Extension of the Lanczos algorithm for simultaneous computation of multiple targeted singular vector sets

• Published in: Quarterly journal of the Royal Meteorological Society Vol. 146; no. 726; pp. 454 - 467
• Main Author: Ono, Kosuke
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.01.2020; Wiley Subscription Services, Inc
• Subjects: Algorithms; Computation; Computer applications; ensemble prediction; Growth rate; Lanczos algorithm; Procedures; singular vector
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.3686

This article discusses an extension of the singular vector (SV) method in the context of an initial perturbation generator for an ensemble prediction system (EPS). In general, multiple SVs targeted at different regions are computed in operational EPSs to extract growing modes with focus on different parts of the EPS domain. However, significant computational cost is associated with running all the procedures of the SV computations multiple times. In this study, the Lanczos algorithm used for SV computation was extended to allow simultaneous computation of multiple targeted SV sets. Algebraic calculations in the algorithm, such as orthonormalizations and eigenvalue problem resolution, are separately implemented for multiple “subdomains” incorporating different targeting areas, and SV sets are computed for individual subdomains. However, forward and backward linearized propagation runs through the whole domain, shared among the SV sets from all the subdomains. As such algebraic operation accounts for a relatively small part of all computation, the computational cost increment brought by the algorithm extension is also small in relation to single SV computations. For verifications, consistency between SVs produced with the original and extended algorithms was examined. Both SV sets spanned the same subspaces with similar linear growth rates, except those derived on subdomain boundaries, where SVs produced using the extended algorithm were truncated by the boundary. To avoid such truncation, it is necessary to set a subdomain large enough to cover the target area and its surrounding region. In this article, some applications of this algorithm in operational situations are suggested. Also, an application of subdomain to the wave‐number space is described. Mixing ratio of water vapour (g · kg−1) and wind (arrows with half‐barbs representing 5 knots, full barbs 10 knots and triangles 50 knots) at 950 hPa, and sea‐level pressure (contour interval: 2 hPa) from the nonlinear trajectory for the TLM and ADM. (a) At T + 0 (1800 UTC 17 September 2017), (b) at T + 6 (0000 UTC 18 September 2017).