arXiv subjects

Valerie N. P. Ho

Publications and source records attributed to Valerie N. P. Ho.

4 recordsLinked to original sources

arXiv2026

Walk on spheres and Array-RQMC

We use Array-RQMC sampling in a walk on spheres (WoS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WoS reduces the Monte Carlo MSE or variance by factors ranging from $71$-fold to $3087$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta data

math.NA↗

arXiv2026

Randomized quasi-Monte Carlo for walk on spheres

We investigate the use of randomized quasi-Monte Carlo (RQMC) in walk on spheres algorithms to solve boundary value problems for functions with Dirichlet boundary conditions in $\mathbb{R}^d$. For harmonic functions with $d=2$, the integrands of interest are periodic indicator functions over regions $Θ$ in the torus $\mathbb{T}^k$. We give conditions for $\partialΘ$ to have $k-1$ dimensional Minkowski content which allows us to use results of He and Wang (2015). The RQMC estimates involve multiple values of $k$. We see sampling variances decreasing with the number $n$ of sample points at slightly better than Monte Carlo rates. The median variance rate in $4$ RQMC methods over $5$ worked examples, including some with $d=3$ and some with nonzero source functions, was slightly better than $O(n^{-1.1})$. The variance reduction factors ranged from $1.8$ to $10.7$ at $n=2^{17}$. None of the four RQMC methods dominated the others. Changes: cite Liu (2025), describe the lattices that were used, describe usage of AI

math.NA↗

arXiv2026

Beta-trees for testing multivariate goodness-of-fit and localizing deviations from a model

We introduce a novel goodness-of-fit (GOF) procedure based on Beta-tree partitions. A Beta-tree produces a data-adaptive partition of the sample space into regions and provides guaranteed finite sample confidence intervals for the probability contents of each region. The proposed test assesses whether the probabilities assigned by a null distribution $F_0$ fall within these intervals, thereby quantifying agreement between the model and the data. A key application is the selection of the number of components in a mixture model, where the null distribution is constructed via $k$-means clustering. In contrast to classical global GOF tests such as Kolmogorov-Smirnov or Anderson-Darling, which quantify the discrepancy through a single global statistic, our method is designed to detect local departures from the null and to identify regions of model misspecification. We demonstrate the efficiency of our test in detecting departures from the null on some simulated and real datasets.

stat.ME↗

arXiv2026

Quasi-Monte Carlo with one categorical variable

We study randomized quasi-Monte Carlo (RQMC) estimation of a multivariate integral where one of the variables takes only a finite number of values. This problem arises when the variable of integration is drawn from a mixture distribution as is common in importance sampling and also arises in some recent work on transport maps. We find that when integration error decreases at an RQMC rate that it is then important to oversample the smallest mixture components instead of using a proportional allocation. This can even improve the rate of convergence. The optimal allocations depend on the possibly unknown convergence rate. Designing the sample with an incorrect assumption on the rate still attains that convergence rate, with an inferior implied constant. The penalty for using a pessimistic rate is typically higher than for using an optimistic one. We also find that for the most accurate RQMC sampling methods, it is advantageous to arrange that our $n=2^m$ randomized Sobol' points split into subsample sizes that are also powers of $2$.

stat.CO↗