O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing

A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument is an element of [-1, 1]. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is O(1). The proposed algorithm also immediately yields an O(1) algorithm for computing an arbitrary Gauss-Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss-Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the O(1) complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss-Legendre quadrature rules

Scalable parallel computation of the translation operator in three dimensions

We propose a novel algorithm for the parallel, distributed-memory computation of the translation operator in the three-dimensional multilevel fast multipole algorithm (MLFMA). Sequential algorithms can compute the translation operator with L multipoles and O(L-2) sampling points in O(L-2) time. State-of-the-art hierarchical parallelization schemes of the MLFMA rely on the distribution of radiation patterns and associated translation operators among P = O(L-2) parallel processes, necessitating the development of distributed-memory algorithms for the computation of the translation operator. Whereas a baseline parallel algorithm computes this translation operator in O(L) time, we propose an algorithm that achieves this in only O(log L) time. For large translation operators and a high number of parallel processes, our algorithm proves to be roughly ten times faster than the baseline algorithm

Weak scalability analysis of the distributed-memory parallel MLFMA

Distributed-memory parallelization of the multilevel fast multipole algorithm (MLFMA) relies on the partitioning of the internal data structures of the MLFMA among the local memories of networked machines. For three existing data partitioning schemes (spatial, hybrid and hierarchical partitioning), the weak scalability, i.e., the asymptotic behavior for proportionally increasing problem size and number of parallel processes, is analyzed. It is demonstrated that none of these schemes are weakly scalable. A nontrivial change to the hierarchical scheme is proposed, yielding a parallel MLFMA that does exhibit weak scalability. It is shown that, even for modest problem sizes and a modest number of parallel processes, the memory requirements of the proposed scheme are already significantly lower, compared to existing schemes. Additionally, the proposed scheme is used to perform full-wave simulations of a canonical example, where the number of unknowns and CPU cores are proportionally increased up to more than 200 millions of unknowns and 1024 CPU cores. The time per matrix-vector multiplication for an increasing number of unknowns and CPU cores corresponds very well to the theoretical time complexity

Performing large full-wave simulations by means of a parallel MLFMA implementation

In this paper large full-wave simulations are performed using a parallel Multilevel Fast Multipole Algorithm (MLFMA) implementation. The data structures of the MLFMA-tree are partitioned according to the so-called hierarchical partitioning scheme, while the radiation patterns are partitioned in a blockwise way. To test the implementation of the algorithm, a full-wave simulation of a canonical example with more than 50 millions of unknowns has been performed

A century of tree line changes in sub-Arctic Sweden shows local and regional variability and only a minor influence of 20th century climate warming

Aim Models project that climate warming will cause the tree line to move to higher elevations in alpine areas and more northerly latitudes in Arctic environments. We aimed to document changes or stability of the tree line in a sub-Arctic model area at different temporal and spatial scales, and particularly to clarify the ambiguity that currently exists about tree line dynamics and their causes. Location The study was conducted in the Tornetrask area in northern Sweden where climate warmed by 2.5 degrees C between 1913 and 2006. Mountain birch (Betula pubescens ssp. czerepanovii) sets the alpine tree line. Methods We used repeat photography, dendrochronological analysis, field observations along elevational transects and historical documents to study tree line dynamics. Results Since 1912, only four out of eight tree line sites had advanced: on average the tree line had shifted 24 m upslope (+0.2 m year-1 assuming linear shifts). Maximum tree line advance was +145 m (+1.5 m year-1 in elevation and +2.7 m year-1 in actual distance), whereas maximum retreat was 120 m downslope. Counter-intuitively, tree line advance was most pronounced during the cooler late 1960s and 1970s. Tree establishment and tree line advance were significantly correlated with periods of low reindeer (Rangifer tarandus) population numbers. A decreased anthropozoogenic impact since the early 20th century was found to be the main factor shaping the current tree line ecotone and its dynamics. In addition, episodic disturbances by moth outbreaks and geomorphological processes resulted in descent and long-term stability of the tree line position, respectively. Main conclusions In contrast to what is generally stated in the literature, this study shows that in a period of climate warming, disturbance may not only determine when tree line advance will occur but if tree line advance will occur at all. In the case of non-climatic climax tree lines, such as those in our study area, both climate-driven model projections of future tree line positions and the use of the tree line position for bioclimatic monitoring should be used with caution