Browsing by Author "Wu, Zhijun"
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Item A Fast Newton's Algorithm for Entropy Maximization in Phase Determination(1999-05) Wu, Zhijun; Phillips, George; Tapia, Richard; Zhang, YinA long-standing problem in X-ray crystallography, known as the phase problem, is to determine the phases for a large set of complex variables, called the structure factors of the crystal, given their magnitudes obtained from X-ray diffraction experiments. We introduce a statistical phase estimation approach to the problem. This approach requires solving a special class of entropy maximization problems repeatedly to obtain the joint probability distribution of the structure factors. The entropy maximization problem is a semi-infinite convex program, which can be solved in a finite dual space by using a standard Newton's method. The Newton's method converges quadratically, but is costly in general, requiring O(n log n) floating point operations in every iteration, where n is the number of variables. We present a fast Newton's algorithm for solving the entropy maximization problem. The algorithm requires only O(n log n) floating point operations for each of its iterates, yet has the same convergence rate as the standard Newton. We describe the algorithm and discuss related computational issues. Numerical results on simple test cases will also be presented to demonstrate the behavior of the algorithm.Item A Geometric Build-Up Algorithm for Soving the Molecular Distance Geometry Problem with Sparse Distance Data(2001-08) Dong, Qunfeng; Wu, ZhijunNuclear magnetic resonance (NMR) structure modeling usually produces a sparse set of inter-atomic distances in protein. In order to calculate the three-dimensional structure of protein, current approaches need to estimate all other "missing" distances to build a full set of distances. However, the estimation step is costly and prone to introducing errors. In this report, we describe a geometric build-up algorithm for solving protein structure by using only a sparse set of inter-atomic distances. Such a sparse set of distances can be obtained by combining NMR data with our knowledge on certain bond lengths and bond angles. It can also include confident estimations on some "missing" distances. Our algorithm utilizes a simple geometric relationship between coordinates and distances. The coordinates for each atom are calculated by using the coordinates of previously determined atoms and their distances. We have implemented the algorithm and tested it on several proteins. Our results showed that our algorithm successfully determined the protein structures with sparse sets of distances. Therefore, our algorithm reduces the need of estimating the "missing" distances and promises a more efficient approach to NMR structure modeling.Item A Linear-Time Algorithm for Solving the Molecular Distance Geometry Problem with Exact Inter-Atomic Distances(2001-06) Dong, Qunfeng; Wu, ZhijunWe describe a linear-time algorithm for solving the molecular distance geometry problem with exact distances between all pairs of atoms. This problem needs to be solved in every iteration of general distance geometry algorithms for protein modeling such as the EMBED algorithm by Crippen and Havel. However, previous approaches to the problem rely on decomposing a distance matrix or minimizing an error function and require O(n2) to O(n3) floating point operations. The linear-time algorithm will provide a much more efficient approach to the problem, especially in large-scale applications. It exploits the problem structure and hence is able to identify infeasible data more easily as well.Item Item A subgradient algorithm for nonlinear integer programming and its parallel implementation(1991) Wu, Zhijun; Dennis, John E., Jr.; Bixby, Robert E.This work concerns efficiently solving a class of nonlinear integer programming problems: min $\{f(x)$: $x \in \{0,1\}\sp{n}\}$ where $f(x)$ is a general nonlinear function. The notion of subgradient for the objective function is introduced. A necessary and sufficient condition for the optimal solution is constructed. And a new algorithm, called the subgradient algorithm, is developed. The algorithm is an iterative procedure, searching for the solution iteratively among feasible points, and in each iteration, generating the next iterative point by solving the problem for a local piecewise linear model of the original problem which is constructed with supporting planes for the objective function at a set of feasible points. Special continuous optimization techniques are used to compute the supporting planes. The problem for each local piecewise linear model is solved by solving an equivalent linear integer program. The fundamental theory for the new approach is built and all related mathematical proofs and derivations such as proofs for convergence properties, the finiteness of the algorithm, as well as the correct formulation of the subproblems are presented. The algorithm is parallelized and implemented on parallel distributed-memory machines. The preliminary numerical results show that the algorithm can solve test problems effectively. To implement the subgradient algorithm, a parallel software system written in EXPRESS C is developed. The system contains a group of parallel subroutines that can be used for either continuous or discrete optimization such as subroutines for QR, LU and Cholesky factorizations, triangular system solvers and so on. A sequential implementation of the simplex algorithm for linear programming also is included. Especially, a parallel branch-and-bound procedure is developed. Different from directly parallelizing the sequential binary branch-and-bound algorithm, a parallel strategy with multiple branching is used for good processor scheduling. Performance results of the system on NCUBE are given.Item Mathematical Modeling of Protein Structure Using Distance Geometry(2000-07) Yoon, Jeong-Mi; Gad, Yash; Wu, ZhijunThis paper reviews methods for structure determination with interatomic distances and explores possible improvement of the methods and ways of combining them with potential energy minimization.Item Parallel Continuous Optimization(2000-01) Dennis, J.E. Jr.; Wu, ZhijunParallel continuous optimization methods are motivated here by applications in science and engineering. The key issues are addressed at different computational levels including local and global optimization as well as strategies for large, sparse versus small but expensive problems. Topics covered include global optimization, direct search with and without surrogates, optimization of linked subsystems, and variable and constraint distribution. Finally, there is a discussion of future research directions.Item Solving the Double Digestion Problem as a Mixed-Integer Linear Program(2001-08) Wu, Zhijun; Zhang, YinThe double digestion problem for DNA restriction mapping is known to be NP-complete. Several approaches to the problem have been used including exhaustive search, simulated annealing, branch-and-bound. In this paper, we consider a mixed-integer linear programming formulation of the problem and show that with this formulation the problem can be solved efficiently to a fairly large size using state-of-the-art integer programming techniques. In particular, we present computational results obtained by using the CPLEX mixed-integer linear programming software on a set of randomly generated, large-scale double digestion problems.Item Strategies for natural language processing using stratified grammar(1988) Wu, Zhijun; Lamb, Sydney M.Using the theory of stratified grammar proposed by Sydney Lamb (1966), I developed a computer system to simulate the human language-using process. A new approach to representing linguistic knowledge, a goal-directed parsing strategy and a method of combining bottom-up processing with the top-down expectation are proposed and implemented. The system can automatically translate English sentences into Chinese sentences in a small task domain. Some linguistic information like syntactic structure, semantic interpretation and dynamic relations of the input sentence can also be obtained by using the system.Item The Bayesian Statistical Approach to the Phase Problem in Protein X-ray Crystallography(1999-04) Wu, Zhijun; Phillips, George; Tapia, Richard; Zhang, YinWe review a Bayesian statistical approach to the phase problem in protein X-ray crystallography. We discuss the mathematical foundations and the computational issues. The introduction to the theory and the algorithms does not require strong background in X-ray crystallography and related physical disciplines.