C.J. Lin, J.J. More, Incomplete Cholesky factorizations with limited memory , SIAM. J. Sci. Comput., Vol. 21 (1999) 24-45  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... be inaccurate (i.e. jjA Gamma LL jj F is large) if d i is chosen too small, the triangular solves with L and L can become unstable (i.e. jjI Gamma L AL jj F is large) Some heuristics for choosing d i can be found in [9, 10] and in the optimization literature [11, 12] see also [13]. Unfortunately, it is frequently the case that even a handful of pivot shifts will result in a poor preconditioner, regardless of the shift strategy used. The problem is that if a nonpositive pivot occurs, the loss of information about the true factors due to dropping has already been so great ....

....the modification of diagonal entries is done dynamically, as dropping occurs in the course of the incomplete factorization process. Another approach, known as diagonally compensated reduction of positive off diagonal entries, consists This point is convincingly argued by Lin and Mor e in [13]. of modifying the matrix A before computing an incomplete factorization. In the simplest variant of this method [17] positive off diagonal entries are set to zero and their original values are added to the corresponding diagonal entries. The resulting matrix satisfies A = A C with C positive ....

Lin CJ, Mor'e JJ. Incomplete Cholesky factorizations with limited memory. SIAM Journal on Scientific Computing 1999; 21:24--45.

....we assume that L = I, whenever we use a constraint of the form (3.6) If the problem is given in general form (L 6= I) it is possible to transform it to the standard form (L = I) by means of the algorithms given in [14] and [37] or by a change of variable in case L is invertible. Lin and Mor e [45] have recently proposed a method that can be applied to quadratically constrained least squares problems, where the constraint is of the form (3.6) They assumed that if L 6= I then L is invertible and they can apply a change of variables. They regard L as a scaling matrix that they compute so ....

C-J. Lin and J.J. Mor'e. Incomplete Cholesky factorizations with limited memory. Technical Report MCS--P682--0897, Mathematics and Computer Science Division, Argonne National Laboratory. Argonne, Illinois, August 1997. 121

....[25] with a sparse Cholesky factorization for the computation of the Newton step (4. 9) The tangential step is obtained by a Steihaug CG method [27] For problems with moderate size we form the reduced Hessian explicitly and use the incomplete Cholesky factorization ICFS of Lin and More [21] as preconditioner. For large problems, we use currently a matrix free Steihaug CG method without preconditioning. If the CG path does not leave the trust region and does not find negative curvature, it is stopped if the current residual r k is reduced to r k 1 # max 10 20 , min 0.05, r k r ....

C.-J. LIN AND J. J. MOR E, Incomplete Cholesky factorizations with limited memory, SIAM J. Sci. Comput., 21 (1999), pp. 24--45.

....problematic in the present context, since control of the ll in is necessary in order to control the work per iteration in the multilevel iteration. Several authors have explored possibilities of controlling the maximum number of ll in elements allowed in each row of the incomplete decomposition [35, 47, 31]. However, for many cases of interest, and in particular for matrices arising from discretizations of partial di erential equations ordered by the minimum degree algorithm, most of the ll in in a complete factorization occurs in the later stages, even of all the rows initially have about the same ....

C.-J. Lin and J. J. Mor e, Incomplete Cholesky factorizations with limited memory, SIAM J. 19 Sci. Comput., 21 (1999), pp. 24-45.

....This is true even when di erent level values are used for the various subdomains and boundaries. Incomplete Cholesky (IC) preconditioners for symmetric problems could be computed with our parallel algorithmic framework using preconditioners proposed by Jones and Plassmann [21] and Lin and Mor e [23] on each subdomain and on the boundaries. The sparsity patterns of these preconditioners are determined by the numerical values in the matrix and by memory constraints. Lin and Mor e have proved that these preconditioners exist for M and H matrices. Parallel IC preconditioners also can be shown ....

C. Lin and J. J. Mor e, Incomplete Cholesky factorizations with limited memory, SIAM J. Sci. Comput., 21 (1999), pp. 24-45.

....norms. See the text for a key to the data. These indicate the limitations of our approach, and for these problems preconditioners which try to mimic the structure of the Hessian without incurring the cost of the fill in such as the limited memory incomplete Cholesky factorization proposed by Lin and Mor e (1997), and the references contained therein are likely to be preferable. The third category contains the harder, highly nonlinear problems CURLYxx, NONCVXUN, SBRYBND, SCOSINE and SCURLYxx. For these problems, the 2 norm is ineffective, and some rescaling is necessary. Interestingly, the modified ....

C.-J. Lin and J. J. Mor'e. Incomplete Cholesky factorizations with limited memory. Technical Report ANL/MCS-P682-0897, Argonne National Laboratory, Illinois, USA, 1997.

....we assume that L = I, whenever a constraint of the form (3.5) is used. If the problem is given in general form (L 6= I) it is possible to transform it to the standard form (L = I) by means of the algorithms given in [13] and [34] or by a change of variable in case L is invertible. Lin and Mor e [43] have recently proposed a method that can be applied to quadratically constrained least squares problems. They assumed that if L 6= I then L is invertible and they can apply a change of variables. Moreover, the regard L as a scaling matrix that they compute so that it clusters the eigenvalues of ....

C-J. Lin and J.J. Mor'e. Incomplete Cholesky factorizations with limited memory. Technical Report MCS--P682--0897, Mathematics and Computer Science Division, Argonne National Laboratory. Argonne, Illinois, August 1997.

....problems. Interesting features of this implementation include the use of projected searches and a preconditioned conjugate gradient method to determine the minor iterates and the use of a limited memory preconditioner. We use the incomplete Cholesky factorization icfs of Lin and More [26] as a preconditioner since this factorization does not require the choice of a drop tolerance, and the amount of storage can be specified in advance. Section 7 presents the results of a comparison between TRON and the LANCELOT [14] and L BFGS B [36] codes. These results show that on the problems ....

....region is violated, a negative curvature direction is generated, or the convergence condition (5.8) is satisfied. As noted in section 5, this condition can be satisfied by choosing the minor iterates so that A(x k,j ) # A(x k,j 1 ) For additional details, see the discussion in Lin and More [26]. In our algorithms we choose D from an incomplete Cholesky factorization. From a theoretical viewpoint, the choice of D is not important, but the numerical results are strongly dependent on the choice of D. We use the incomplete Cholesky factorization icfs of Lin and More [26] The icfs ....

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C.-J. Lin and J. J. Mor e, Incomplete Cholesky factorizations with limited memory, SIAM J. Sci. Comput., 21 (1999), pp. 24--45.

....problems. Interesting features of this implementation include the use of projected searches and a preconditioned conjugate gradient method to determine the minor iterates, and the use of a limited memory preconditioner. We use the incomplete Cholesky factorization icfs of Lin and Mor e [26] as a preconditioner since this factorization does not require the choice of a drop tolerance, and the amount of storage can be specified in advance. Section 7 presents the results of a comparison between TRON and the LANCELOT [14] and L BFGS B [36] codes. These results show that on the problems ....

....region is violated, a negative curvature direction is generated, or the convergence condition (5.8) is satisfied. As noted in Section 5, this condition can be satisfied by choosing the minor iterates so that A(x k;j ) ae A(x k;j 1 ) For additional details, see the discussion in Lin and Mor e [26]. In our algorithms we choose D from an incomplete Cholesky factorization. From a theoretical viewpoint, the choice of D is not important, but the numerical results are strongly dependent on the choice of D. We use the incomplete Cholesky factorization icfs of Lin and Mor e [26] The icfs ....

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C.-J. Lin and J. J. Mor' e, Incomplete Cholesky factorizations with limited memory, Preprint MCS-P682-0897, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 1997. To appear in the SIAM Journal on Scientific Computing.

....by a similar procedure as well. Thus, L should be sparse and be generated without requiring the whole matrix ( A(S Gamma1 k Omega X k ) A T ) After testing several preconditioners, we found it is possible to modify a version of the Lin and Mor e incomplete Cholesky factorization (ICF) [8]. Their basic idea is from Saad [14, 15] and Jones and Plassmann [6] the procedure is in Algorithm 3.1. Note that in this algorithm, the (i; j) component of ( A(S Gamma1 k Omega X k ) A T ) is represented as a(i,j) The incomplete factor L is calculated column by column. After the jth ....

....replace negative diagonal elements with positive numbers. Another approach is the shifted incomplete factorization of Manteuffel [10] It adds ffI to the original matrix, where I is an identity matrix. Eventually, if ff is big enough, the matrix will be diagonal dominant and ICF will succeed. In [8], these two approaches are discussed, and the shifted incomplete factorization appears more stable. Algorithm 3.2 thus is the actual procedure for calculating ICF of a matrix ( A(S Gamma1 k Omega X k ) A T ) by using the shifted incomplete factorization. In this algorithm, D is a ....

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C.-J. Lin and J. J. Mor'e. Incomplete Cholesky factorizations with limited memory. Preprint MCS-P682-0897, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 1997.

....SBMIN algorithm rely on the trust region philosophy,the codes are quite different. We detail some of the differences in Section 3. In this paper we consider a preconditioner based on the incomplete Cholesky factorization of Jones and Plassmann [13] with the modifications proposed by Chin and Mor e [14] for the solution of trust region subproblems. This version of the incomplete Cholesky factorization has predictable storage requirements, a dynamic sparsity pattern, and does not require the specification of a drop tolerance. These advantages make this incomplete Cholesky factorization an ....

....Cholesky factorization an attractive ingredient in an optimization code. The SBMIN code, on the other hand, uses the incomplete Cholesky factorization of Munksgaard [19] that has unpredictable storage requirements and requires the specification of a drop tolerance. As shown by Chin and Mor e [14], codes that require the specification of a drop tolerance suffer from erratic behavior. The test problems described in Section 5 come from the MINPACK 2 test problem collection [1] since this collection is representative of large scale optimization problems arising from applications. We ....

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C.-J. Lin and J. J. Mor' e, Incomplete cholesky factorizations with limited memory, Preprint MCS-P682-0897, Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, 1997.

....where a second threshold must be decided. In addition, this approach uses only the information of the original matrix but not sub matrices during the factorization. Saad [30, 31] and Jones and Plassmann [14] keep only the largest elements (in magnitude) in the preconditioner L. Lin and Mor e [18] used this idea and proposed a preconditioner based on Algorithm 2.1. In Algorithm 2.1, the incomplete factor L is calculated column by column. After the jth column is obtained, some of the largest (in magnitude) elements are stored back to L (see the last line of Algorithm 2.1) For sparse ....

....is also O(m 2 ) In Section 3, 5 and 6, we will discuss the practical computational time for our test matrices. Based on the implementation, after selecting the p largest elements, there may be a need to sort the row (column) indices of L to the ascending order. For example, in [14] and [18], the matrix L is stored in the compressed sparse column format. Hence there is an array which stores row indices of columns of L. Their implementations require that the row indices of each column is in an ascending order. If that is the case, the cost would be O(p log 2 p) for each column. Then ....

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C.-J. Lin and J. J. Mor'e. Incomplete Cholesky factorizations with limited memory. Preprint MCS-P682-0897, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 1997. To appear in SIAM Journal on Scientific Computing.

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C.J. Lin, J.J. More, Incomplete Cholesky factorizations with limited memory , SIAM. J. Sci. Comput., Vol. 21 (1999) 24-45

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