Improving computational efficiency in large linear inverse problems: an example from carbon dioxide flux estimation V. Yadav and A. M. MichalakDepartment of Global Ecology, Carnegie Institution for Science, Stanford, California, 94305, USA

Received: 07 Sep 2012 – Published in Geosci. Model Dev. Discuss.: 19 Oct 2012

Revised: 07 Mar 2013 – Accepted: 26 Mar 2013 – Published: 03 May 2013

Abstract. Addressing a variety of questions within Earth science disciplines entails
the inference of the spatiotemporal distribution of parameters of interest
based on observations of related quantities. Such estimation problems often
represent inverse problems that are formulated as linear optimization
problems. Computational limitations arise when the number of observations
and/or the size of the discretized state space becomes large, especially if
the inverse problem is formulated in a probabilistic framework and therefore
aims to assess the uncertainty associated with the estimates. This work
proposes two approaches to lower the computational costs and memory
requirements for large linear space–time inverse problems, taking the
Bayesian approach for estimating carbon dioxide (CO_{2}) emissions and uptake
(a.k.a. fluxes) as a prototypical example. The first algorithm can be used
to efficiently multiply two matrices, as long as one can be expressed as a
Kronecker product of two smaller matrices, a condition that is typical when
multiplying a sensitivity matrix by a covariance matrix in the solution of
inverse problems. The second algorithm can be used to compute a posteriori uncertainties
directly at aggregated spatiotemporal scales, which are the scales of most
interest in many inverse problems. Both algorithms have significantly lower
memory requirements and computational complexity relative to direct
computation of the same quantities (O(*n*^{2.5}) vs. O(*n*^{3})). For an
examined benchmark problem, the two algorithms yielded massive savings in
floating point operations relative to direct computation of the same
quantities. Sample computer codes are provided for assessing the
computational and memory efficiency of the proposed algorithms for matrices
of different dimensions.

**Citation:** Yadav, V. and Michalak, A. M.: Improving computational efficiency in large linear inverse problems: an example from carbon dioxide flux estimation, Geosci. Model Dev., 6, 583-590, doi:10.5194/gmd-6-583-2013, 2013.