2 ms·

Very cool! My only gripe of low discrepancy sequences is how they are used in Quasi Monte-Carlo integration: 1/n \sum_1^N f(a_n), so each point gets equal

Very cool! My only gripe of low discrepancy sequences is how they are used in Quasi Monte-Carlo integration: 1/n \sum_1^N f(a_n), so each point gets equal weight. One can generalize this that every point gets it's own weight: \sum_1^N w_{N,n} f(a_n). I have never seen this generalization elsewhere. I only study Quasi Monte-Carlo as a hobby, maybe I just missed it. Once I had significantly improved discrepancies (close to theoretical minimum) with a modified Van der Corput sequence and linearly diminishing weights. It's all in one dimension though. My other idea for two dimensions were to use the Hilbert-curve (the infinite limit) to map a one-dimensional sequence to a square area, but AFAIK this was done by others before.

extremelearning 8y ago
Yes, using an open (infinite) one dimensional low-discrepancy sequence in conjunction with a space-filling curve such as the Hilbert-curve has been explored. For example by Owen [1] Niederreiter himself has a beautiful paper that includes a section on weighted low discrepancy sequence [2] where he also cites several references within it that might be of interest to you. [1] http://statweb.stanford.edu/~owen/reports/extgrid.pdf http://statweb.stanford.edu/~owen/reports/extgrid.pdf [2] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.614.2202&rep=rep1&type=pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.614...