On POD-based Deflation Vectors for DPCG applied to porous media problems

Cortes, G.B. Diaz; Vuik, C.

doi:10.1016/j.cam.2017.06.032

Cited by 11 publications

(17 citation statements)

References 23 publications

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“…Deflation is used to annihilate the effect of the most unfavourable eigenvalues of the spectrum of on the convergence of the PCG method by setting these unfavourable eigenvalues to 0 [ 20 ]. The deflation is performed by introducing a second-level preconditioner, , also called deflation matrix, into the preconditioned linear system of equations, , as follows [ 19 – 21 ]: where is the vector of deflated solutions and is related to the vector of solutions of the system of equations as ; the deflation matrix is equal to ; the matrix is the deflation-subspace matrix of rank that contains columns, called deflation vectors; and the matrix is a symmetric positive definite matrix, called Galerkin or coarse matrix [ 28 ], which can be easily computed and inverted, or factored if it is too large.…”

Section: Methodsmentioning

confidence: 99%

“…The deflation vectors of the deflation-subspace matrix can be defined following several techniques based on, e.g., approximating eigenvectors [ 29 ], recycling information of previous Krylov subspaces [ 21 ], or subdomain deflation vectors [ 22 ]. All these approaches have their own advantages and disadvantages [ 20 , 28 ]. For example, some advantages of the subdomain deflation approach are that is sparse, and that additional computations for the DPCG method (in comparison to the PCG method) can be implemented efficiently [ 22 ].…”

Section: Methodsmentioning

confidence: 99%

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Deflated preconditioned conjugate gradient method for solving single-step BLUP models efficiently

et al. 2018

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BackgroundThe single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) method, such as single-step genomic BLUP (ssGBLUP), simultaneously analyses phenotypic, pedigree, and genomic information of genotyped and non-genotyped animals. In contrast to ssGBLUP, SNP effects are fitted explicitly as random effects in the ssSNPBLUP model. Similarly, principal components associated with the genomic information can be fitted explicitly as random effects in a single-step principal component BLUP (ssPCBLUP) model to remove noise in genomic information. Single-step genomic BLUP is solved efficiently by using the preconditioned conjugate gradient (PCG) method. Unfortunately, convergence issues have been reported when solving ssSNPBLUP by using PCG. Poor convergence may be linked with poor spectral condition numbers of the preconditioned coefficient matrices of ssSNPBLUP. These condition numbers, and thus convergence, could be improved through the deflated PCG (DPCG) method, which is a two-level PCG method for ill-conditioned linear systems. Therefore, the first aim of this study was to compare the properties of the preconditioned coefficient matrices of ssGBLUP and ssSNPBLUP, and to document convergence patterns that are obtained with the PCG method. The second aim was to implement and test the efficiency of a DPCG method for solving ssSNPBLUP and ssPCBLUP.ResultsFor two dairy cattle datasets, the smallest eigenvalues obtained for ssSNPBLUP (ssPCBLUP) and ssGBLUP, both solved with the PCG method, were similar. However, the largest eigenvalues obtained for ssSNPBLUP and ssPCBLUP were larger than those for ssGBLUP, which resulted in larger condition numbers and in slow convergence for both systems solved by the PCG method. Different implementations of the DPCG method led to smaller condition numbers, and faster convergence for ssSNPBLUP and for ssPCBLUP, by deflating the largest unfavourable eigenvalues.ConclusionsPoor convergence of ssSNPBLUP and ssPCBLUP when solved by the PCG method are related to larger eigenvalues and larger condition numbers in comparison to ssGBLUP. These convergence issues were solved with a DPCG method that annihilates the effect of the largest unfavourable eigenvalues of the preconditioned coefficient matrix of ssSNPBLUP and of ssPCBLUP on the convergence of the PCG method. It resulted in a convergence pattern, at least, similar to that of ssGBLUP.Electronic supplementary materialThe online version of this article (10.1186/s12711-018-0429-3) contains supplementary material, which is available to authorized users.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Deflated preconditioned conjugate gradient method for solving single-step BLUP models efficiently

et al. 2018

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“…On the other side, in [5], Graham and Hagger applied it to the development of additive Schwarz domain decomposition preconditioners on unstructured meshes for such cases. More recent contributions incorporate the use of deflation techniques [6,7,8], model order reduction [2,8] and multigrid preconditioning [9].…”

Section: Introductionmentioning

confidence: 99%

On Preconditioning Variable Poisson Equation with Extreme Contrasts in the Coefficients

Alsalti-Baldellou¹,

Trias²,

Gorobets³

et al. 2021

14th WCCM-ECCOMAS Congress

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It is well known that the solution by means of iterative methods of very ill-conditioned systems leads to very poor convergence rates. In this context, preconditioning becomes crucial in order to modify the spectrum of the system being solved and improve the performance of the solvers. A proper balance between the reduction in the number of iterations and the overhead of the construction and application of the preconditioner needs to be sought to actually decrease the total execution time of the solvers. This is particularly important when considering variable coefficients matrices as, in general, its preconditioners will also be variable and need to be updated regularly at an affordable cost. In this work we present a family of variable preconditioners designed for the effective solution of variable Poisson equation with extreme contrasts in the coefficients, which represents a particularly challenging case as it translates into a variable and extremely ill-conditioned system arising in many situations such as with multiphase flows presenting high density ratios or in the presence of highly-stretched adaptive mesh refinements. Finally, the results of the numerical experiments performed are presented and discussed, confirming our preconditioners as extremely affordable, highly-parallelizable and easy-to-implement alternatives to the more standard (and usually unfeasible) preconditioners, still showing great improvements in the rate of convergence of the solvers without requiring the variable coefficients matrix to be explicitly rebuilt at each iteration.

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“…the same well configurations but different flow rates or bottom hole pressures (bhp) [35,40,48]. The basis can also be computed on-the-fly, using, e.g., the solution of the latest time steps [16,40,42]. With this approach, the basis has to be adapted during the simulation.…”

Section: Introductionmentioning

confidence: 99%

“…Another approach was developed by Pasetto et al [10], who suggested constructing a preconditioner for the CG method, based on a POD basis for the solution of groundwater flow models. The use of the POD basis within a deflation procedure to accelerate the CG method was introduced by Diaz Cortes et al [16] for single-phase flow simulation problems.…”

Section: Introductionmentioning

confidence: 99%

Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods

Cortes

Vuik

2021

Comput Geosci

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We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

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On POD-based Deflation Vectors for DPCG applied to porous media problems

Cited by 11 publications

References 23 publications

Deflated preconditioned conjugate gradient method for solving single-step BLUP models efficiently

Deflated preconditioned conjugate gradient method for solving single-step BLUP models efficiently

On Preconditioning Variable Poisson Equation with Extreme Contrasts in the Coefficients

Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods

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