The high-dimensional space of human diseases built from diagnosis records and mapped to genetic loci.

Human diseases are traditionally studied as singular, independent entities, limiting researchers' capacity to view human illnesses as dependent states in a complex, homeostatic system. Here, using time-stamped clinical records of over 151 million unique Americans, we construct a disease representation as points in a continuous, high-dimensional space, where diseases with similar etiology and manifestations lie near one another. We use the UK Biobank cohort, with half a million participants, to perform a genome-wide association study of newly defined human quantitative traits reflecting individuals' health states, corresponding to patient positions in our disease space. We discover 116 genetic associations involving 108 genetic loci and then use ten disease constellations resulting from clustering analysis of diseases in the embedding space, as well as 30 common diseases, to demonstrate that these genetic associations can be used to robustly predict various morbidities.

HLIBCov: Parallel hierarchical matrix approximation of large covariance matrices and likelihoods with applications in parameter identification.

We provide more technical details about the HLIBCov package, which is using parallel hierarchical (H-) matrices to: •Approximate large dense inhomogeneous covariance matrices with a log-linear computational cost and storage requirement.•Compute matrix-vector product, Cholesky factorization and inverse with a log-linear complexity.•Identify unknown parameters of the covariance function (variance, smoothness, and covariance length). These unknown parameters are estimated by maximizing the joint Gaussian log-likelihood function. To demonstrate the numerical performance, we identify three unknown parameters in an example with 2,000,000 locations on a PC-desktop.

Reconstructing parameters of the FitzHugh-Nagumo system from boundary potential measurements.

We consider distributed parameter identification problems for the FitzHugh-Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton-Krylov type that combines Newton's method for numerical optimization with Krylov subspace solvers for the resulting Karush-Kuhn-Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions.