Recursive filter based GPU algorithms in a Data Assimilation scenario

Abstract

Data Assimilation process is generally used to estimate the best initial state of a system in order to improve accuracy of future states prediction. This powerful technique has been widely applied in investigations of the atmosphere, ocean, and land surface. In this work, we deal with the Gaussian convolution operation which is a central step of the Data Assimilation approach, as well as in several data analysis procedures. In particular, we consider the use of recursive filters to approximate the Gaussian convolution. In [1] we presented an accelerated first-order recursive filter to compute the Gaussian convolution kernel, in a very fast way. We present theory and results, and we provide a new GPU-parallel implementation which is based on the third order recursive filter. To observe the benefits in terms of performance, tests and experiments complete our work.

Introduction

Data Assimilation (DA) is a prediction-correction method for combining a physical model with observations. Assimilation techniques are used for atmospheric and ocean modeling. Main common approaches, like Kalman filters and Bayesian techniques, are based on statistical interpolation [2]. Other techniques, for example variational methods such as 3D-Var and 4D-Var, are based on minimization approaches [3], [4], [5]. In general, an almost complete overview of Data Assimilation methods can be found in [6].

Among the most important applications of DA there is the Machine Learning (ML) field [7]. Machine Learning algorithms try to provide an adequate forecast for predicting and understanding a multitude of phenomena. In other words, these two areas of investigation are at the same level, in statistical physics problems [8], [9]. In fact, many developments in DA have been used in the in ML field, producing alternative and innovative methods [10], [11].

In order to perform a correct ML, a suitable classifier is used in the analyze phase of ML process. Indeed, the variational approach of the Data Assimilation process, characterized by a cost function minimization, is a good choice for that classification. Numerically, this means to apply an iterative procedure using a covariance matrix defined by measuring the error between predictions and observed data. Here, we are interested in those numerical issues. In particular, since the error covariance matrix presents, in general, a Gaussian correlation structure, the Gaussian convolution process plays a key role in such a problem. Furthermore, it should be noted that, beyond its fundamental role in the Data Assimilation field, the convolution operation is always significant in the computational process of most big-data analysis problems.

Because of the need to process large amount of data, parallel approaches and High-Performance Computing (HPC) architectures, as multicore or Graphics Processing Units (GPUs), are mandatory [12], [13]. Moreover, parallel computing turned out to be very helpful during the modelling step of a suitable domain decomposition (DD) process of a specific algorithm [14]. Some papers deal with parallel Data Assimilation [15], [16], [17], and recently, an ad-hoc combination of parallel methods with different space reduction techniques (e.g. AutoEncoders, Truncated SVD, etc.) could be good in order to achieve appreciable results in space and time terms [18]. In this paper, we limit our attention to the basic step represented by a parallel implementation for the Gaussian convolution.

In particular, we propose an accelerated procedure to approximate the Gaussian convolution which is based on the recursive filters (RFs). In fact, Gaussian RFs have been designed to provide an accurate and efficient approximated Gaussian convolution [19], [20], [21], [22]. Since the use of RFs is mainly suitable in case of large execution time, many parallel implementations have been presented (see survey in [23]).

In [1] we presented a GPU implementation of the K-iterated first-order RF, together with a performance analysis in terms of execution times, by varying the parallel configurations and the iteration number K. Here we recall theory and results given in [1] and we also consider the third order recursive filter in order to study how much it is possible to improve the calculation speed, as the size of the problem increases.

In other words, we propose a novel parallel implementation, for the third order RF, that exploits the computational power of the GPUs, as we already did for the first-order RF in [1]. This choice of this parallelization environment is because GPUs are very useful for solving numerical problems in several application fields [24], [25] and to manage big size input data.

The rest of the paper is organized as follows. Section 2 recalls the variational Data Assimilation problem and introduces the computational kernel of Gaussian convolution. In Section 3, the use of the recursive filters, to approximate the discrete Gaussian convolution, is described together issues related to the boundary conditions. In particular, we exhibit features about the first-order recursive filter and the third order one. In Section 4 a detailed description of our GPU-parallel algorithms is provided: the domain decomposition strategy and the implementations, related to the first-order recursive filter and the third order one, are presented. The experiments shown in Section 5, allow us to make interesting considerations about the performance of the algorithms. Finally, conclusions are drawn in Section 6.