Project

Funding: FWF
Funding period: Aug 2026 - Jul 2030
Project number: PAT1173425

 

Team

  • Friedrich Pillichshammer (PI, project leader, JKU, Aug 2026 - )
  • Peter Kritzer (PI, national research partner, RICAM, Aug 2026 - )
  • Nicolas Nagel (PostDoc, JKU, Aug 2026 - )

 

Abstract

This project studies how to efficiently solve high‑dimensional numerical problems—such as approximating a function or computing its integral—using quasi‑Monte Carlo (QMC) methods. These problems often involve many variables, which makes classical approaches too slow or inaccurate. QMC methods address this by evaluating a function at a carefully chosen set of points that cover the space evenly, improving accuracy over standard random sampling while keeping costs manageable.

Two goals guide the research:

  • More universal algorithms: Many current QMC methods are tuned to very specific assumptions about the problem, such as how smooth the function is or how important each variable is. These tailored methods are excellent when their assumptions are correct, but they often perform poorly when the assumptions are off—which is common in practice, since these parameters are rarely known precisely. The project aims to design algorithms that work reliably across a wider range of situations without needing fine‑tuned parameters.

  • Practical feasibility: The focus is not only on proving that a method exists, but also on ensuring it can be implemented with reasonable computational effort and tested numerically.

A key theme is using and comparing “figures of merit” that guide how QMC point sets are constructed. Traditional criteria depend strongly on the assumed function space parameters, limiting universality. The project instead emphasizes more robust criteria that depend less on unknown details. Two important examples are:

  • The R‑criterion for lattice rules (a structured family of QMC methods).

  • Discrepancy, which measures how uniformly points fill the unit cube. Low discrepancy is directly linked to small integration error through the Koksma–Hlawka inequality, a cornerstone result connecting point distribution to QMC accuracy.

By developing QMC rules optimized with such general measures—and by clarifying how different figures of merit relate—the project aims to deliver algorithms that are broadly applicable, theoretically sound, and practical for modern high‑dimensional tasks in areas like data science, machine learning, and simulation.

Publications

  • J. Dick, T. Goda, G. Larcher, F. Pillichshammer, K. Suzuki. On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences. To appear in: Math. Comp., 2026. https://arxiv.org/abs/2502.06202
  • P. Kritzer, N. Nagel, F. Pillichshammer. Infinite sequences with optimal diaphony, periodic L2-discrepancy, and beyond. Submitted, 2026. https://arxiv.org/abs/2606.05482
  • M. Haase, N. Nagel. Precise asymptotics and exact formulas for tensor product energies of Fibonacci lattices. Submitted, 2026. https://arxiv.org/abs/2605.20895

Talks

  • N. Nagel. Infinite sequences with optimal worst-case error decay of QMC-rules for periodic functions. Workshop and Summer School on Applied Analysis 2026. Chemnitz, Germany, September 2026. 
  • F. Pillichshammer. Curse of dimensionality vs. tractability for discrepancy - a survey. Eighth International Conference on Uniform Distribution Theory. Lisbon, Portugal, September 2026. 
  • N. Nagel. Precise asymptotics and explicit formulas for Fibonacci lattices. Eighth International Conference on Uniform Distribution Theory. Lisbon, Portugal, September 2026.