folding features from them. In this paper, we model each folding trajectory as a multi-dimensional curve. We then develop an effective multiple curve comparison (MCC) algorithm, called the enhanced partial order (EPO) algorithm, to extract features from a set of diverse folding trajectories, including both successful and unsuccessful simulation runs. Our EPO algorithm addresses several new challenges presented by comparing high dimensional curves coming from folding trajectories. A detailed case study of applying our algorithm to a miniprotein Trp-cage²⁴ demonstrates that our algorithm can detect similarities at rather low level, and extract biologically meaningful folding events.