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requires preconditioning the A matrix. Preconditioning is done by the MATLAB program cholinc (Incomplete Cholesky Preconditioner). The cholinc precondi-. This include the Jacobi, Gauss-Siedel, Incomplete LU factorization, and the conjugate gradient methods. The chapter also introduces the algebraic multigrid.

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The reverse Cuthill—McKee RCM ordering algorithm is the most simple but not very efficient method for reordering a sparse matrix to minimize its profile width, followed by a uniform block processor partitioning in the reordered matrix. The remaining vertices are combined in the bordered block to be processed individually. Inside the independent blocks, the arithmetic operations are independent and can be performed in parallel.

The internal vertices of each block can be ordered by the RCM algorithm to enhance the efficiency of the triangular decomposition. Another positive effect of such an ordering consists in the following: for a certain class of matrices, the complete decomposition has an almost minimally possible number of nonzero elements.

Generally, the search for an optimal permutation is an NP-complete problem. There exist other matrix ordering algorithms for the most optimal distribution of matrices among processors. Many of them are in open access. In independent blocks, the arithmetic operations are completely independent and can be performed in parallel without data exchange among the processors in use.

The only disadvantage consists in the fact that, for the threshold based decompositions, the number of arithmetic operations may be different for different processors, which may lead to unbalanced computations. The last block containing the separators of all blocks can be processed in a head processor if the number of the processors in use is small; otherwise, the separators should processed in parallel.

The above algorithms for the matrix ordering and for the identification of independent blocks can be used for the separators. The same approach can be employed at the next step with developing a hierarchical or nested algorithm. In the case of the threshold based decomposition, the above approaches can be applied to the Schur complement. In the case of the structural decompositions, there exist algorithms for multilevel ordering with the minimum number of fill-in elements and data exchanges.

The approaches described above belong to the class of "direct" or "explicit" decompositions when the reduction of fill-ins during the factorization is based only on the absolute values of fill-in elements. In these approaches, the structural properties are of minor importance and do not influence the dropping of elements.

An alternative approach is based on the following strategy: the structural properties are the only reason for dropping the fill-in elements to enhance the parallelism resources. For example, it is reasonable to firstly distribute the matrix rows among the processors and, before the decomposition, to drop all the elements that connect a processor with the others.

I this case the decomposition can be performed independently within each of the blocks. Any one of the structural or threshold based incomplete versions of the Cholesky decomposition can be used inside each block. The formation of such a preconditioner is said to be the non-overlapping block Jacobi method. Such a preconditioning is most simple and can be performed in parallel with no data exchange; however, the convergence of this process is not very high and is almost independent of the quality of decomposition inside each of the blocks.

Compared to the Jacobi method, the additive Schwarz method provides the decompositions of much higher quality. This method is also known as the overlapping block Jacobi method. Its essence consists in the following: the structure of each of the matrix blocks is expanded by the addition of several layers of adjacent rows. The triangular decomposition is made for the expanded matrix.

As a result, the expanded subproblem is solved on each of the processors using the data from other processors. When a subproblem is solved, the nonlocal components of the solution are dropped on each processor. Such a version of the method is known as the restricted additive Schwarz method.

The convergence of the additive Schwarz method is usually much higher than that of the Jacobi method and is monotonically increasing with an increase in the overlapping size. Although this method requires additional operations and data exchanges, its computational cost may be considerably less on parallel computers. Another version of the additive decomposition is based on the backward overlapping of the blocks in the direction of lesser numbers of rows and provides a slightly higher rate of convergence.

This version is called the block incomplete inverse Cholesky abbreviated as BIIC and is also known as the incomplete inverse triangular decomposition. Note that it can be used for nonsymmetric matrices. Within each of the blocks, this method can be combined with the second-order incomplete symmetric triangular decomposition known as the IC2 decomposition.

This combination is called the block incomplete inverse Cholesky decomposition of second order and is abbreviated as BIIC2. The idea of this method was first proposed by I. Kaporin without consideration of its parallelization. Its parallel implementation is known as the Kaporin-Konshin method. Linear systems with sparse lower upper triangular matrices are solved in a similar manner as is done in the case of dense matrices.

The forward and backward substitutions are performed using only nonzero elements on the basis of approaches for sparse matrices. Linear systems with complex lower upper triangular matrices are solved in a similar manner as is done in the case of real matrices. The arithmetic operations are performed according to the complex arithmetic rules as is done for the factorization of Hermitian matrices. The following peculiarity should be mentioned: the linear systems with block bordered triangular matrices can be solved in parallel, since, in each of the blocks, the arithmetic operations are independent of the other blocks.

Please enable JavaScript to pass antispam protection! Antispam by CleanTalk. Main authors: Igor Konshin Contents. Categories : Method level Finished articles. Navigation menu Personal tools Create account Log in. Namespaces Page Discussion. Views Read View source View history. Navigation Main page Forum Recent changes. Connect and share knowledge within a single location that is structured and easy to search. I have a problem in finding the numerical material that describing in detail for incomplete Cholesky combined with conjugate gradient method by using Matlab.

Someone can help me? Many thank in advance. I'm not really sure what the "numerical material" means but if you'd like to use the incomplete Cholesky preconditioner with conjugate gradients in MATLAB, you might consider using doc cholinc and doc pcg commands for detailed information. Sign up to join this community.

The best answers are voted up and rise to the top. Stack Overflow for Teams — Start collaborating and sharing organizational knowledge. Create a free Team Why Teams? Learn more. Incomplete Cholesky decomposition conjugate gradient method in Matlab Ask Question. Asked 7 years, 6 months ago.

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Hi Mikhail, RCI solver compute lower triangular matrix L and upper triangular matrix U that has sparse format and their multiplication is near initial matrix A. Ok, several more questions: Do I need to provide my symmetric matrix in full non-symmetric format for dcsrilu0 routine? How do the L and U matrices stored in the only output parameter bilu0 of this routine? Hi, You need to provide full matrix format if you want to obtain incomplete Cholesky decomposition for this matrix.

If you provide only lower or upper part of initial matrix as input parameter for dcsrilu0 routine you'd gotten incomplete Cholesky decomposition only for this part of matrix. About second question: L and U matrices stored in one matrix described by arrays ia, ja andbilu0.

I did all as you said, but it doesn't seem to work. At least, convergance rate is much worse than for simple Jacobi preconditioner, so I assume something went wrong. I'm not very expirienced in numerical methods, so maybe I'm missing something obvious. Mikhail, Incomplete Cholesky preconditioner could be better than Jacobi in condition number sense in several cases but could be worse than it in other.

There is not any theoretical results on quality of such preconditioner in general case. Moreover, incomplete Cholesky preconditioner is unsymmetrical so you can't use it with CG. Yes, I cannot use LU factorization, that's from where my questions did arise. Royi Novice. Could you please fix that? The Incomplete Cholesky is very important. That's sad. Is there a chance it will be introduced in the upcoming releases?

You could do it using private answer. Post Reply. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search.

I have a problem in finding the numerical material that describing in detail for incomplete Cholesky combined with conjugate gradient method by using Matlab. Someone can help me? Many thank in advance. I'm not really sure what the "numerical material" means but if you'd like to use the incomplete Cholesky preconditioner with conjugate gradients in MATLAB, you might consider using doc cholinc and doc pcg commands for detailed information.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Start collaborating and sharing organizational knowledge. Create a free Team Why Teams? Learn more. Incomplete Cholesky decomposition conjugate gradient method in Matlab Ask Question. Asked 7 years, 6 months ago. Modified 7 years, 6 months ago. Viewed times.

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