Hendrickson, Bruce. The Molecule Problem: Determining Conformation from Pairwise Distances, PhD Thesis, Computer Science Department, Cornell University. Technical Report 90-1159.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that even in this case we employ the square root of the quantities used to define D affine ) k Before we investigate a subspace adaptation of TIR, we demonstrate the effectiveness of our reflection idea and affine scaling technique. We consider random problem instances of molecule minimization [9, 10], which minimize a quartic subject to bounds on the variables. Table 1 and 2 list the average number of iterations (over ten random test problem instances for each entry) required for the different techniques under comparison. The notation in front of a number indicates that the average number ....

....( 6] We used twenty different unconstrained nonlinear test problems. All but four are test problems described in [12] but with all the bound constraints removed. The problems EROSENBROCK and EPOWELL are taken from [13] The last two problems, molecule problems MOLE1 and MOLE3, are described in [9, 10]. For all problems, the number of variables r is 260. The minimization algorithm terminates when Ilgll2 10 6. We use the parameter = 0.0005 in both FIG. 11 and FIG. 12. Tables 3 and 4 compare the Steihaug and subspace methods described above in terms of the number of minimization iterations ....

Bruce A. Hendrickson. The molecule problem: Determining conformation from pairwise distances. Cornell University Ph.D thesis, Computer Science, 1991.

....ij ) 2 : Hence, f( is zero precisely when the v i s provide a realization of the partial matrix A. This optimization problem is hard to solve (as it may have many local optimum solutions) Several algorithms have been proposed in the literature; see, in particular, CH88] GHR93] Ha91] [He90, He95], KTO93] MW97] PL97] They are based on general techniques for global optimization like tabu and pattern search [PL97] the continuation approach (which consists of transforming the original function f( into a smoother function having fewer local optimizers, MW96, MW97] or ....

.... and pattern search [PL97] the continuation approach (which consists of transforming the original function f( into a smoother function having fewer local optimizers, MW96, MW97] or divide and conqueer strategies aiming to break 6 the problem into a sequence of smaller or easier subproblems [CH88, He90, He95]. In [He90, He95] the basic step consist of finding principal submatrices having a unique realization, treating each of them separately and then trying to combine the solutions. Thus arises the problem of identifying principal submatrices having a unique realization, which turns out to be NP hard ....

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B. Hendrickson. The Molecule Problem: Determining Conformation from Pairwise Distances. Ph.D. thesis, Technical Report 90-1159, Cornell University, Department of Computer Science, Ithaca, NY, 1990.

....by solving a distance geometry problem using the NMR distance data. They can be formulated as global nonlinear least squares optimization problems (Crippen and Havel [30] or, from a graph theoretic point of view, 10 they are a class of NP complete graph embedding problems (Hendrickson [41], Saxe [79, 80] Mor e and Wu [64] Recent attempts to solve these problems on parallel high performance architectures are by (Hendrickson [42] Mor e and Wu [66] Byrd, Schnabel et al. [97] etc) A simple version of the distance geometry problem is to find a set of points to realize a given set ....

B. A. Hendrickson. The molecule problem: Determining conformation from pairwise distances. PhD thesis, Cornell University, Ithaca, New York, 1991.

....are surprisingly many applications for this problem, sometimes called the molecule problem. These applications include NMR data, determination of protein structure, surveying, satellite ranging, and molecular conformation; e.g. the survey [23] and the discussion in [50] and the related papers [49, 89, 114, 117, 44]. We now consider the approximate EDMCP and follow the approach in [1] where the reader will nd all the proofs and details omitted here. Let A be a pre distance matrix and let H be an n n symmetric matrix with nonnegative elements (weights) Consider the objective function f(D) kH (A ....

B. HENDRICKSON. The molecule problem: Determining conformation from pairwise distances. PhD thesis, Cornell University, Ithaca, New York, 1991.

....from NMR experiments. Another interesting issue is the dependence of the structures on the distance data. From a mathematical viewpoint, we do not know when structures can be determined uniquely with exact, but incomplete distance data. For some results in this direction, see Hendrickson [13, 14]. Acknowledgments Our work has been influenced, in particular, by conversations with Paul Bash, Gordon Crippen, and Teresa Head Gordon. Gail Pieper, as usual, deserves special thanks for her comments on the manuscript. 14 ....

B. A. Hendrickson, The molecule problem: Determining conformation from pairwise distances, PhD thesis, Cornell University, Ithaca, New York, 1991.

.... Gamma u 2 i;j u 2 i;j ; 0 ) 2:6) Clearly, x = fx 1 ; xm g solves the distance geometry problem if and only if x is a global minimizer of f and f(x) 0. Special optimization algorithms have been developed for solving the distance geometry problem (2. 1) For example, Hendrickson [21, 22] used a graph theoretic viewpoint to develop algorithms that test the uniqueness and rigidity of the distance graph. These algorithms can be used to reduce the problem into smaller, easier subproblems. Glunt, Hayden, and Raydan [12, 13] have proposed a special gradient method for determining a ....

B. A. Hendrickson, The molecule problem: Determining conformation from pairwise distances, PhD thesis, Cornell University, Ithaca, New York, 1991.

....our impression that better results are obtained by starting from the upper bounds than from the lower bounds. We currently prefer to set Delta 0 = 25 Delta L L :75 Delta U U . Example 2 Subsequent numerical experiments were performed on a 63 atom molecular fragment studied by Hendrickson [16], who provided a set of 236 (exact) interatomic distances. Given a partial distance matrix with these 236 distances specified, the problem is to complete the partial matrix to a 3 dimensional distance matrix. Mor e and Wu [26] attempted to solve a closely related (and presumably easier) problem ....

....[ ij ; u ij ] 1 Sigma ffl)d ij , for various choices of ffl. It should be emphasized that none of these problems is realistic in the sense that the bounds might be derived, either theoretically or experimentally, if the molecule s 3 dimensional structure was unknown. Furthermore, Hendrickson [16] determined that a structure with the 236 specified distances is nearly rigid and quite difficult to determine. Nevertheless, studying these problems yielded some fascinating insights. Following Crippen and Havel [6] Mor e and Wu [26] formulated the embedding problem as an unconstrained ....

B. A. Hendrickson. The Molecule Problem: Determining Conformation from Pairwise Distances. PhD thesis, Cornell University, 1991.

....to know exact latitude, longitude and altitude. Unfortunately, many exisiting measurements are in error and these errors propagate over time if new points are added that are located based on old faulty data. Such problems also arise in surveying. The traditional approach to this problem [11, 10] is to attempt solve a system of millions of nonlinear equations in order to find locations for the points that minimize an error function. The approach in this paper is quite different but tries to achieve the same end result with a much simpler algorithm. Similar issues arise in the analysis of ....

....effective ways of combining the approach here with the techniques of distance geometry. Within the area of rigidity theory when a sparse, error free, set of distances is known the work most closely associated with problems of reconstruction is that dealing with global rigidity [4, 9, 10]. Here, the issue is to determine when a set of exact distances is sufficient to uniquely determine a point set up to congruence. Finally, we note related work of Saxe [15] proving that it is NP hard to decide whether a sufficiently sparse set of distances is consistent with a set of points in R ....

B. Hendrickson, The Molecule Problem: Determining Conformation from Pairwise Distances, Ph.D. thesis, Cornell University Dept. of Computer Science, 1990.

....(see [20] 12] and [5] for global optimization background) in the search for a global minimum of f , but these general algorithms do not take advantage of the structure in the distance geometry problem. Other algorithms used in the solution of distance geometry problems (for example, Hendrickson [10, 11], Havel [9] and Glunt, Hayden, and Raydan [7, 8] must also rely on general techniques, such as multistarts or simulated annealing, to claim convergence to a global minimizer. The continuation approach for global optimization hinges on the ability to gradually transform the original function into ....

B. A. Hendrickson, The molecule problem: Determining conformation from pairwise distances, PhD thesis, Cornell University, Ithaca, New York, 1991.

....optimization problem. The objective function for the distance geometry problem is defined so that the constraints are satisfied at a global minimizer of the problem. Special optimization techniques for this class of problems have been developed by Crippen and Havel [4] Havel [10] Hendrickson [11, 12], Glunt, Hayden, and Raydan [8, 9] and Mor e and Wu This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W 31 109 Eng 38 and by the Argonne ....

....show that the continuation method found the optimal solutions to the problems successfully. On the other hand, the multistart method failed to find a solution in all cases. We also tested GMIN with a 63 atom protein fragment. For this problem, 236 pairwise distances are specified. Hendrickson [11] showed that the molecule with this set of distance data is nearly rigid and difficult to determine. Our numerical results confirm this finding. We ran GMIN from 100 starting points for = 0:01; 0:02; 0:04; 0:06; 0:08; 0:1. In all cases we set 0 = 1 and p = 20 for continuation. The results from ....

B. A. Hendrickson, The molecule problem: Determining conformation from pairwise distances, Ph.D. thesis, Cornell University, Ithaca, New York, 1991.

....be possible to position subgraphs first and then piece them together. Since the running time grows rapidly with problem size this could lead to a substantial reduction of computational effort. In fact, an approach to the molecule problem using precisely this approach has recently been proposed [21]. For this approach to be infallible we would need to develop sufficiency conditions for a graph to have a unique realization. Unfortunately, the necessary conditions developed in this paper are not sufficient. Connelly has identified a class of bipartite graphs that satisfy the conditions ....

B. Hendrickson, The Molecule Problem: Determining Conformation from Pairwise Distances, PhD thesis, Cornell University, Dept. of Computer Science, Ithaca, NY, 1990. Technical Report 90-1159.

....Various heuristics have been developed that rely in various ways upon knowledge about chemical structures. A survey of this previous work is beyond the scope of this paper, but a good overview can be found in Chapter 6 of [10] A more detailed description of the current work is provided in [14]. This paper is structured as follows. The characterization of uniquely realizable graphs is the topic of the next section. This is followed in x3 by an algorithm to identify uniquely realizable subgraphs, step (1) in Fig. 1. In x4 we describe ABBIE s technique for breaking a large, uniquely ....

....than 15 vertices. All larger subgraphs were divided into pieces using the small vertex separator heuristic from x4. Also, the stress test to verify unique realizability was turned off. Besides the intrinsic reduction in effort, this allowed for some economy in the redundant rigidity calculation [14]. If a subgraph passed the necessary tests, but wasn t truly uniquely realizable, disabling the stress test could lead to incorrect coordinates being computed for the subgraph. However, this would be detected when the subgraph would be used in later optimizations since it would be unable to fit ....

B. Hendrickson, The Molecule Problem: Determining Conformation from Pairwise Distances, PhD thesis, Cornell University, Dept. of Computer Science, Ithaca, NY, 1990. Technical Report 90-1159.

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Hendrickson, Bruce. The Molecule Problem: Determining Conformation from Pairwise Distances, PhD Thesis, Computer Science Department, Cornell University. Technical Report 90-1159.

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