Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Zur Konvergenz der Randpunktmethode

Das von Maz'ya eingeführte Approximationsverfahren, die Methode der näherungsweisen Näherungen (Approximate Approximations), kann auch zur numerischen Lösung von Randintegralgleichungen verwendet werden (Randpunktmethode). In diesem Fall hängen die Komponenten der Matrix des resultierenden Gleichungssystems zur Berechnung der Näherung für die Dichte nur von der Position der Randpunkte und der Richtung der äußeren Einheitsnormalen in diesen Punkten ab. Dieses numerisches Verfahren wird am Beispiel des Dirichlet Problems für die Laplace Gleichung und die Stokes Gleichungen in einem beschränkten zweidimensionalem Gebiet untersucht. Die Randpunktmethode umfasst drei Schritte: Im ersten Schritt wird die unbekannte Dichte durch eine Linearkombination von radialen, exponentiell abklingenden Basisfunktionen approximiert. Im zweiten Schritt wird die Integration über den Rand durch die Integration über die Tangenten in Randpunkten ersetzt. Für die auftretende Näherungspotentiale können sogar analytische Ausdrücke gewonnen werden. Im dritten Schritt wird das lineare Gleichungssystem gelöst, und eine Näherung für die unbekannte Dichte und damit auch für die Lösung der Randwertaufgabe konstruiert. Die Konvergenz dieses Verfahrens wird für glatte konvexe Gebiete nachgewiesen.

Enlarging a training set for genomic selction by imputation of un-genotyped animals in populations of varying genetic architecture

Background: The most common application of imputation is to infer genotypes of a high-density panel of markers on animals that are genotyped for a low-density panel. However, the increase in accuracy of genomic predictions resulting from an increase in the number of markers tends to reach a plateau beyond a certain density. Another application of imputation is to increase the size of the training set with un-genotyped animals. This strategy can be particularly successful when a set of closely related individuals are genotyped. ----- Methods: Imputation on completely un-genotyped dams was performed using known genotypes from the sire of each dam, one offspring and the offspring’s sire. Two methods were applied based on either allele or haplotype frequencies to infer genotypes at ambiguous loci. Results of these methods and of two available software packages were compared. Quality of imputation under different population structures was assessed. The impact of using imputed dams to enlarge training sets on the accuracy of genomic predictions was evaluated for different populations, heritabilities and sizes of training sets. ----- Results: Imputation accuracy ranged from 0.52 to 0.93 depending on the population structure and the method used. The method that used allele frequencies performed better than the method based on haplotype frequencies. Accuracy of imputation was higher for populations with higher levels of linkage disequilibrium and with larger proportions of markers with more extreme allele frequencies. Inclusion of imputed dams in the training set increased the accuracy of genomic predictions. Gains in accuracy ranged from close to zero to 37.14%, depending on the simulated scenario. Generally, the larger the accuracy already obtained with the genotyped training set, the lower the increase in accuracy achieved by adding imputed dams. ----- Conclusions: Whenever a reference population resembling the family configuration considered here is available, imputation can be used to achieve an extra increase in accuracy of genomic predictions by enlarging the training set with completely un-genotyped dams. This strategy was shown to be particularly useful for populations with lower levels of linkage disequilibrium, for genomic selection on traits with low heritability, and for species or breeds for which the size of the reference population is limited.