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On The Performance of Prefix-Sum Parallel Kalman Filters and Smoothers on GPUs

Simo Särkkä, Ángel F. García-Fernández

TL;DR

This work empirically evaluates temporally parallel Kalman filtering and smoothing on GPUs using all-prefix-sum (scan) algorithms, comparing multiple prefix-sum strategies and introducing a novel parallel two-filter smoother (PTFS). It demonstrates that prefix-sum choice strongly influences practical performance, with Blelloch- and Ladner-Fischer-based scans yielding favorable speedups, and that a two-GPU PTFS can outperform single-GPU counterparts. The authors provide Julia-based Metal and CUDA implementations and show GPU speedups up to about 750× (Metal) and 500× (CUDA) relative to sequential Kalman filtering. The results validate that GPU-accelerated, temporally parallel Bayesian estimation can significantly accelerate real-time state estimation tasks while remaining close to theoretical operation counts.

Abstract

This paper presents an experimental evaluation of parallel-in-time Kalman filters and smoothers using graphics processing units (GPUs). In particular, the paper evaluates different all-prefix-sum algorithms, that is, parallel scan algorithms for temporal parallelization of Kalman filters and smoothers in two ways: by calculating the required number of operations via simulation, and by measuring the actual run time of the algorithms on real GPU hardware. In addition, a novel parallel-in-time two-filter smoother is proposed and experimentally evaluated. Julia code for Metal and CUDA implementations of all the algorithms is made publicly available.

On The Performance of Prefix-Sum Parallel Kalman Filters and Smoothers on GPUs

TL;DR

This work empirically evaluates temporally parallel Kalman filtering and smoothing on GPUs using all-prefix-sum (scan) algorithms, comparing multiple prefix-sum strategies and introducing a novel parallel two-filter smoother (PTFS). It demonstrates that prefix-sum choice strongly influences practical performance, with Blelloch- and Ladner-Fischer-based scans yielding favorable speedups, and that a two-GPU PTFS can outperform single-GPU counterparts. The authors provide Julia-based Metal and CUDA implementations and show GPU speedups up to about 750× (Metal) and 500× (CUDA) relative to sequential Kalman filtering. The results validate that GPU-accelerated, temporally parallel Bayesian estimation can significantly accelerate real-time state estimation tasks while remaining close to theoretical operation counts.

Abstract

This paper presents an experimental evaluation of parallel-in-time Kalman filters and smoothers using graphics processing units (GPUs). In particular, the paper evaluates different all-prefix-sum algorithms, that is, parallel scan algorithms for temporal parallelization of Kalman filters and smoothers in two ways: by calculating the required number of operations via simulation, and by measuring the actual run time of the algorithms on real GPU hardware. In addition, a novel parallel-in-time two-filter smoother is proposed and experimentally evaluated. Julia code for Metal and CUDA implementations of all the algorithms is made publicly available.

Paper Structure

This paper contains 35 sections, 2 theorems, 39 equations, 15 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Parallel all-prefix-sum algorithms
  3. Complexity measures of parallel programs
  4. Parallel computation of all-prefix-sums
  5. Hillis--Steele algorithm
  6. Blelloch's algorithm
  7. Ladner's and Fischer's circuit
  8. Hybrid algorithms of Sengupta et al.
  9. Other algorithms
  10. Sequential Bayesian and Kalman filtering and smoothing
  11. Bayesian filtering
  12. Kalman filtering
  13. Smoothing
  14. Forward-backward smoother
  15. Two-filter smoother
  16. ...and 20 more sections

Key Result

Lemma 3

Given two elements $\left(A_{i},b_{i},C_{i},\eta_{i},J_{i}\right)$ and $\left(A_{j},b_{j},C_{j},\eta_{j},J_{j}\right)$ the binary operator $\otimes$ for filtering returns an element $\left(A_{i,j},b_{i,j},C_{i,j},\eta_{i,j},J_{i,j}\right)$ where

Figures (15)

  • Figure 1: Illustration of the operation of the algorithm of Hillis and Steele (Alg. \ref{['alg:hillis-steele']}) Hillis:1986 with 16 elements. Each column is a data storage element with $\otimes$ being the associative operator between two elements. The arrows indicate increasing time. At the final time step, the data elements contain the prefix sums.
  • Figure 2: Illustration of the operation of Blelloch's algorithm (Alg. \ref{['alg:blelloch']}) Blelloch:1990 with 16 elements.
  • Figure 3: Illustration of the operation of the in-place Ladner and Fischer's algorithm (Alg. \ref{['alg:inplace-lafi']}) Ladner:1980 with 16 elements.
  • Figure 4: Sketch of Julia implementation of generic up-sweep GPU kernel of a parallel associative scan.
  • Figure 5: Functions to compute the index and stride on Metal and CUDA.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Lemma 4