Space-filling Curves Research Articles

Convergence to closed-form distribution for the backward [formula omitted] at some random times and the phase transition at [formula omitted

We study a one-dimensional SDE that we obtain by performing a random time change of the backward Loewner dynamics in H. The stationary measure for this SDE has a closed-form expression. We show the convergence towards its stationary measure for this SDE, in the sense of random ergodic averages. The precise formula of the density of the stationary law gives a phase transition at the value κ=8 from integrability to non-integrability, that happens at the same value of κ as the change in behavior of the SLEκ trace from non-space filling to space-filling curve. Using convergence in total variation for the law of this diffusion towards stationarity, we identify families of random times on which the law of the arguments of points under the backward SLEκ flow converge to a closed form expression measure. For κ=4, this gives precise characterization for the random times on which the law of the arguments of points under the backward SLEκ flow converge to the uniform law.

Investigation of Structural Energy Absorption Performance in 3D-Printed Polymer (Tough 1500 Resin) Materials with Novel Multilayer Thin-Walled Sandwich Structures Inspired by Peano Space-Filling Curves.

Inspired by Peano space-filling curves (PSCs), this study introduced the space-filling structure design concept to novel thin-walled sandwich structures and fabricated polymer samples by 3D printing technology. The crushing behaviors and energy absorption performance of the PSC multilayer thin-walled sandwich structures and the traditional serpentine space-filling curve (SSC) multilayer thin-walled sandwich structures were investigated using quasi-static compression experiments and numerical analysis. Taking the initial peak crushing force (IPF), specific energy absorption (SEA), and crushing force efficiency (CFE) as evaluation criteria, the effects of geometric parameters, including the curve order, layer height, septa thickness, and wall thickness, on energy absorption performance were comprehensively examined. The results indicated that the energy absorption capacity of the PSC structure was significantly enhanced due to its complex hierarchy. Specifically, the second-order PSC structure demonstrated a 53.2% increase in energy absorption compared to the second-order SSC structure, while the third-order PSC structure showed more than a six-fold increase in energy absorption compared to the third-order SSC structure. Furthermore, a multi-objective optimization method based on the response surface method and the NSGA-II algorithm were employed to optimize the wall thickness and layer height of the proposed novel PSC structures. The optimal solutions suggested that a reasonable wall thickness and layer height were two important factors for designing PSC structures with better energy absorption performance. The findings of this study provide an effective guide for using the space-filling concept with Peano curves for the design of a novel polymer thin-walled energy absorber with high energy absorption efficiency.

An application of space-filling curves to improve results of turbulent aerodynamics modeling with convolutional neural networks

When carrying out calculations for turbulent flow simulation, one inevitably has to face the choice between accuracy and speed of calculations. In order to simultaneously obtain both a computationally efficient and more accurate model, a surrogate model can be built on the basis of some fast special model and knowledge of previous calculations obtained by more accurate base models from various test bases or some results of serial calculations. The objective of this work is to construct a surrogate model which allows to improve the accuracy of turbulent calculations obtained by a special model on unstructured meshes. For this purpose, we use 1D Convolutional Neural Network (CNN) of the encoder-decoder architecture and reduce the problem to a single dimension by applying space-filling curves. Such an approach would have the benefit of being applicable to solutions obtained on unstructured meshes. In this work, a non-local approach is applied where entire flow fields obtained by the special and base models are used as input and ground truth output respectively. Spalart-Allmaras (SA) model and Near-wall Domain Decomposition (NDD) method for SA are taken as the base and special models respectively. The efficiency and accuracy of the obtained surrogate model are demonstrated in a case of supersonic flow over a compression corner with different values for angle α and Reynolds number Re. We conducted an investigation into interpolation and extrapolation by Re and also into interpolation by α.

Florets for Chiplets: Data Flow-aware High-Performance and Energy-efficient Network-on-Interposer for CNN Inference Tasks

Recent advances in 2.5D chiplet platforms provide a new avenue for compact scale-out implementations of emerging compute- and data-intensive applications including machine learning. Network-on-Interposer (NoI) enables integration of multiple chiplets on a 2.5D system. While these manycore platforms can deliver high computational throughput and energy efficiency by running multiple specialized tasks concurrently, conventional NoI architectures have a limited computational throughput due to their inherent multi-hop topologies. In this paper, we propose Floret, a novel NoI architecture based on space-filling curves (SFCs). The Floret architecture leverages suitable task mapping, exploits the data flow pattern, and optimizes the inter-chiplet data exchange to extract high performance for multiple types of convolutional neural network (CNN) inference tasks running concurrently. We demonstrate that the Floret architecture reduces the latency and energy up to 58% and 64%, respectively, compared to state-of-the-art NoI architectures while executing datacenter-scale workloads involving multiple CNN tasks simultaneously. Floret achieves high performance and significant energy savings with much lower fabrication cost by exploiting the data-flow awareness of the CNN inference tasks.

Efficient Quality Diversity Optimization of 3D Buildings through 2D Pre-optimization.

Quality diversity algorithms can be used to efficiently create a diverse set of solutions to inform engineers' intuition. But quality diversity is not efficient in very expensive problems, needing 100.000s of evaluations. Even with the assistance of surrogate models, quality diversity needs 100s or even 1000s of evaluations, which can make it use infeasible. In this study we try to tackle this problem by using a pre-optimization strategy on a lower-dimensional optimization problem and then map the solutions to a higher-dimensional case. For a use case to design buildings that minimize wind nuisance, we show that we can predict flow features around 3D buildings from 2D flow features around building footprints. For a diverse set of building designs, by sampling the space of 2D footprints with a quality diversity algorithm, a predictive model can be trained that is more accurate than when trained on a set of footprints that were selected with a space-filling algorithm like the Sobol sequence. Simulating only 16 buildings in 3D, a set of 1024 building designs with low predicted wind nuisance is created. We show that we can produce better machine learning models by producing training data with quality diversity instead of using common sampling techniques. The method can bootstrap generative design in a computationally expensive 3D domain and allow engineers to sweep the design space, understanding wind nuisance in early design phases.

Improving image classification of one-dimensional convolutional neural networks using Hilbert space-filling curves

Improving image classification of one-dimensional convolutional neural networks using Hilbert space-filling curves

Planar substitutions to Lebesgue type space-filling curves and relatively dense fractal-like sets in the plane

We present an algorithm for generating curves filling the unit square; i.e. space-filling curves, from any given planar substitution satisfying a mild condition. The proposed algorithm is mimicking construction steps of Lebesgue's curve and is based on linear interpolation. Generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are satisfied.

A Knowledge Graph-Based Deep Learning Framework for Efficient Content Similarity Search of Sustainable Development Goals Data

ABSTRACT Sustainable development denotes the enhancement of living standards in the present without compromising future generations’ resources. Sustainable Development Goals (SDGs) quantify the accomplishment of sustainable development and pave the way for a world worth living in for future generations. Scholars can contribute to the achievement of the SDGs by guiding the actions of practitioners based on the analysis of SDG data, as intended by this work. We propose a framework of algorithms based on dimensionality reduction methods with the use of Hilbert Space Filling Curves (HSFCs) in order to semantically cluster new uncategorised SDG data and novel indicators, and efficiently place them in the environment of a distributed knowledge graph store. First, a framework of algorithms for insertion of new indicators and projection on the HSFC curve based on their transformer-based similarity assessment, for retrieval of indicators and load-balancing along with an approach for data classification of entrant-indicators is described. Then, a thorough case study in a distributed knowledge graph environment experimentally evaluates our framework. The results are presented and discussed in light of theory along with the actual impact that can have for practitioners analysing SDG data, including intergovernmental organizations, government agencies and social welfare organizations. Our approach empowers SDG knowledge graphs for causal analysis, inference, and manifold interpretations of the societal implications of SDG-related actions, as data are accessed in reduced retrieval times. It facilitates quicker measurement of influence of users and communities on specific goals and serves for faster distributed knowledge matching, as semantic cohesion of data is preserved.

Messing up to clean up: Semi-randomized frequency selective space-filling curves to suppress physiological signal fluctuations in MRI.

Rapid 3D steady-state sequences are widely used but are also known to be sensitive to semi-periodic physiological signal fluctuations due to, for example, cardiac pulsation, breathing, and eye/eyelids movement. This semi-periodicity results in repeating artifacts in the image whose intensity depends on the scan parameters. The purpose of this study is to design a reordering of the 2D phase encodes (within the 3D acquisition) that reduces these artifacts. A randomized order of the phase encodes can suppress repeating artifact but may also introduce its own apparent noise, for example, in cases of slow subject movement or gradual changes in eddy currents. In a new design a semi-randomized space-filling curve is generated by scrambling the local order of the phase encodes to achieve a controlled frequency selective effect, that is, eliminating artifacts above a chosen (fluctuation) frequency threshold while leaving lower frequencies untouched, thus overcoming the limitations of a randomized order. The method was characterized in simulations and substantiated by human brain imaging at 7 T using two steady-state gradient echo variants that suffer from pulsation, either near blood vessels or near the ventricles. The simulations with a point source show that the maximum artifact intensity can be reduced by factors of 10-50 depending on the scan parameters. In human scanning, the new approach drastically reduced physiologically induced artifacts and was superior in this regard to both full randomization and a generalized Hilbert curve, another semi-randomized approach. The phase-encodes reordering presented here effectively removes artifacts arising from local fluctuations.

Real-space solution to the electronic structure problem for nearly a million electrons.

We report a Kohn-Sham density functional theory calculation of a system with more than 200 000 atoms and 800 000 electrons using a real-space high-order finite-difference method to investigate the electronic structure of large spherical silicon nanoclusters. Our system of choice was a 20nm large spherical nanocluster with 202 617 silicon atoms and 13 836 hydrogen atoms used to passivate the dangling surface bonds. To speed up the convergence of the eigenspace, we utilized Chebyshev-filtered subspace iteration, and for sparse matrix-vector multiplications, we used blockwise Hilbert space-filling curves, implemented in the PARSEC code. For this calculation, we also replaced our orthonormalization + Rayleigh-Ritz step with a generalized eigenvalue problem step. We utilized all of the 8192 nodes (458 752 processors) on the Frontera machine at the Texas Advanced Computing Center. We achieved two Chebyshev-filtered subspace iterations, yielding a good approximation of the electronic density of states. Our work pushes the limits on the capabilities of the current electronic structure solvers to nearly 106 electrons and demonstrates the potential of the real-space approach to efficiently parallelize large calculations on modern high-performance computing platforms.

Solving the electronic structure problem for over 100 000 atoms in real space

Using a real-space high-order finite-difference approach, we investigate the electronic structure of large spherical silicon nanoclusters. Within Kohn-Sham density functional theory and using pseudopotentials, we report the self-consistent field convergence of a system with over 100 000 atoms: a ${\mathrm{Si}}_{107,641}{\mathrm{H}}_{9,084}$ nanocluster with a diameter of 16 nm. Our approach uses Chebyshev-filtered subspace iteration to speed up the convergence of the eigenspace, and blockwise Hilbert space-filling curves to speed up sparse matrix-vector multiplications, all of which are implemented in the parsec code. For the largest system, we utilized 2048 nodes (114 688 cores) on the Frontera machine in the Texas Advanced Computing Center. Our quantitative analysis of the electronic structure shows how it gradually approaches its bulk counterpart as a function of nanocluster size. The band gap is enlarged due to quantum confinement in nanoclusters, but decreases as the system size increases, as expected. Our work serves as a proof of concept for the capacity of the real-space approach in efficiently parallelizing very large calculations using high-performance computer platforms, which can straightforwardly be replicated in other systems with more than ${10}^{5}$ atoms.

Microbiome maps: Hilbert curve visualizations of metagenomic profiles.

Abundance profiles from metagenomic sequencing data synthesize information from billions of sequenced reads coming from thousands of microbial genomes. Analyzing and understanding these profiles can be a challenge since the data they represent are complex. Particularly challenging is their visualization, as existing techniques are inadequate when the taxa number is in the thousands. We present a technique, and accompanying software, for the visualization of metagenomic abundance profiles using a space-filling curve that transforms a profile into an interactive 2D image. We created Jasper, an easy to use tool for the visualization and exploration of metagenomic profiles from DNA sequencing data. It orders taxa using a space-filling Hilbert curve, and creates a "Microbiome Map", where each position in the image represents the abundance of a single taxon from a reference collection. Jasper can order taxa in multiple ways, and the resulting microbiome maps can highlight "hot spots" of microbes that are dominant in taxonomic clades or biological conditions. We use Jasper to visualize samples from a variety of microbiome studies, and discuss ways in which microbiome maps can be an invaluable tool to visualize spatial, temporal, disease, and differential profiles. Our approach can create detailed microbiome maps involving hundreds of thousands of microbial reference genomes with the potential to unravel latent relationships (taxonomic, spatio-temporal, functional, and other) that could remain hidden using traditional visualization techniques. The maps can also be converted into animated movies that bring to life the dynamicity of microbiomes.

GPU-parallelisation of Haar wavelet-based grid resolution adaptation for fast finite volume modelling: application to shallow water flows

Abstract Wavelet-based grid resolution adaptation driven by the ‘multiresolution analysis’ (MRA) of the Haar wavelet (HW) allows to devise an adaptive first-order finite volume (FV1) model (HWFV1) that can readily preserve the modelling fidelity of its reference uniform-grid FV1 counterpart. However, the MRA entails an enormous computational effort as it involves ‘encoding’ (coarsening), ‘decoding’ (refining), analysing and traversing modelled data across a deep hierarchy of nested, uniform grids. GPU-parallelisation of the MRA is needed to handle its computational effort, but its algorithmic structure (1) hinders coalesced memory access on the GPU and (2) involves an inherently sequential tree traversal problem. This work redesigns the algorithmic structure of the MRA in order to parallelise it on the GPU, addressing (1) by applying Z-order space-filling curves and (2) by adopting a parallel tree traversal algorithm. This results in a GPU-parallelised HWFV1 model (GPU-HWFV1). GPU-HWFV1 is verified against its CPU predecessor (CPU-HWFV1) and its GPU-parallelised reference uniform-grid counterpart (GPU-FV1) over five shallow water flow test cases. GPU-HWFV1 preserves the modelling fidelity of GPU-FV1 while being up to 30 times faster. Compared to CPU-HWFV1, it is up to 200 times faster, suggesting that the GPU-parallelised MRA could be used to speed up other FV1 models.

A knowledge graph-based deep learning framework for efficient content similarity search of Sustainable Development Goals data

Abstract Sustainable development denotes the enhancement of living standards in the present without compromising future generations’ resources. Sustainable Development Goals (SDGs) quantify the accomplishment of sustainable development and pave the way for a world worth living in for future generations. Scholars can contribute to the achievement of the SDGs by guiding the actions of practitioners based on the analysis of SDG data, as intended by this work. We propose a framework of algorithms based on dimensionality reduction methods with the use of Hilbert Space Filling Curves (HSFCs) in order to semantically cluster new uncategorised SDG data and novel indicators, and efficiently place them in the environment of a distributed knowledge graph store. First, a framework of algorithms for insertion of new indicators and projection on the HSFC curve based on their transformer-based similarity assessment, for retrieval of indicators and load-balancing along with an approach for data classification of entrant-indicators is described. Then, a thorough case study in a distributed knowledge graph environment experimentally evaluates our framework. The results are presented and discussed in light of theory along with the actual impact that can have for practitioners analysing SDG data, including intergovernmental organizations, government agencies and social welfare organizations. Our approach empowers SDG knowledge graphs for causal analysis, inference, and manifold interpretations of the societal implications of SDG-related actions, as data are accessed in reduced retrieval times. It facilitates quicker measurement of influence of users and communities on specific goals and serves for faster distributed knowledge matching, as semantic cohesion of data is preserved.

LMSFC: A Novel Multidimensional Index Based on Learned Monotonic Space Filling Curves

The recently proposed learned indexes have attracted much attention as they can adapt to the actual data and query distributions to attain better search efficiency. Based on this technique, several existing works build up indexes for multi-dimensional data and achieve improved query performance. A common paradigm of these works is to (i) map multi-dimensional data points to a one-dimensional space using a fixed space-filling curve (SFC) or its variant and (ii) then apply the learned indexing techniques. We notice that the first step typically uses a fixed SFC method, such as row-major order and z -order. It definitely limits the potential of learned multi-dimensional indexes to adapt variable data distributions via different query workloads. In this paper, we propose a novel idea of learning a space-filling curve that is carefully designed and actively optimized for efficient query processing. We also identify innovative offline and online optimization opportunities common to SFC-based learned indexes and offer optimal and/or heuristic solutions. Experimental results demonstrate that our proposed method, LMSFC, outperforms state-of-the-art non-learned or learned methods across three commonly used real-world datasets and diverse experimental settings.

Patterns of the radiation properties for Peano antennas

This paper studies metallic microstrip antennas with air as a substrate in the UHF band, patterned after space-filling, self-avoiding, and self-similar (FASS) Peano curves. Our novel study is based on context-free grammar and genetic programming as computing tools to unravel the role of geometry on both; the Voltage Standing Wave Ratio (VSWR<2) and frequency resonance patterns for Peano antennas. We use in our approach the numeric method of moments (MoM) implemented in Matlab 2021a to solve the corresponding Maxwell equations. Novel equations for the patterns of both features (resonance frequencies and frequencies such that the VSWR<2) are provided as functions of the characteristic length L. Antennas spanning a Ltimes L area (Lle 0.1,m), feeding points set at seven places, and three widths of the metallic strip are introduced as instances of our approach. Finally, a Python 3.7 application is constructed to facilitate the extension and use of our results.

Efficient private key generation from iris data for privacy and security applications

Private key generation based on biometric data is gaining popularity in several cryptographic applications. Some key generation approaches using fingerprint and face data are known for this purpose. The existing approaches, in general, follow direct features from raw data, which may have an issue of revealing users’ biometric data. Further, iris biometrics being a better source for stable and secure key generation, is yet to be explored. In this work, an iris biometric-based key generation approach is proposed. In the proposed method, first, an ensemble L0 Gradient Minimization and an edge-preserving filter are used to extract iris structure from the iris images. Secondly, the iris texture data is normalized, and the consistent region is selected based on a novel statistical method. Thirdly, feature vectors are generated using the ensemble local space-filling curve descriptors and their variants. Fourth, a discriminant feature vector is generated using hybrid feature selection and neighbourhood component analysis. Finally, an interval-based encoding scheme is used to generate a key from the discriminant feature vector. To substantiate the efficacy of the proposed approach, extensive experiments have been carried out with near-infrared images, visible wavelength images, at-a-distance, and on-move images. Further, distinctiveness, dissimilarity, randomness, and security analyses have been carried out to validate the resilience of keys against different attacks. More significantly, a unique key can be generated dynamically, that is, from an online image. The proposed approach is applicable to different cryptographic applications, such as data storage security, remote authentication protocol, blockchain system security, etc.

Towards Designing and Learning Piecewise Space-Filling Curves

To index multi-dimensional data, space-filling curves (SFCs) have been used to map the data to one dimension, and then a one-dimensional indexing method such as the B-tree is used to index the mapped data. The existing SFCs all adopt a single mapping scheme for the whole data space. However, a single mapping scheme often does not perform well on all the data space. In this paper, we propose a new type of SFC called piecewise SFCs, which adopts different mapping schemes for different data subspaces. Specifically, we propose a data structure called Bit Merging tree (BMTree), which can generate data subspaces and their SFCs simultaneously and achieve desirable properties of the SFC for the whole data space. Furthermore, we develop a reinforcement learning based solution to build the BMTree, aiming to achieve excellent query performance. Extensive experiments show that our proposed method outperforms existing SFCs in terms of query performance.

Online Evasive Strategy for Aerial Survey using Sierpinski curve

This paper deals with the aerial survey of a closed region using the Space-Filling curve, particularly Sierpinski curve. The specified region is triangulated, and the Sierpinski curve is used to explore each triangular region. The entire region may have one or more obstacles. An algorithm is presented which suggests evasive maneuver if an obstacle is detected. The algorithm is online; that is, it does not require prior knowledge of the location of obstacles and can be used while the robotic system traverses the designated path. The fractal nature of the Sierpinski curve and simple geometric observations were used to formulate and validate the algorithm. The non-uniform coverage and multiple obstacle problems are also dealt with towards the end.

Borel measurable Hahn-Mazurkiewicz theorem

It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a continuous image of the unit interval. We prove that an assignment, which takes as an input a Peano continuum and produces as an output a continuous mapping whose range is the Peano continuum, can be realized in a Borel measurable way. Similarly, we find a Borel measurable assignment which takes any nonempty compact metric space and assigns a continuous mapping from the Cantor set onto that space. To this end we use the Burgess selection theorem. Finally, a Borel measurable way of assigning an arc joining two selected points in a Peano continuum is found.