Deflation techniques applied on mixed model equations

In this paper, we consider a block Jacobi preconditioner and various deflation techniques applied in the Deflated Preconditioned Conjugate Gradient (DPCG) method for solving a sparse system of linear equations derived from a statistical linear mixed model that analyses simultaneously phenotypic and pedigree information of genotyped and ungenotyped animals with Single Polymorphism Nucleotide genotypes of genotyped animals. In livestock production systems, evaluating the genetic merit of the animals through such a model is a key process to ensure an improvement of animals for some characteristics of interest at each generation. First, we propose to define the deflation vectors using a subdomain deflation approach that considers some biological properties of the genotypes. Using simulated data, this approach reduces the number of iterations by up to 87% in comparison to a Preconditioned Conjugate Gradient method with a Jacobi preconditioner. Furthermore, compared to a DPCG method with same number of subdomains but defined randomly, this approach reduces the number of iterations by up to 20% for the same computational costs of one DPCG iteration. The properties of the resulting systems show that this approach annihilates the largest eigenvalues of the preconditioned coefficient matrix. Second, we propose the use of solution vectors of 12 systems of equations that include between 0.25% and 3% less data, as deflation vectors. For reducing the computational costs, we also consider a Proper Orthogonal Decomposition-reduced set of these 12 vectors. The properties of the resulting systems show that this recycling information approach annihilates the smallest eigenvalues of the preconditioned coefficient matrix, and results in a reduction of up to 39% in comparison to the PCG method. Finally, based on our experiment, the combination of the subdomain deflation approach relying on biological properties and of the POD-based approach to recycle previous solution vectors, for defining the deflation vectors, results in annihilating both the smallest and largest eigenvalues, and in a reduction of up to 88 % of the number of iterations in comparison to the PCG method.