Two-level preconditioning applied on the ssSNPBLUP model

Two-level preconditioning applied on the ssSNPBLUP model

Nguyen, Buu-Van (TU Delft Electrical Engineering, Mathematics and Computer Science)

Vuik, Cornelis (mentor)
Vandenplas, Jérémie (mentor)
Lin, H.X. (graduation committee)

Delft University of Technology

Applied Mathematics

2021-07-19

The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) model can potentially be used in animal breeding for genetic evaluations. It has been reported that this model has convergence issues when the Preconditioned Conjugate Gradient (PCG) method is applied. This is due to the linear system being ill-conditioned. Moreover, in a recent research a subspace decomposition deflation method has been proposed. Unfortunately, the method requires too many deflation vectors, which results in an implementation of the Deflated PCG (DPCG) algorithm that requires a high demand of RAM. In this thesis a different subdomain decomposition strategy is proposed. This subdomain decomposition method utilises the $k$-means algorithm applied on the matrix of correlation among the single nucleotide polymorphisms (SNPs). This method has been applied on a simulated data set of dairy cattle. This method results in halving the number of deflation vectors required for the same rate of convergence compared with the previous subdomain decomposition deflation method.

In practice for genetic evaluations the data sets increases over time by incorporating new information. In this thesis a specific initial solution has been investigated. This initial solution closely resembles the solution of the updated system to solve the ssSNPBLUP model efficiently. Also a deflation method that utilises the previous solutions of genetic evaluations can be applied. The Proper Orthogonal Decomposition (POD) based deflation method is proposed in this thesis to full fill this requirement. Moreover, only a few deflation vectors are needed to improve the rate of convergence.

Furthermore, the subdomain decomposition deflation method deflates the largest eigenvalues of the preconditioned coefficient matrix and the POD deflation method deflates the smallest eigenvalues of the preconditioned matrix. Combining the deflation vectors of both methods results in an improvement in the rate of convergence compared with the methods on their own.

Based on the results the best performing method is a combination of selecting the initial solution as the previous genetic evaluation, the POD deflation method and the k-means++ clustering applied on the subdomain decomposition deflation method. In this research a reduction of 81% in iteration time and 19% in total computation time has been observed. Note that the initial solution as the previous genetic evaluation approach can only be employed if that solution is available. Likewise, the POD method can only be applied if multiple solutions of genetic evaluations are available.

ssSNPBLUP
Conjugate Gradient
CG
Preconditioned Conjugate Gradient
PCG
Deflated Preconditioned Conjugate Gradient
DPCG
Subdomain Decomposition
k-means
k-means clustering
Proper Orthogonal Decomposition
POD
Genomics
Iterative solvers
Simulated data
Big Data

http://resolver.tudelft.nl/uuid:ea4f1db4-47fa-461c-a200-bf0f1b5ae7ad

Student theses

master thesis

© 2021 Buu-Van Nguyen