Earticle

이 논문에서 우리는 CRAY-2에서 편미분방정식에서 발생하는 대형희귀 연립방정 식의 효과적인 벡터준비행렬을 만들기 위한 재배열방법을 제시한다. 이 재배열방법은 종래의 빨강/검정 배열의 선형 형태로써, ILU 준비행렬의 변형에 사용될 경우 필인 (fill-in)을 크게 하면 종래의 빨강/검정 재배열의 약점이던 수렴율의 감소를 극복할 수 있다. 우리는 CRAY-2에서 여러 가지 실험을 통해 우리의 주장을 입증한다. 또, 에러 행렬의 후로베니우스 놈을 계산한 결과도 우리의 주장과 일치한다.

In this paper we present a reordering scheme that could lead to efficient vectorization of the preconditioners for the large sparse linear systems arising from partial differential equations on the CRAY-2, This reordering scheme is a line version of the conventional red/black ordering. This reordering scheme, coupled with a variant of ILU(Incomplete LU) preconditioning, can overcome the poor rate of convergence of the conventional red/black reordering, if relatively large number of fill-ins were used. We substantiate our claim by conducting various experiments on the CRAY-2 machine. Also, the computation of the Frobenius norm of the error matrices agree with our claim.

The main idea of Regular Sequences of Similar Algorithm based on Sparse Linear (HR Algorithm) is to calculation the near optimal solution from the solution of sparse integer programming model. The advantages of HR algorithm are simplifying the problem of huge solution space of integer programming, reducing the number of goal constraints, and turning the problem into a simplified integer programming. The near optimal solution, which is hard to apply accurate method, can be found under limited time by using HR algorithm based on regular sequences.

The sparse linear method can reduce or expanse the solution range of integer programming problem by using former enumeration method lists all solution space, and then constraining them into a scale, after that picking up optimal solution form this scale. Calculation time will be reduced by using a revised integer programming model based on sparse linear, which can help reduce the number of solution space.

It is important to solve the large sparse linear system appeared in many application field such as $AA^Ty={\beta}$ efficiently. In solving this linear system, the sparse solver using the splitting method for the relatively dense column is experimentally better than the direct solver using the Cholesky method.

In this paper we propose a block-type parallel preconditioner for solving large sparse nonsymmetric linear systems, which we expect to be scalable. It is Multi-Color Block SOR preconditioner, combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. We compared it with four other preconditioners, which are ILU(0)-wavefront ordering, ILU(0)-Multi-Color ordering, SPAI(SParse Approximate Inverse), and SSOR preconditioner. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024 x 1024, and for an ill-conditioned matrix from the shell problem from the Harwell-Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR converges.

Burman (1987) and Hall and Titterington (1987) studied kernel smoothing for sparse multinomial data in detail. Both of their estimators for cell probabilities are sparse asymptotic consistent under some restrictive conditions on the true cell probabilities. Dong and Simonoff (1994) adopted boundary kernels to relieve the restrictive conditions. We propose a local linear kernel estimator which is popular in nonparametric regression to estimate cell probabilities. No boundary adjustment is necessary for this estimator since it adapts automatically to estimation at the boundaries. It is shown that our estimator attains the optimal rate of convergence in mean sum of squared error under sparseness. Some simulation results and a real data application are presented to see the performance of the estimator.

Local linear smoothing enjoys several excellent theoretical and numerical properties, an in a range of applications is the method most frequently chosen for fitting curves to noisy data. Nevertheless, it suffers numerical problems in places where the distribution of design points(often called predictors, or explanatory variables) is spares. In the case of univariate design, several remedies have been proposed for overcoming this problem, of which one involves adding additional ″pseudo″ design points in places where the orignal design points were too widely separated. This approach is particularly well suited to treating sparse bivariate design problem, and in fact attractive, elegant geometric analogues of unvariate imputation and interpolation rules are appropriate for that case. In the present paper we introduce and develop pseudo dta rules for bivariate design, and apply them to real data.

We consider a sparse high-dimensional linear regression model. Penalized methods using LASSO or non-convex penalties have been widely used for variable selection and estimation in high-dimensional regression models. In penalized regression, the selection and prediction performances depend on which penalty function is used. For example, it is known that LASSO has a good prediction performance but tends to select more variables than necessary. In this paper, we propose an additive sparse penalty for variable selection using a combination of LASSO and minimax concave penalties (MCP). The proposed penalty is designed for good properties of both LASSO and MCP.We develop an efficient algorithm to compute the proposed estimator by combining a concave convex procedure and coordinate descent algorithm. Numerical studies show that the proposed method has better selection and prediction performances compared to other penalized methods.

This thesis presents a parallel implementation of an iterative method for large scale, sparse linear system and gives result of computational experiments performed on both single transputer and multi transputer parallel computers. To solve linear system, we use conjugate gradient method and develope data storage techinique, data communication scheme. In addition to the explanation of conjugate gradient method, the result of computational experiment is summarized.

오늘날 관찰값의 개수에 비해 변수의 개수가 큰 희박한 데이터셋은 다양한 분야에서 쉽게 찾아볼 수 있고, 통계학에서 그러한 데이터셋에 대한 분석은 하나의 도전이 되어 왔다. 그러한 희박한 데이터에 대한 분류를 위해 판별분석모형들이 최근에 개발되었다. 그러한 판별분석모형들 중 하나의 접근법은 그룹들을 잘 구분해주는 차원들을 찾기를 시도하는데, 그러한 차원들은 데이터의 변수의 개수보다 훨씬 적다. 그러한 모형에서 차원의 수는 예측과 자료의 시각화를 위해 중요한 역할을 하고 일반적으로 K-묶음 교차타당성 방법에 의해 결정된다. 하지만, 희박한 데이터의 경우 K-묶음 교차타당성 방법 적용시 각 묶음에 대한 관찰값의 개수가 매우 적을 수 있기 때문에 교차타당성에 의한 차원 수 결정은 신뢰성이 떨어질 수 있다. 따라서, 본 연구에서는 그러한 희박판별분석모형에 의해 찾아진 차원들에서 각 그룹들의 평균 간의 표준화된 거리에 근거한 측도를 사용하여 최적의 차원 수를 결정하는 방법을 제안하고, 제안된 방법은 모의실험을 통해 검증된다.

Datasets with small n and large p are often found in various fields and the analysis of the datasets is still a challenge in statistics. Discriminant analysis models for such datasets were recently developed in classification problems. One approach of those models tries to detect dimensions that distinguish between groups well and the number of the detected dimensions is typically smaller than p. In such models, the number of dimensions is important because the prediction and visualization of data and can be usually determined by the K-fold cross-validation (CV). However, in sparse data scenarios, the CV is not reliable for determining the optimal number of dimensions since there can be only a few observations for each fold. Thus, we propose a method to determine the number of dimensions using a measure based on the standardized distance between the mean values of each group in the reduced dimensions. The proposed method is verified through simulations.