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in general there is no formula for updating the cholesky factorization of the covariance matrix when the hyperparameter is changed. There are other techniques for speeding up the computations though. For instance the method SKI (http://proceedings.mlr.press/v37/wilson15.pdf) can be used to compute matrix-vector products with the covariance matrix. Lanczos ...


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The question of selection of solver and preconditioner is difficult and to a great degree depends on how much you actually know about where the matrix comes from and what kind of operator it represents. You might benefit from going through some of the considerations I lay out in videos 34 and following here: https://www.math.colostate.edu/~bangerth/videos....


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