Gain coefficients for scrambled Halton points

Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of $\int_{[0,1]^d}f(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}$ with a variance that is $o(1/n)$ for any $f\in L^2[0,1]^d$. It also satisfies some non-asymptotic bounds where the variance is no larger than some $\Gamma<\infty$ times the ordinary Monte Carlo variance. For scrambled Sobol' points, this quantity $\Gamma$ grows exponentially in $d$. For scrambled Faure points, $\Gamma \leqslant \exp(1)\doteq 2.718$ in any dimension, but those points are awkward to use for large $d$. This paper shows that certain scramblings of Halton sequences have gains below an explicit bound that is $O(\log d)$ but not $O( (\log d)^{1-\epsilon})$ for any $\epsilon>0$ as $d\to\infty$. For $6\leqslant d\leqslant 10^6$, the upper bound on the gain coefficient is never larger than $3/2+\log(d/2)$

Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy

Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015) and let points $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^d$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^d$. An analogous non-positive local discrepancy (NPLD) property provides a computable lower bound. It has been known since Gabai (1967) that the two dimensional Hammersley points in any base $b\ge2$ have non-negative local discrepancy. Using the probabilistic notion of associated random variables, we generalize Gabai's finding to digital nets in any base $b\ge2$ and any dimension $d\ge1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\ge3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\ge2$ either fails to be projection regular or has all its points on the main diagonal

Terrain Diffusion Network: Climatic-Aware Terrain Generation with Geological Sketch Guidance

Sketch-based terrain generation seeks to create realistic landscapes for virtual environments in various applications such as computer games, animation and virtual reality. Recently, deep learning based terrain generation has emerged, notably the ones based on generative adversarial networks (GAN). However, these methods often struggle to fulfill the requirements of flexible user control and maintain generative diversity for realistic terrain. Therefore, we propose a novel diffusion-based method, namely terrain diffusion network (TDN), which actively incorporates user guidance for enhanced controllability, taking into account terrain features like rivers, ridges, basins, and peaks. Instead of adhering to a conventional monolithic denoising process, which often compromises the fidelity of terrain details or the alignment with user control, a multi-level denoising scheme is proposed to generate more realistic terrains by taking into account fine-grained details, particularly those related to climatic patterns influenced by erosion and tectonic activities. Specifically, three terrain synthesisers are designed for structural, intermediate, and fine-grained level denoising purposes, which allow each synthesiser concentrate on a distinct terrain aspect. Moreover, to maximise the efficiency of our TDN, we further introduce terrain and sketch latent spaces for the synthesizers with pre-trained terrain autoencoders. Comprehensive experiments on a new dataset constructed from NASA Topology Images clearly demonstrate the effectiveness of our proposed method, achieving the state-of-the-art performance. Our code and dataset will be publicly available

A Robotic Arm Based Design Method for Modular Building in Cold Region

The robotic arm has emerged as an essential tool for the rapid construction of high-quality buildings due to its ability to repeat instructions, achieve precise positioning and fine operations. The robotic arm can also effectively replace workers to complete building construction under low temperature and short daylight conditions. Thus, it can be forward-looking to construct modular buildings in cold region, but how to realize the construction of modular buildings through human-machine coordination and remote operation is a significant issue. This article discusses the feasibility of robotic arm assembly design methods in the field of architecture by simulating the complete design and construction process of modular buildings. According to parameterized module design, a three-dimensional computer model is transformed into an electronic file that directs the action of the robotic arm to complete assembly via a custom processing program. This article details a theoretical and practical exploration of the construction of modular manipulators and has certain guiding significance for the intelligent design and construction of future buildings

A Robotic Arm Based Design Method for Modular Building in Cold Region

The robotic arm has emerged as an essential tool for the rapid construction of high-quality buildings due to its ability to repeat instructions, achieve precise positioning and fine operations. The robotic arm can also effectively replace workers to complete building construction under low temperature and short daylight conditions. Thus, it can be forward-looking to construct modular buildings in cold region, but how to realize the construction of modular buildings through human-machine coordination and remote operation is a significant issue. This article discusses the feasibility of robotic arm assembly design methods in the field of architecture by simulating the complete design and construction process of modular buildings. According to parameterized module design, a three-dimensional computer model is transformed into an electronic file that directs the action of the robotic arm to complete assembly via a custom processing program. This article details a theoretical and practical exploration of the construction of modular manipulators and has certain guiding significance for the intelligent design and construction of future buildings