Priv.-Doz. Mag. Dr.

# Peter Kritzer

Multivariate Algorithms and Quasi-Monte Carlo Methods

#### Contact

Telephone: +43 (732) 2468 - 5239

https://orcid.org/0000-0002-7919-7672

#### Biographical sketch

• 2003: Master degree in Mathematics, University of Salzburg, Austria.
• 2005: Doctoral degree in Mathematics, Sub Auspiciis Praesidentis, University of Salzburg, Austria.
• 2011: Recipient of the Information-Based Complexity Young Researcher Award.
• 2012: Habilitation in Mathematics, Johannes Kepler University Linz, Austria.
• 2013: Recipient of the Kardinal Innitzer Förderungspreis, Vienna, Austria.
• 2015: Recipient of the Prize for Achievement in Information-Based Complexity.
• At RICAM since October 2015.

#### FORMER AND CURRENT POSITIONS

• 2003-2008: PhD student and Post-Doc, University of Salzburg, Austria.
• 2008-2009: Post-Doc, University of New South Wales, Australia.
• 2009-2010: Portfolio Manager, Schoellerbank, Austria.
• 2010-2015: Senior Post-Doc, Johannes Kepler University Linz, Austria.
• Since 2015: Senior Scientist, RICAM. Linz, Austria.

#### Research interests

• High-dimensional algorithms, information-based complexity, quasi-Monte Carlo methods, discrepancy theory.

#### Complete publication list

##### Books
• J. Dick, P. Kritzer, F. Pillichshammer. Lattice Rules. Springer, Cham, 2022.
##### Peer reviewed journal publications
• A. Ebert, P. Kritzer, O. Osisiogu, T. Stepaniuk. Component-by-component digit-by-digit construction of good polynomial lattice rules in weighted Walsh spaces. Constructive Approximation 56, 75-119, 2022.
• A. Ebert, P. Kritzer, O. Osisiogu, T. Stepaniuk. Construction of good polynomial lattice rules in weighted Walsh spaces by an alternative component-by-component construction. Mathematics and Computers in Simulation, 192, 399-419, 2022.
• A. Ebert, P. Kritzer, D. Nuyens, O. Osisiogu. Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness. Journal of Complexity 66, 101555, 2021.
• P. Kritzer, F. Pillichshammer, G.W. Wasilkowski. On quasi-Monte Carlo methods in weighted ANOVA spaces. Mathematics of Computation 329, 1381-1406, 2021.
• P. Kritzer, F. Pillichshammer, L. Plaskota, G.W. Wasilkowski. On alternative quantization for doubly weighted approximation and integration over unbounded domains. Journal of Approximation Theory 256, 105433, 2020.
• P. Kritzer, F. Pillichshammer, L. Plaskota, G.W. Wasilkowski. On efficient weighted integration via a change of variables. Numerische Mathematik 146, 545-570, 2020.
• P. Kritzer, F. Pillichshammer, H. Wozniakowski. Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitary linear functionals. Journal of Complexity 61, 101501, 2020.
• A. Ebert, P. Kritzer. Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm. Journal of Computational and Applied Mathematics 351, 77-100, 2019.
• A. Hinrichs, P. Kritzer, F. Pillichshammer, G. Wasilkowski.Truncation dimension for linear problems on multivariate function spaces. Numerical Algorithms 80, 661-685, 2019.
• P. Kritzer, F.Y. Kuo, D. Nuyens, M. Ullrich. Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-\alpha-1/2})$ error independently of the dimension. Journal of Approximation Theory 240, 96-113 2019.
• P. Kritzer, G. Leobacher, M. Szölgyenyi, S. Thonhauser. Approximation methods for piecewise deterministic Markov processes and their costs. Scandinavian Actuarial Journal 2019:4, 308-335, 2019.
• P. Kritzer, F. Pillichshammer, G.W. Wasilkowski.Truncation in average and worst case settings for special classes of $\infty$-variate functions. Mathematics and Computers in Simulation, 161, 52-65, 2019.
• P. Kritzer, H. Wozniakowski. Notes on tractability conditions for linear multivariate problems. Journal of Complexity 51, 110-128, 2019.
• C. Irrgeher, P. Kritzer, F. Pillichshammer. Integration and approximation in cosine spaces of smooth functions. Mathematics and Computers in Simulation, 143, 35-45, 2018.
• P. Kritzer, H. Laimer, F. Pillichshammer. Tractability of $L_2$-approximation in hybrid function spaces. Functiones et Approximatio Commentarii Mathematici 58, 89-104, 2018.
• P. Kritzer, F. Pillichshammer, G.W. Wasilkowski. A note on equivalence of anchored and ANOVA spaces; lower bounds. Journal of Complexity 38, 31-38, 2017.
• P. Kritzer, F. Pillichshammer, H. Wozniakowski. $L_\infty$-approximation in Korobov spaces with exponential weights. Journal of Complexity 41, 102-125, 2017.
• J. Dick, P. Kritzer. On a projection-corrected component-by-component construction. Journal of Complexity 32, 74-80, 2016.
• P. Hellekalek, P. Kritzer, F. Pillichshammer. Open type quasi-Monte Carlo integration based on Halton sequences in weighted Sobolev spaces. Journal of Complexity 33, 169-189, 2016.
• C. Irrgeher, P. Kritzer, F. Pillichshammer, H. Wozniakowski. Approximation in Hermite spaces of smooth functions. Journal of Approximation Theory 207, 98-126, 2016.
• C. Irrgeher, P. Kritzer, F. Pillichshammer, H. Wozniakowski. Tractability of multivariate approximation defined over Hilbert spaces with exponential weights. Journal of Approximation Theory 207, 301-338, 2016.
• P. Kritzer, H. Niederreiter. Mixed orthogonal arrays, $(u,m,e,s)$-nets, and $(u,e,s)$-sequences. Discrete Mathematics 339, 2199-2208, 2016.
• P. Kritzer, F. Pillichshammer, G.W. Wasilkowski. Very low truncation dimension for high dimensional integration under modest error demand. Journal of Complexity 35, 63-85, 2016.
• J. Dick, P. Kritzer, G. Leobacher, F. Pillichshammer. A reduced fast component-by-component construction of lattice points for integration in weighted spaces with fast decreasing weights. Journal of Computational and Applied Mathematics 276, 1-15, 2015.
• J. Dick, P. Kritzer, G. Leobacher, F. Pillichshammer. Numerical integration in log-Korobov and log-cosine spaces. Numerical Algorithms 70, 753-775, 2015.
• H. Faure, P. Kritzer, F. Pillichshammer From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules. Indagationes Mathematicae 26, 760-822, 2015.
• C. Irrgeher, P. Kritzer, G. Leobacher, F. Pillichshammer. Integration in Hermite spaces of analytic functions. Journal of Complexity 31, 380-404, 2015.
• P. Kritzer, H. Niederreiter. Propagation rules for $(u,m,e,s)$-nets and $(u,e,s)$-sequences. Journal of Complexity 31, 457-473, 2015.
• P. Kritzer, F. Pillichshammer. Component-by-component construction of shifted Halton sequences. Uniform Distribution Theory 10 (2), 45-63, 2015.
• J. Dick, P. Kritzer, F. Pillichshammer, H. Wozniakowski. Approximation of analytic functions in Korobov spaces. Journal of Complexity 30, 2-28, 2014.
• P. Kritzer, G. Larcher, F. Pillichshammer. Discrepancy estimates for index-transformed uniformly distributed sequences. Functiones et Approximatio Commentarii Mathematici 51, 197-220, 2014.
• P. Kritzer, F. Pillichshammer, H. Wozniakowski. Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces. Mathematics of Computation 83, 1189-1206, 2014.
• H. Faure, P. Kritzer. New star discrepancy bounds for $(t,m,s)$-nets and $(t,s)$-sequences. Monatshefte für Mathematik 172, 55-75, 2013.
• P. Kritzer, G. Larcher. On the arrangement of point sets in the unit interval. Manuscripta Mathematica 140, 377-391, 2013.
• P. Kritzer, F. Pillichshammer. On the existence of low-diaphony sequences made of digital sequences and lattice points. Mathematische Nachrichten 286, 224-235, 2013.
• J. Dick, P. Kritzer. A higher order Blokh-Zyablov propagation rule for higher order nets. Finite Fields and Their Applications 18, 1169-1183, 2012.
• P. Hellekalek, P. Kritzer. On the diaphony of some finite hybrid point sets. Acta Arithmetica 156, 257-282, 2012.
• F.J. Hickernell, P. Kritzer, F.Y. Kuo, D. Nuyens. Weighted compound integration rules with higher order convergence for all $N$. Numerical Algorithms 59, 161-183, 2012.
• P. Kritzer. On an example of finite mixed quasi-Monte Carlo point sets. Monatshefte für Mathematik 168, 443-459, 2012.
• P. Kritzer, F. Pillichshammer. Low discrepancy polynomial lattice point sets. Journal of Number Theory 132, 2510-2534, 2012.
• R. Hofer, P. Kritzer. On hybrid sequences built from Niederreiter-Halton sequences and Kronecker sequences. Bulletin of the Australian Mathematical Society 84, 238-254, 2011.
• P. Kritzer. A note on the extreme discrepancy of the Hammersley net in base 2. Uniform Distribution Theory 6 (1), 9-19, 2011.
• P. Kritzer, F. Pillichshammer. A lower bound on a quantity related to the quality of polynomial lattices. Functiones et Approximatio Commentarii Mathematici 45, 125-137, 2011.
• P. Kritzer, F. Pillichshammer. On the component by component construction of polynomial lattice point sets for numerical integration in weighted Sobolev spaces. Uniform Distribution Theory 6 (1), 79-100, 2011.
• J. Dick, P. Kritzer. Duality theory and propagation rules for generalized digital nets. Mathematics of Computation 79, 993-1017, 2010.
• J. Baldeaux, J. Dick, P. Kritzer. On the approximation of smooth functions using generalized digital nets. Journal of Complexity 25, 544-567, 2009.
• R. Hofer, P. Kritzer, G. Larcher, F. Pillichshammer. Distribution properties of generalized van der Corput-Halton sequences and their subsequences. International Journal of Number Theory 5, 719-746, 2009.
• X. Zeng, P. Kritzer, F.J. Hickernell. Spline methods using integration lattices and digital nets. Constructive Approximation 30, 529-555, 2009.
• B. Doerr, M. Gnewuch, P. Kritzer, F. Pillichshammer. Component-by-component construction of low-discrepancy point sets of small size. Monte Carlo Methods and Applications 14, 129-149, 2008.
• J. Dick, P. Kritzer, F.Y. Kuo, I.H. Sloan. Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels. Journal of Complexity 23, 752-772, 2007.
• J. Dick, P. Kritzer, G. Leobacher, F. Pillichshammer. Constructions of general polynomial lattice rules based on the weighted star discrepancy. Finite Fields and Their Applications 13, 1045-1070, 2007.
• J. Dick, P. Kritzer, F. Pillichshammer, W.Ch. Schmid. On the existence of higher order polynomial lattices based on a generalized figure of merit. Journal of Complexity 23, 581-593, 2007.
• P. Kritzer, G. Larcher, F. Pillichshammer. A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets. Annali di Matematica Pura ed Applicata 186, 229-250, 2007.
• P. Kritzer, F. Pillichshammer. Constructions of general polynomial lattices for multivariate integration. Bulletin of the Australian Mathematical Society 76, 93-110, 2007.
• P. Kritzer, F. Pillichshammer. On the weighted dyadic diaphony of digital $(t,s)$-sequences. Proceedings in Applied Mathematics and Mechanics 7, 1026601-1026602, 2007.
• P. Kritzer, F. Pillichshammer. Point sets with low $L_p$-discrepancy. Mathematica Slovaca 57, 11-32, 2007.
• J. Dick, P. Kritzer. A best possible upper bound on the star discrepancy of $(t,m,2)$-nets. Monte Carlo Methods and Applications 12, 1-17, 2006.
• P. Kritzer. Improved upper bounds on the star discrepancy of $(t,m,s)$-nets and $(t,s)$-sequences. Journal of Complexity 22, 336-347, 2006.
• P. Kritzer. On some remarkable properties of the two-dimensional Hammersley point set in base 2. Journal de Theorie des Nombres de Bordeaux 18, 203-221, 2006.
• P. Kritzer, F. Pillichshammer. An exact formula for the $L_2$ discrepancy of the shifted Hammersley point set. Uniform Distribution Theory 1, 1-13, 2006.
• J. Dick, P. Kritzer. Star discrepancy estimates for digital $(t,m,2)$-nets and digital $(t,2)$-sequences over $\mathbb{Z}_2$. Acta Mathemica Hungarica 109, 239-254, 2005.
• P. Kritzer. A new upper bound on the star discrepancy of $(0,1)$-sequences. Integers 5, #A11, 9 pp. (electronic), 2005.
• P. Kritzer, F. Pillichshammer. Improvements of the discrepancy of the van der Corput sequence. Mathematica Pannonica 16, 179-198, 2005.

##### Conference Contribution: Publication in Proceedings
• A. Ebert, P. Kritzer, D. Nuyens. Constructing QMC finite element methods for elliptic PDEs with random coefficients by a reduced CBC construction. In: P. L'Ecuyer, B. Tuffin (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2018, 183-205, Springer, Cham, 2020.
• F.J. Hickernell, P. Kritzer, H. Wozniakowski. Exponential tractability of linear tensor product problems. In: D.R. Wood, J. DeGier, C. Praeger, T. Tao (eds.), MATRIX Annals, 61-78,  Springer, Cham, 2019.
• P. Kritzer, F. Pillichshammer. Tractability of multivariate integration in hybrid function spaces. In: R. Cools, D. Nuyens (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2014, 437-454, Springer, Berlin, 2016.
• P. Kritzer, G. Leobacher, F. Pillichshammer. Component-by-component construction of hybrid point sets based on Hammersley and lattice point sets. In: J. Dick, F.Y. Kuo, G.W. Peters, I.H. Sloan (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2012, 501-515, Springer, Heidelberg, 2013.
• J. Dick, P. Kritzer, F.Y. Kuo. Approximation of functions using digital nets. In: A. Keller, S. Heinrich, H. Niederreiter (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2006, 275-297, Springer, Berlin, 2008.
• P. Kritzer, F. Pillichshammer. The weighted dyadic diaphony of digital sequences. In: A. Keller, S. Heinrich, H. Niederreiter (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2006, 549-560, Springer, Berlin, 2008.
• P. Kritzer. On the star discrepancy of digital nets and sequences in three dimensions. In: H. Niederreiter, D. Talay (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2004, 273-287, Springer, Berlin, 2006.
##### Contribution in Collection
• Y. Ding, F.J. Hickernell, P. Kritzer, S. Mak. Adaptive approximation for multivariate linear problems with inputs lying in a cone. In: F.J. Hickernell, P. Kritzer (eds.), Multivariate Algorithms and Information-Based Complexity, 109-145, DeGruyter, Berlin/Boston, 2020.
• P. Kritzer, H. Niederreiter, F. Pillichshammer. Ian Sloan and lattice rules. In: J. Dick, F.Y. Kuo, H. Wozniakowski (eds.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 741-769, Springer, Cham, 2018.
• P. Kritzer, F. Pillichshammer, G.W. Wasilkowski. Truncation dimension for function approximation. In: J. Dick, F.Y. Kuo, H. Wozniakowski (eds.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 771-792, Springer, Cham, 2018.
• H. Faure, P. Kritzer. Discrepancy bounds for low-dimensional point sets. In: G. Larcher, F. Pillichshammer, A. Winterhof, C.P. Xing (eds.), Applied Algebra and Number Theory, 58-90, Cambridge University Press, Cambridge, 2014.
• P. Kritzer, F. Pillichshammer, H. Wozniakowski. Tractability of multivariate analytic problems. In: P. Kritzer, H. Niederreiter, F. Pillichshammer A. Winterhof (eds.), Uniform Distribution and Quasi-Monte Carlo Methods, 147-170, Radon Series in Computational and Applied Mathematics, DeGruyter, Berlin, 2014.

##### Editorship
• F.J. Hickernell, P. Kritzer. Multivariate Algorithms and Information-Based Complexity, Radon Series in Computational and Applied Mathematics, DeGruyter, Berlin/Boston, 2020.
• E. Buckwar, P. Kritzer, G. Leobacher, F. Pillichshammer, A. Winterhof. Special Issue of Mathematics and Computers in Simulation: Tenth IMACS Seminar on Monte Carlo Methods (MCM 2015), Elsevier, 2015.
• P.~Kritzer, H.~Niederreiter, F.~Pillichshammer, A.~Winterhof. Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, DeGruyter, Berlin, 2014.