1,194 research outputs found
Reconstructing Kernel-based Machine Learning Force Fields with Super-linear Convergence
Kernel machines have sustained continuous progress in the field of quantum
chemistry. In particular, they have proven to be successful in the low-data
regime of force field reconstruction. This is because many physical invariances
and symmetries can be incorporated into the kernel function to compensate for
much larger datasets. So far, the scalability of this approach has however been
hindered by its cubical runtime in the number of training points. While it is
known, that iterative Krylov subspace solvers can overcome these burdens, they
crucially rely on effective preconditioners, which are elusive in practice.
Practical preconditioners need to be computationally efficient and numerically
robust at the same time. Here, we consider the broad class of Nystr\"om-type
methods to construct preconditioners based on successively more sophisticated
low-rank approximations of the original kernel matrix, each of which provides a
different set of computational trade-offs. All considered methods estimate the
relevant subspace spanned by the kernel matrix columns using different
strategies to identify a representative set of inducing points. Our
comprehensive study covers the full spectrum of approaches, starting from naive
random sampling to leverage score estimates and incomplete Cholesky
factorizations, up to exact SVD decompositions.Comment: 18 pages, 12 figures, preprin
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
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