## Quasi-Monte Carlo methods for multidimensional numerical integration.(English)Zbl 0662.65021

Numerical integration III, Proc. Conf. Oberwolfach/FRG 1987, ISNM 85, 157-171 (1988).
[For the entire collection see Zbl 0641.00023.]
The paper contains a survey on the recent researches concerning the integration in several variables by quasi-Monte Carlo methods. It is well known that we can estimate an integral $$\int_{I_ k}f(X)dX=J$$ by the relation $$(1)\quad J=N^{-1}\sum^{N}_{n=1}f(X_ n)$$ where $$I_ k=[0,1]^ k$$, and $$X_ n\in I_ k$$. If the $$X_ n$$ points are an independent random sample from the uniform distribution on $$I_ k$$, (1) gives a statistical estimation of J (the classical Monte Carlo method) for which the expected value of the error is O(1/$$\sqrt{N})$$. In quasi- Monte Carlo methods the points $$X_ n$$ are selected on the contrary such that the deterministic error can satisfy a lower bound. Sequences of sets of N points are known which, for increasing N, allow to attain the bound O((lg N)$${}^{k-1}/N)$$, under very weak conditions on f(X).
Sequences with this property can be obtained by various techniques. The author considers the three most important methods: 1) the method of low- discrepancy sequences (quasi-random sequences, 2) the method of good lattice points, 3) method based on pseudorandom numbers, and exposes exhaustively the researches developed about these methods after 1978. The paper contains also a new result (Theorem 4) concerning the generation of low discrepancy sequences by use of hyperderivatives. The final list of references quotes seventysix papers.
Reviewer: M.Cugiani

### MSC:

 65D32 Numerical quadrature and cubature formulas 65C05 Monte Carlo methods 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis

Zbl 0641.00023

### Software:

TOMS659; Algorithm 647; TESTPACK