Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.06021

A Sub-linear Low-Rank Solver for Poisson's Equation using Machine Learning Frameworks for GPU Acceleration

Måns I. Andersson·Daniel Appelö
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we explore a fast Poisson solver for problems with a solution that is known to be low-rank. We use an adaptive and warm started cross approximation called Cross-DEIM that iterates between index selection and and cross approximation to generate a low-rank solution. This paper focuses on leveraging a modern machine learning framework, PyTorch, as a general purpose array language to implement low-rank solvers based on Cross-DEIM. PyTorch enables native access to GPUs and accelerators but with a user-friendly high-level interface. We investigate statistical leverage scores for the index selection for the cross approximation due to the cost associated with the pivoted algorithms used with the discrete empirical interpolation methods (DEIM and QDEIM) which are historically preferred. The cross approximation is naturally paired with a Discrete Sine Transform (DST) Poisson solver. This allows the Fast Fourier Transform (FFT) to be evaluated in batches along dimensions independently without any global transpose even in higher dimensions. We present performance results running on a A100 GPU and AMD EPYC CPU demonstrating the usefulness of the approach that enables problems sizes that previously were not feasible.

Explore related subjects

cs.PFcs.NAmath.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Måns I. Andersson, Daniel Appelö. 2026-07-07. A Sub-linear Low-Rank Solver for Poisson's Equation using Machine Learning Frameworks for GPU Acceleration. https://arxiv.org/abs/2607.06021

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

SARA: SLO-Aware Resource Allocation for Disaggregated Agentic LLM Services

Recent advances in large language models (LLMs) are driving the emergence of multi-modal and agentic services for mobile users through cloud and edge infrastructures, where long-context workloads pose daunting challenges for inference latency. Existing disaggregated LLM serving systems largely rely on hardware profiling, configuration enumeration, or heuristic scheduling, offering limited analytical guidance for cost-efficient resource allocation. In this paper, we propose SARA, a Service level objectives (SLOs)-Aware Resource Allocation framework for disaggregated agentic LLM serving systems, which maximizes goodput under a deployment cost constraint and a series of quantile-based SLO constraints. By capitalizing on queuing theory, we first model the prefill, KV cache transfer, and decode stages as an M/G/k queue, an M/G/1 queue, and a generalized birth-death process, respectively. The analysis reveals that the prefill and decode stages are dominantly limited by computational capacity and high-bandwidth memory (HBM) resources, respectively. With these mathematical models, we further derive tractable tail behaviors of different stage-wise service level metrics for both light- and heavy-tailed workloads. These characterizations explicitly map workload, model architecture, and hardware parameters to stage-wise SLO constraints and minimum resource requirements. Finally, we develop an effective resource allocation framework to maximize system goodput under limited cost budgets. Simulation and hardware results demonstrate that the proposed framework accurately predicts the stage-wise SLO with mean errors below 5%, and improves system goodput by 26.6% on average over state-of-the-art baseline methods under the same deployment cost.

cs.PF↗
arXiv2026

Performance Analysis of Low-Order, GPU-accelerated Finite Element Kernels using Kokkos

We study performance portability for low-order, matrix-free finite element kernels, using the example of a vectorial, variable-coefficient PDE operator originating in geophysical models. Written in Kokkos, the kernel is compared on NVIDIA H100, AMD MI250X, AMD MI300A and Intel PVC Max 1550 GPUs. Owing to its low order and to optimizations that reduce the arithmetic, the kernel has a low arithmetic intensity, so that its performance is determined by how the finite element assembly is mapped onto the memory hierarchy. This is a dimension in which the architectures differ even within one vendor family, causing different performance characteristics. We examine how Kokkos' hierarchical parallelism and shared scratch memory, which are used for the shared degrees of freedom of the conforming discretization, behave on each device. Finally, we show how portability gaps can be narrowed with tuning levers such as the size of the thread groups, the balance between occupancy and register use, and the atomic accumulation strategy at the end of the kernel.

cs.PF↗
arXiv2026

Measured Joules, Learned Routes: Learning to Route for Energy-Efficient LLM Serving

Large language models (LLMs) and agentic AI systems are creating rapidly growing inference energy demands as model sizes grow and reasoning trajectories extend. While in practice, many queries do not require the capabilities of the largest available model, and routinely directing such queries to a high-capability model can introduce unnecessary, considerable computation and energy consumption. In this paper, we investigate whether adaptive routing across a heterogeneous pool of LLMs can reduce this energy burden without substantially compromising task performance. We design a language-model-based router that reads in each query and selects an answer model from a fixed candidate pool. The candidate models are first profiled through an offline tournament that records their correctness, latency, power, and GPU energy for each query. Using these measurements, the router is trained through supervised fine-tuning followed by group relative policy optimization (GRPO) with the tailored paradigms. Results demonstrate that learned routing can selectively allocate expensive model capacity based on query context and improve the accuracy-energy tradeoff in multi-LLM serving. Across seven benchmark tasks, we also observe a sharp accuracy-energy phase transition among routers, providing practical insights into improving energy efficiency while maintaining LLM performance.

cs.PF↗