T. F. Coleman and J. J. Mor'e [1983], "Estimation of sparse Jacobian matrices and graph coloring problems ", SIAM J. Numer. Anal. 20, pp. 187-209.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....performances temps et m emoire de P . En r eponse, on peut exploiter le fait que certaines jacobiennes sont creuses, et les manipuler en cons equence. On peut aussi calculer des d eriv ees directionnelles (mode direct ) suivant certaines directions bien choisies, puis reconstruire la matrice [30] [73] 32] 85] adifor exploite cette propri et e au moment de l ex ecution, en sous traitant syst ematiquement les op erations sur les jacobiennes a une biblioth eque sp ecialis ee dans les matrices creuses. On peut aussi tenter de trouver la structure creuse de la jacobienne par une ex ecution ....

Coleman T.F., Mor e J.J. "Estimation of sparse Jacobian matrices and graph coloring problems". SIAM journal on numerical analysis, 20:187-209, 1983.

....Science, Old Dominion University, Norfolk, VA 23529 0162 USA. pothen cs.odu.edu and ICASE, NASA Langley Research Center, Hampton, VA 23681 2199 USA. pothen icase.edu Since the early 70 s e#cient computation of Jacobian and Hessian matrices using FD techniques have been studied extensively [2, 3, 4, 6, 8, 12, 13]. Curtis, Powell and Reid [6] were the first to observe that sparsity can be exploited in reducing the number of function evaluations required in the estimation of Jacobian matrices. In the Hessian case, Powell and Toint [13] were the first to exploit symmetry. Several authors later extended these ....

....number of function evaluations required in the estimation of Jacobian matrices. In the Hessian case, Powell and Toint [13] were the first to exploit symmetry. Several authors later extended these works and studied the matrix estimation problems from a graph theoretic viewpoint. Coleman and More [3], the first to adopt this view, showed that the estimation of a sparse Jacobian matrix using a direct method can be seen as a distance 1 graph coloring problem. Later, McCormick [12] showed that the approximation of a Hessian matrix using a direct method is equivalent to a distance 2 graph ....

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T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, February 1983.

....for estimating these matrices via finite di#erences (FD) or automatic di#erentiation (AD) are needed. It is known that the problem of minimizing the number of function evaluations (or AD passes) required in the computation of these matrices can be formulated as variants of graph coloring problems [1, 2, 3, 9, 13]. The particular coloring problem di#ers with the optimization context: whether the Jacobian or the Hessian matrix is to be computed; whether a direct or a substitution method is employed; and whether only columns, or only rows, or both columns and rows are to be used to evaluate the matrix ....

....we focus on direct methods that use column partitioning. A partition of the columns of a nonsymmetric matrix A is said to be consistent with the direct determination of A if whenever a ij is a non zero element of A then the group containing column j has no other column with a non zero in row i [1]. Similarly, a partition of the columns of a symmetric matrix A is called symmetrically consistent with the direct determination of A if whenever a ij is a non zero element of A then either (i) the group containing column j has no other column with a non zero in row i, or (ii) the group containing ....

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T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, February 1983.

....AD passes) ## This author s research was supported by NSF grant DMS 9807172, DOE ASCI level2 subcontract B347882 from Lawrence Livermore National Lab; and by DOE SCIDAC grant DE FC02 01ER25476. required in the computation of these matrices can be formulated as variants of graph coloring problems [1 3, 8, 11]. The particular coloring problem di#ers with the optimization context: whether the Jacobian or the Hessian matrix is to be computed; whether a direct or a substitution method is employed; and whether only columns, or only rows, or both columns and rows are to be used to evaluate the matrix ....

....we focus on direct methods that use column partitioning. A partition of the columns of an unsymmetric matrix A is said to be consistent with the direct determination of A if whenever a ij is a non zero element of A then the group containing column j has no other column with a non zero in row i [1]. Similarly, a partition of the columns of a symmetric matrix A is called symmetrically consistent with the direct determination of A if whenever a ij is a non zero element of A then either (i) the group containing column j has no other column with a non zero in row i, or (ii) the group containing ....

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T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, February 1983.

....of sets of column indices defining a consistent partition of the columns of J. For a consistent partition, the unknowns in each row of J is obtained (read off) from the associated diagonal linear system. A graph theoretic interpretation of the consistent column partitioning problem is given in [4]. It has been shown that finding a consistent column partition of J is equivalent to coloring the vertices of an associated graph. The coloring obtained is dependent on the order in which the vertices are considered during the coloring procedure. An often cited example [4] shows that it may be ....

....problem is given in [4] It has been shown that finding a consistent column partition of J is equivalent to coloring the vertices of an associated graph. The coloring obtained is dependent on the order in which the vertices are considered during the coloring procedure. An often cited example [4] shows that it may be advantageous to consider a more general problem where segments of columns are grouped together. In [18] we have proposed new techniques to estimate sparse Jacobian matrices. In this approach both columns and rows are grouped together and the resulting segments of columns are ....

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T. F. Coleman and J. J. Mor. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187-209, 1983.

....the number of colors used. The GCP arises in a number of scientific computing and engineering applications. Examples include time tabling and scheduling [11] frequency assignment [6] register allocation [3] printed circuit testing [8] parallel numerical computation [1] and optimization [4]. Col oring a general graph with the minimum number of colors is known to be an NP hard problem [7] thus one often relies on heuristics to compute a solution. In a parallel application a graph coloring is usually performed in order to partition the work associated with the vertices into ....

....fast and minimizing the number of colors used becomes less important. For this purpose there exist several linear time, or close to linear time, sequential greedy coloring heuristics. These heuristics have been found to be effective in coloring graphs that arise from a number of appli cations [4, 10]. This paper deals mainly with the later problem of developing fast sublinear parallel coloring algorithms. Previous work on developing such algorithms has been performed on distributed memory computers using explicit messsage passing. The speedup obtained so far has been discouraging [1] The ....

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T.F. Coleman and J.J. More. Estimation of sparse jacobian matrices and graph coloring problems. SIAM Journal on Numerical Anal- ysis, 20(1):187-209, 1983.

....so that the number of colours are minimised. If found, the minimum possible number of colours is known as the chromatic number of the graph G and denoted (G) The GCP is well studied and has many applications including scheduling, timetabling and the solution of sparse linear systems, see e.g. [4, 9, 23, 24]. However it is also known to be one of the most difficult combinatorial optimisation problems, e.g. 18] Not only is the problem of finding (G) NP hard, 11] but Lund Yannakakis have even shown that, for some 0, approximate graph colouring within a factor of N is also NP hard, 25] ....

T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, 1983.

....given a graph G(V; E) assign a colour to each vertex in V such that no two adjacent vertices have the same colour and so that the number of colours is minimised. The GCP is well studied and has many applications including scheduling, timetabling and the solution of sparse linear systems, see e.g. [7, 13, 35, 36]. However it is also often cited as one of the most difficult combinatorial optimisation problems, e.g. 27] The chromatic number of G, denoted (G) is the minimum number of colours required to colour the graph. Not only is the problem of finding (G) NP hard, 17] but Lund Yannakakis have ....

T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, 1983.

....W such that the nonzero elements of J can easily be extracted from the calculated pair (W T J, JV ) The motivation for solving above problem comes from the following two observations on the problem of computing a sparse Jacobian matrix. Sparse finitedi #erencing literature [CPR74,CGM84,CM84a,CM84b,CGM85,CC86] provides a solution based on partitioning of columns, to define a matrix V such that J can be determined from the product JV . However, the matrix V is not guaranteed to be thin, even if J has a lot of sparsity: consider a sparse matrix J with a single dense row. Alternatively, a ....

....bi partition problems can also be expressed in terms of graphs and graph coloring. This graph view is important in that it more readily exposes the relationship of the bi partition problems with the combinatorial approaches used in the sparse finite di#erencing literature, e.g. CPR74,CGM84,CM84a,CM84b,CGM85, CC86] 2.1.2 Algorithms for direct and substitution bi coloring The two combinatorial problems we face, corresponding to direct determination and determination by substitution, can both be approached in the following way. First, permute and partition the structure of J : J = P J Q ....

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Thomas F. Coleman and Jorge J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. on Numerical Analysis, 20(1):187--209, 1984.

....real world problems, among others, time tabling and scheduling, frequency assignment, register allocation, and efficient estimation of sparse matrices in optimization, have successfully been modeled using the graph coloring problem. See Lewandowski (1994) Gamst (1986) Chaitin et al. 1981) and Coleman and More (1983) for some of the works in each of these applications respectively. Besides modeling real world problems, graph coloring plays a crucial role in the field of parallel computation. In particular, when a computational task is modeled using a graph where the vertices represent the subtasks and the ....

....The graph coloring problem is known to be NP complete (see Garey and Johnson (1979) making heuristic approaches inevitable in practice. There exist a number of sequential graph coloring heuristics that are quite effective in coloring graphs encountered in practical applications. See Coleman and More (1983) for some of the popular heuristics. However, due to their inherent sequential nature, these heuristics are difficult to parallelize. In fact, in Greenlaw et al. 1995) coloring the vertices of a graph in a given order where each vertex is assigned the smallest color that has not been given to ....

Coleman, T. and More, J. (1983). Estimation of sparse jacobian matrices and graph coloring problems. SIAM Journal on Numerical Analysis, 20(1):187-- 209.

....the number of colors used. The GCP arises in a number of scientific computing and engineering applications. Examples include time tabling and scheduling [11] frequency assignment [6] register allocation [3] printed circuit testing [8] parallel numerical computation [1] and optimization [4]. Coloring a general graph with the minimum number of colors is known to be an NP hard problem [7] thus one often relies on heuristics to compute a solution. In a parallel application a graph coloring is usually performed in order to partition the work associated with the vertices into ....

....fast and minimizing the number of colors used becomes less important. For this purpose there exist several linear time, or close to linear time, sequential greedy coloring heuristics. These heuristics have been found to be effective in coloring graphs that arise from a number of applications [4, 10]. This paper deals mainly with the latter problem of developing fast sublinear parallel coloring algorithms. Previous work on developing such algorithms has been performed on distributed memory computers using explicit messsage passing. The speedup obtained so far has been discouraging [1] The ....

[Article contains additional citation context not shown here]

T.F. Coleman and J.J. More. Estimation of sparse jacobian matrices and graph coloring problems. SIAM Journal on Numerical Analysis, 20(1):187--209, 1983.

....is the size of the Jacobian matrix, and of all the intermediate Jacobian matrices during the computation. However, the Jacobian matrix is often sparse. One question is how to use this sparsity. One can compute the derivatives following some well chosen directions, and then rebuild the Jacobian [30] [73] 32] 85] adifor exploits sparsity in a dioeerent way, linking operations on the Jacobian matrices to a specialized sparse matrix library. One can also nd the sparsity of the matrix by iobservingj an execution of P . This is indeed very close to the inspector executor methods [69] 20] used ....

Coleman T.F., Mor# J.J. "Estimation of sparse Jacobian matrices and graph coloring problems". SIAM journal on numerical analysis, 20:187209, 1983.

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T. F. Coleman and J. J. Mor'e [1983], "Estimation of sparse Jacobian matrices and graph coloring problems ", SIAM J. Numer. Anal. 20, pp. 187-209.

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T. F. Coleman and J. J. Mor'e, "Estimation of sparse Jacobian matrices and graph coloring problems," SIAM J. Numer. Anal. 20, 1983, pp. 187-209.

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Coleman, T. F. and Mor, J. J. (1983): Estimation of Sparse Jacobian Matrices and Graph Coloring Problems. In: SIAM Journal of Numerical Analysis, Vol 20, pp 187-209.

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T. F. Coleman and J. J More. Estimation of sparse jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 1(20):187--209, 1983.

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COLEMAN, T. F., and MORE , J. J., Estimation of Sparse Jacobian Matrices and Graph Coloring Problems, SIAM Journal on Numerical Analysis, Vol. 20, pp. 187--209, 1983.

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T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, 1983.

No context found.

T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, 1983.

No context found.

T.F. Coleman and J.J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM Journal on Numerical Analysis, 20(1):187--209, 1983.

No context found.

T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, 1983.

No context found.

T.F. Coleman and J.J. More. Estimation of sparse jacobian matrices and graph coloring problems. SIAM Journal on Numerical Analysis, 20(1):187--209, 1983.

No context found.

T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, 1983.

No context found.

T. F. Coleman and J. J. More. Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal., 20(1):187--209, February 1983.

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Coleman, T. F. and Mor'e, J. J., Estimation of Sparse Jacobian Matrices and Graph Coloring Problems, SIAM J. Numer. Anal., Vol. 20, pp 187-209, (1983).

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