Research

The guiding question of my research is how hard high-dimensional problems really are: how the intrinsic difficulty of numerical integration and approximation scales with dimension, when the curse of dimensionality can be broken, and how far random information can take us compared with optimal, hand-crafted constructions.

Main research topics

• Complexity of high-dimensional problems: tractability and the curse of dimensionality
• Power of random information: random vs. optimal data for approximation and integration
• Sampling recovery: function values, least squares, and discretization
• Numerical integration: mixed smoothness, Frolov cubature, Monte Carlo, applied number theory
• Discrepancy & dispersion of high-dimensional point sets
• Geometry of Banach spaces, esp. s-numbers of operators
• Large random matrices & asymptotic geometric analysis
• Foundations of machine learning: neural network approximation, learning theory

A few results from my work that I consider the most important.

• Bounds between all s-numbers. The approximation numbers of a bounded linear operator are dominated by every s-number sequence: \(a_n(S)\le e(n+1)\,s_n(S)\), which is optimal up to the constant. This settles questions of Mityagin, Henkin, Carl and Pietsch dating back to 1963; a similar bound holds for widths of convex sets. Ullrich, Adv. Oper. Theory (2024)Ullrich, arXiv:2608.05024 (2026)Lean formalization, machine-checked
• Optimal \(L_2\)-approximation in reproducing kernel Hilbert spaces. Function values are almost as powerful as arbitrary linear information whenever the approximation numbers are square-summable: weighted least squares at random points reach the optimal rate up to a logarithmic oversampling factor, and a subsequent subsampling argument removes this factor to give a sharp bound on the sampling numbers. Krieg & Ullrich, Found. Comput. Math. (2021)Dolbeault, Krieg & Ullrich, Appl. Comput. Harmon. Anal. (2023)
• Optimal sampling recovery in more general norms. Optimal recovery of functions in the uniform (\(L_\infty\)) norm and in \(L_p\)-norms, with sharp bounds obtained through the Christoffel function, sampling projections and entropy numbers. Krieg, Pozharska, T. Ullrich & Ullrich, Math. Comp. (2025)Krieg, Pozharska, T. Ullrich & Ullrich, J. Math. Anal. Appl. (2026)Ullrich, J. Complexity (2026)
• Discretization of continuous norms. Near-optimal bounds on the number of points needed to discretize the continuous uniform and \(L_p\)-norms of functions from general finite-dimensional spaces (Marcinkiewicz-type discretization). We also obtained a constructive algorithm to generate such point sets for reproducing kernel Hilbert spaces. Krieg, Pozharska, T. Ullrich & Ullrich, J. Math. Anal. Appl. (2026)Chkifa, Dolbeault, Krieg & Ullrich, arXiv:2602.18719 (2026)
• Learning a vector from few nonlinear measurements. Any vector in \(\mathbb{R}^m\) can be recovered to arbitrary precision from only \(\lceil\log_2(m+1)\rceil+1\) adaptively chosen continuous measurements, with consequences for infinite-dimensional problems. Krieg, Novak & Ullrich, arXiv:2412.06468 (2024)
• Approximation by deep ReLU networks. Nonlocal techniques that give approximation rates for deep ReLU neural networks: we construct a Riesz basis of “smooth” functions consisting of small (deep) neural networks. Schneider, Ullrich & Vybral, J. Mach. Learn. Res. (2026)
• The power of random information. A theory of when i.i.d. random information is almost optimal, and when it is far from optimal, for approximation and integration in the worst-case setting, using function values or Gaussian measurements. Hinrichs, Krieg, Novak, Prochno & Ullrich, Trans. Amer. Math. Soc. (2021)Hinrichs, Krieg, Novak, Prochno & Ullrich, Multivariate Algorithms & IBC, De Gruyter (2020)Sonnleitner & Ullrich, J. Appl. Numer. Anal. (2023), survey
• Adaption, randomization and noisy information. How much adaptive, randomized and noisy nonlinear measurements can improve on deterministic linear information for approximation problems. Krieg, Novak & Ullrich, Forum Math. Sigma (2025)Krieg, Novak, Plaskota & Ullrich, J. Fourier Anal. Appl. (2026)
• The curse of dimensionality for integration. Numerical integration of smooth functions suffers the curse of dimensionality, both for classical smooth classes and on general domains, answering long-standing questions. Hinrichs, Prochno & Ullrich, J. Complexity (2019), Best Paper AwardHinrichs, Novak, Ullrich & Woźniakowski, Math. Comp. (2014)
• Optimal Monte Carlo integration. A randomized algorithm that integrates multivariate smooth functions at the optimal rate, beating every deterministic method. Ullrich, SIAM J. Numer. Anal. (2017)
• Frolov cubature for mixed smoothness. A detailed analysis of the optimality of Frolov's cubature, a number-theoretic construction based on algebraic lattices. T. Ullrich & Ullrich, SIAM J. Numer. Anal. (2016)
• Rapid mixing of Swendsen–Wang. Proved rapid mixing of the Swendsen–Wang dynamics in two dimensions, settling a central question on the efficiency of this Markov-chain Monte Carlo method. Ullrich, Dissertationes Math. (2014)