Serinv: A Scalable Library for the Selected Inversion of Block-Tridiagonal with Arrowhead Matrices

2025-03-21

AI Summary

Selective inversion—computing only specific entries of the inverse—of large-scale symmetric positive definite block tridiagonal arrowhead matrices arises in climate modeling and materials science, but existing CPU-based shared-memory solvers face severe scalability limitations. Method: We present the first scalable, distributed GPU-accelerated library for selective inversion and Cholesky factorization, built on MPI and GPU parallelism. Our approach integrates customized block Cholesky decomposition, efficient Schur complement updates, and optimized task scheduling to exploit hierarchical memory and inter-GPU communication. Contribution/Results: On 16 GPUs, our solver achieves up to two orders-of-magnitude speedup over PARDISO and MUMPS. It attains strong and weak scaling efficiencies of 32.3% and 47.2%, respectively, significantly enhancing the feasibility and performance of large-scale sparse matrix inverse computations.

Technology Category

• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Blockchains and distributed ledgers

Abstract

The inversion of structured sparse matrices is a key but computationally and memory-intensive operation in many scientific applications. There are cases, however, where only particular entries of the full inverse are required. This has motivated the development of so-called selected-inversion algorithms, capable of computing only specific elements of the full inverse. Currently, most of them are either shared-memory codes or limited to CPU implementations. Here, we introduce Serinv, a scalable library providing distributed, GPU-based algorithms for the selected inversion and Cholesky decomposition of positive-definite, block-tridiagonal arrowhead matrices. This matrix class is highly relevant in statistical climate modeling and materials science applications. The performance of Serinv is demonstrated on synthetic and real datasets from statistical air temperature prediction models. In our numerical tests, Serinv achieves 32.3% strong and 47.2% weak scaling efficiency and up to two orders of magnitude speedup over the sparse direct solvers PARDISO and MUMPS on 16 GPUs.

Problem

• Efficient inversion of block-tridiagonal arrowhead matrices
• Distributed GPU-based selected inversion computation
• Performance improvement over existing sparse solvers

Innovation

• Distributed GPU-based selected inversion algorithm
• Cholesky decomposition for block-tridiagonal arrowhead matrices
• Scalable library for statistical climate modeling