Search NASA⌕ Search
Home/Archive/DOE OSTI

DOE OSTI · 2420555

A posteriori superlinear convergence bounds for block conjugate gradient

Schaerer, Christian E.·Szyld, Daniel B.·Torres, Pedro J.
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we extend to the block case the a posteriori bound showing superlinear convergence of the conjugate gradient method developed by van der Vorst and Vuik in [J. Comput. Applied Math., 48 (1993), pp. 327–341]. That is, we obtain similar bounds but now for the block conjugate gradient method. We also present a series of computational experiments, illustrating the validity of the bound developed here as well as the bound by Simoncini and Szyld from [SIAM Review, 47 (2005), pp. 247–272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity and how this onset depends on the eigenvalue distribution and the block size.

Explore related subjects

97 MATHEMATICS AND COMPUTINGA posteriori analysisBlock conjugate gradient methodMathematicsSuperlinear convergence

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Schaerer, Christian E., Szyld, Daniel B., Torres, Pedro J.. 2022-12-22. A posteriori superlinear convergence bounds for block conjugate gradient. https://doi.org/10.1553/etna_vol58s115

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

KEEP EXPLORING

Related reports

DOE OSTI2026

TANTE: Time-adaptive operator learning via neural Taylor expansion

Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress in recent years, enabling efficient approximation of complex spatiotemporal dynamics. However, most existing methods rely on fixed time step sizes during rollout, which limits their ability to adapt to varying temporal complexity and often leads to error accumulation. In this work, we propose the Time-Adaptive Transformer with Neural Taylor Expansion (TANTE), a novel operator-learning framework that produces continuous-time predictions with adaptive step sizes. TANTE predicts future states by performing a Taylor expansion at the current state, where neural networks learn both the higher-order temporal derivatives and the local radius of convergence. This allows the model to dynamically adjust its rollout based on the local behavior of the solution, thereby reducing cumulative error and improving computational efficiency. We demonstrate the effectiveness of TANTE across a wide range of PDE benchmarks, achieving superior accuracy and adaptability compared to fixed-step baselines, delivering accuracy gains of 60-80 % and speed-ups of 30-40 % at inference time.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Structured illumination for surface-resolved grazing-incidence X-ray scattering

Grazing-incidence (GI) scattering techniques are widely used to characterize thin films, offering high surface sensitivity and insight into morphology and structure. However, these approaches typically provide statistical averaged information due to elongated footprint or limited spatial resolution due to beam size. Here we introduce a method that combines structured illumination with GI X-ray scattering and leverages our computational imaging approach to resolve local structural details. We demonstrate that our method captures local features of an organic semiconductor thin film without the need for sample rotation as in tomography. The method expands GI techniques from statistical averaging to high-resolution imaging, thereby providing the capability for detailed analysis of local material properties, such as domain shape, orientation and polymorphism, which are critical for advancing material design towards more efficient and tailored materials.

97 MATHEMATICS AND COMPUTING↗