Geomechanical simulations can give essential insights into subsurface processes, but typically require solving large, ill-conditioned linear systems. An important method for solving these linear systems is the Conjugate Gradient method, but applying this method to ill-conditioned matrices can result in slow convergence. To improve the convergence of the Conjugate Gradient method, the iterative solver is preconditioned using the Algebraic Multigrid method. In Algebraic Multigrid methods, a hierarchy of matrices of different sizes is derived. When applying these methods as a preconditioner to the Conjugate Gradient method, on each level of the multigrid hierarchy fast convergence is observed in particular components of the residual. This leads to much fewer iterations being required in the Conjugate Gradient method, at the cost of the iterations being computationally more expensive. These Algebraic Multigrid methods do require a more problem-specific setup configuration than more simple preconditioners like the Jacobi preconditioner. In this research, the Conjugate Gradient method preconditioned with various Algebraic Multigrid methods is studied and compared with the Jacobi preconditioned Conjugate Gradient method. For this, the Conjugate Gradient method, preconditioned with both the Jacobi and Algebraic Multigrid-based methods, is applied to linear problems derived from geomechanical simulations. Using Algebraic Multigrid preconditioners can reduce the number of iterations required for convergence of the Conjugate Gradient method by a factor of 80. While a single iteration with an Algebraic Multigrid preconditioner is more time-expensive than an iteration with a Jacobi preconditioner, significant reductions, of up to five times, are observed in the runtimes of the linear solver. This comes at the cost of a higher peak memory requirement in the application of the linear solver. The vast reduction of the runtime of the linear solvers makes the studied geomechanical simulations significantly faster. This makes the Algebraic Multigrid preconditioners a valuable addition to these simulations.
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In agricultural studies it is often important to predict the performance of genetically different plants. To make sure predictions are done well, it is necessary to make sure they are not influenced by effects of the field on which they are planted. These field effects or spatial effects are in practice often quite complicated and can be due to a wide variety of reasons. To get a better view of these field effects a good mathematical model is desired. In this paper a model is presented which helps to find these field effects. This model tries to estimate the field effect by comparing data of the same plant on different positions of the field. Data is obtained in a finite amount of positions, which means that the model finds the field effect in a finite amount of positions as well. This field effect is found using a cross-validation technique obtained from Tikhonov regularization. The field effect in a finite amount of positions is extended to a field effect in every position of the field. To do this in a good way a kernel method is used, the advantage of which is that it does not depend on a mesh. This kernel method is here applied with a kernel function that is based on Gaussian distribution. This model is applied to several fields of crops to get a view of the performance of the model on real data. ... Read more