With the advent of massive data sets much of the computational science and engineering community has moved toward data-intensive approaches in regression and classification. However, these present significant challenges due to increasing size, complexity and dimensionality of the problems. In particular, covariance matrices in many cases are numerically unstable and linear algebra shows that often such matrices cannot be inverted accurately on a finite precision computer. A common ad hoc approach to stabilizing a matrix is application of a so-called nugget. However, this can change the model and introduce error to the original solution. It is well known from numerical analysis that ill-conditioned matrices cannot be accurately inverted. In this paper we develop a multilevel computational method that scales well with the number of observations and dimensions. A multilevel basis is constructed adapted to a kD-tree partitioning of the observations. Numerically unstable covariance matrices with large condition numbers can be transformed into well conditioned multilevel ones without compromising accuracy. Moreover, it is shown that the multilevel prediction exactly solves the Best Linear Unbiased Predictor (BLUP) and Generalized Least Squares (GLS) model, but is numerically stable. The multilevel method is tested on numerically unstable problems of up to 25 dimensions. Numerical results show speedups of up to 42,050 times for solving the BLUP problem, but with the same accuracy as the traditional iterative approach. For very ill-conditioned cases the speedup is infinite. In addition, decay estimates of the multilevel covariance matrices are derived based on high dimensional interpolation techniques from the field of numerical analysis. This work lies at the intersection of statistics, uncertainty quantification, high performance computing and computational applied mathematics.

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