The quasi-Monte Carlo (QMC) methods use deterministic, usually low-discrepancy sequences, to improve the order of convergence when compared with Monte Carlo methods dealing with the same problem. Although this improved convergence depends on the smoothness properties of the functions under consideration and sometimes cannot be established theoretically, good practical results were obtained in many areas, notably in Financial Mathematics. Most of the QMC methods do not depend on particular properties of the sequences used beyond their good distribution in a suitable multidimensional space. Many families of low-discrepancy sequences have been studies, with the family of the Sobol sequences being not only one of the oldest but also the most popular in practical applications. Since these sequences have many free parameters, it is desirable to have methods to obtain a good set of parameters, suitable for a wide class of problems. In this work we propose a fully deterministic method for optimisation of the direction numbers of the Sobol sequences that is flexible in taking into account smoothness properties of the potential sub-integral functions. We demonstrate how by implementing this algorithm on modern GPUs it is possible to obtain good sets of direction numbers within a reasonable timeframe. The numerical tests demonstrate the significant advantage in terms of accuracy that is observed, which makes the corresponding QMC methods even more competitive.
This supplemental paper to MUT v5.8 provides a deterministic explanation for Quantum Entanglement as a phase-locking phenomenon within the background scalar field. Superluminal transport of the scalar field flux at high scaling factors
z allows entangled particles to synchronize instantaneously across arbitrary distances. The paper bridges Quantum Mechanics and Special Relativity within a unified deterministic framework, eliminating metaphysical ambiguity. This version (v5.8.8) builds upon previous MUT supplements, integrating scalar field dynamics, derived constants (
α,
ξ,
G), and high-scaling flux transport.
We show that the Schrödinger equation describes the ensemble mean dynamics of solitons in a Galilean invariant field theory where we interpret solitons as particles. On a zero background, solitons move classically, following Newton's second law; however, on a nonzero amplitude chaotic background, their momentum and position fluctuate fulfilling an exact uncertainty relation, which gives rise to the emergence of quantum phenomena. The Schrödinger equation for the ensemble of solitons is obtained from this exact uncertainty relation, and the amplitude of the background fluctuations is what corresponds to the value of ℏ . We confirm our analytical results running simulations of solitons moving against a potential barrier and comparing the ensemble probabilities with the predictions of the time-dependent Schrödinger equation, providing a deterministic version of the quantum tunneling effect. We conclude with a discussion of how our theory does not present statistical independence between measurement and experiment outcomes.