Parallel-in-Time Kalman Smoothing Using Orthogonal Transformations

2025-02-17

AI Summary

Linear Kalman smoothers suffer from inherent sequentiality along the time dimension, severely limiting parallel scalability for large-scale temporal state estimation. To address this, this paper proposes a novel, numerically stable, and fully time-parallel Kalman smoother. Our method introduces three key innovations: (1) a highly parallel QR decomposition algorithm tailored for structured sparse triangular matrices arising in smoothing; (2) a selective covariance inversion (SelInv) scheme specifically adapted to the smoothing problem’s sparsity structure; and (3) a fine-grained, TBB-based multithreaded implementation. Evaluated on 64-core Intel and ARM servers, our smoother achieves up to 47× speedup over serial execution—significantly outperforming Särkkä et al.’s 2021 approach—and demonstrates superior strong scaling. Crucially, single-thread performance degrades by only 1.8–2.5×, preserving numerical robustness while attaining high parallel efficiency.

Technology Category

• Intelligent Robots: State Estimation
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We present a numerically-stable parallel-in-time linear Kalman smoother. The smoother uses a novel highly-parallel QR factorization for a class of structured sparse matrices for state estimation, and an adaptation of the SelInv selective-inversion algorithm to evaluate the covariance matrices of estimated states. Our implementation of the new algorithm, using the Threading Building Blocks (TBB) library, scales well on both Intel and ARM multi-core servers, achieving speedups of up to 47x on 64 cores. The algorithm performs more arithmetic than sequential smoothers; consequently it is 1.8x to 2.5x slower on a single core. The new algorithm is faster and scales better than the parallel Kalman smoother proposed by S""arkk""a and Garc'{i}a-Fern'andez in 2021.

Problem

• Parallel-in-time Kalman smoother
• Structured sparse matrices
• Multi-core server scalability

Innovation

• Parallel-in-time Kalman smoother
• Highly-parallel QR factorization
• SelInv selective-inversion algorithm