## Section: New Results

### MIcrowave Data Analysis for petaScale computers

Participants : Laura Grigori, Mikolaj Szydlarski, Meisam Shariffy.

Generalized least square problems with non-diagonal weights arise frequently in an estimation of two dimensional images from data of cosmological as well as astro- or geo- physical observations. As the observational data sets keep growing at Moore's rate, with their volumes exceeding tens and hundreds billions of samples, the need for fast and efficiently parallelizable iterative solvers is generally recognized.

In this work [36] we propose a new iterative algorithm for solving generalized least square systems with weights given by a block-diagonal matrix with Toeplitz blocks. Such cases are physically well motivated and correspond to measurement noise being piece-wise stationary – a common occurrence in many actual observations. Our iterative algorithm is based on the conjugate gradient method and includes a parallel two-level preconditioner (2lvl-PCG) constructed from a limited number of sparse vectors estimated from the coefficients of the initial linear system.

Our prototypical application is the map-making problem in the Cosmic Microwave Background data analysis. We show experimentally that our parallel implementation of 2lvl-PCG outperforms by a factor of up to 6 the standard one-level PCG in terms of both the convergence rate and the time to solution on up to $12,228$ cores of NERSC's Cray XE6 (Hopper) system displaying nearly perfect strong and weak scaling behavior in this regime.