In a series of papers by the authors and several co-authors a method for the numerical integration of high-dimensional Walsh series was developed. Walsh series play an important role in several areas as for example in image- and signalprocessing. The theory is based on the concept of (t; m; s)-nets and (T; s)-sequences developed by H. Niederreiter. These are finite point sets respectively infinite sequences in the s-dimensional unit-cube which are extremely well distributed. It is the aim of this paper to give a concise introduction in this field of quasiMonte Carlo integration, to announce some new results on the topic which are not yet published, and to state the most important problems which are still unsolved and which are of relevance for our theory.
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505
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Random Number Generation and QuasiMonte Carlo Methods
– Niederreiter
- 1992
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135
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Uniform Distribution of Sequences
– Kuipers, Niederreiter
- 1974
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119
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Lattice Methods for Multiple Integration
– Sloan, Joe
- 1990
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79
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Distribution of points in a cube and approximate evaluation of integrals (in
– Sobol
- 1967
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78
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On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals
– Halton
- 1960
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62
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Point sets and sequences with small discrepancy. Monatshefte für Mathematik 104
– Niederreiter
- 1987
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55
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Discrépance de Suites Associées à un Système de Numération (en Dimension s
– Faure
- 1982
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53
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Fourier Analysis on Groups
– Rudin
- 1960
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34
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Applications of Walsh and Related Functions
– Beauchamp
- 1984
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33
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Low-discrepancy and low-dispersion sequences
– Niederreiter
- 1988
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32
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Number Theoretic Methods in Approximate Analysis. Fizmatgiz
– Korobov
- 1963
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31
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The approximate computation of multiple integrals
– Korobov
- 1959
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25
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On irregularities of distribution
– Roth
- 1954
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21
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Funktionen von beschrankter Variation in der Theorie der Gleichverteilung
– Hlawka
- 1961
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12
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Zur angenaherten Berechnung mehrfacher Integrale
– Hlawka
- 1962
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12
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On the numerical integration of Walsh series by number-theoretic methods
– Larcher, Traunfellner
- 1994
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8
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Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration, Monatsh
– Schmid
- 1996
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8
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Multivariate Walsh series, digital nets and quasi-Monte Carlo integration
– Schmid
- 1994
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8
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Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series
– Schmid, Wolf
- 1994
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8
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A metrical theorem in diophantine approximation
– Schmidt
- 1960
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8
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On irregularities of distribution vii
– Schmidt
- 1972
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8
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Series. An Introduction to Dyadic Harmonic Analysis
– Walsh
- 1990
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7
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Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series
– Schmid, Wolf
- 1996
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6
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A class of generalized Walsh functions
– Chrestenson
- 1955
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6
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Low-discrepancy point sets obtained by digital constructions over finite fields
– Niederreiter
- 1992
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6
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Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes
– Niederreiter
- 1992
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5
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Good parameters for a class of node sets in quasi-monte carlo integration
– Hansen, Mullen, et al.
- 1993
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5
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On the computation of optimal coefficients
– Korobov
- 1982
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5
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Numerical Integration of High-Dimensional Walsh Series and Concrete Parallel Calculations
– Schmid
- 1994
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5
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Point sets with uniformity properties and orthogonal hypercubes
– Mullen, Whittle
- 1992
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4
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Probabilistic diophantine approximation, I kronecker sequences
– Beck
- 1994
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4
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Optimale Koeffizienten bezuglich zusammengesetzter Zahlen. Monatshefte fur Mathematik
– Larcher
- 1985
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4
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On the distribution of sequences connected with good lattice points, Monatsh
– Larcher
- 1986
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4
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Optimal polynomials for (t; m; s)-nets and numerical integration of multivariate Walsh series
– Schmid
- 1996
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4
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Optimal coefficients modulo prime powers in the threedimensional case
– Larcher, Niederreiter
- 1989
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4
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NIEDERREITER Kronecker-type sequences and nonarchimedean diophantine approximations
– Larcher, Niederreiter
- 1993
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4
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Existence of good lattice points in the sense of Hlawka
– Niederreiter
- 1978
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4
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A combinatorial problem for vector spaces over finite fields, Discrete Mathematics 96
– Niederreiter
- 1991
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4
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Constructions of lowdiscrepancy point sets and sequences, manuscript and lecture notes
– Niederreiter
- 1991
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4
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Metrical theorems on the fractional parts of sequences
– Schmidt
- 1964
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4
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A new and extensive table of good lattice points
– Wesp
- 1994
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4
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Hyperbolic filtering of Walsh series; a new method for data compression
– Zinterhof
- 1994
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3
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Generalized (t; s)-sequences, Kronecker-type sequences, and diophantine approximations of formal Laurent series
– Larcher, Niederreiter
- 1995
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3
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Die endliche Fourier- und Walsh-Transformation mit einer Einfuhrung in die Bildverarbeitung
– Niederdrenk
- 1982
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2
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Series and Transforms: Theory and Applications
– Walsh
- 1991
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2
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A best lower bound for good lattice points
– Larcher
- 1987
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2
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A class of low-discrepancy point-sets and its application to numerical integration by number-theoretical methods
– Larcher
- 1993
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2
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Nets obtained from rational functions over finite fields
– Larcher
- 1993
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2
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On the distribution of an analog to classical Kronecker-sequences
– Larcher
- 1995
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2
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Nets constructed over finite fields and the numerical integration of multivariate Walsh series, Finite Fields
– Larcher, Wolf
- 1996
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