Generation of soft-random numbers by multi-focus approximation

Abstract We propose soft-random numbers (SRNs), a novel class of random sequences designed to interpolate between quasi-random and pseudo-random regimes through a tunable softening parameter in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> [0,1] . Values near zero yield low-discrepancy quasi-random numbers, while values close to one generate pseudo-random sequences, thus enabling flexible control over discrepancy. The discrepancy behavior of SRNs is analyzed across multiple sample sizes and distributions, with statistical validity assessed using two goodness-of-fit tests. Comparative evaluations demonstrate that SRNs achieve competitive low-discrepancy properties while maintaining computational efficiency, outperforming or complementing established methods such as scrambled Sobol sequences and jittered sampling. Applications in Monte Carlo integration, uncertainty quantification, and stochastic decision processes illustrate the practical relevance of SRNs. Overall, the proposed approach provides a computationally efficient and flexible framework for random number generation, bridging the gap between deterministic quasi-random and stochastic pseudo-random techniques.