SobolSequenzen

Sobol sequences are a type of low-discrepancy sequence, also known as quasi-random sequences. They are designed to fill a multidimensional space more uniformly than purely random sampling. This uniformity is achieved by carefully constructing the coordinates of the sample points, aiming to minimize gaps and clusters. The core idea behind Sobol sequences is to use a specific set of direction numbers and a base number to generate successive points.

These sequences are particularly useful in numerical integration and Monte Carlo methods, especially in high dimensions. Traditional random sampling can require a very large number of points to achieve accurate results in many dimensions. Sobol sequences, by providing a more even distribution of points, can often achieve comparable or better accuracy with fewer samples. This makes them computationally more efficient for certain applications.

The generation process involves a bitwise operation. For each dimension, a new point's coordinate is calculated by taking a previously generated value and XORing it with a specific direction number. The choice of direction numbers is crucial and is typically based on primitive roots modulo powers of 2. The quality of a Sobol sequence is often measured by its discrepancy, a mathematical quantity that quantifies how evenly the points are distributed. Lower discrepancy values indicate a more uniform distribution.

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