Discrete Space Research Articles

A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests on (1) discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and (2) discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses. Numerical experiments complete the theoretical results.

ZeroGrads: Learning Local Surrogates for Non-Differentiable Graphics

Gradient-based optimization is now ubiquitous across graphics, but unfortunately can not be applied to problems with undefined or zero gradients. To circumvent this issue, the loss function can be manually replaced by a "surrogate" that has similar minima but is differentiable. Our proposed framework, ZeroGrads , automates this process by learning a neural approximation of the objective function, which in turn can be used to differentiate through arbitrary black-box graphics pipelines. We train the surrogate on an actively smoothed version of the objective and encourage locality, focusing the surrogate's capacity on what matters at the current training episode. The fitting is performed online, alongside the parameter optimization, and self-supervised, without pre-computed data or pre-trained models. As sampling the objective is expensive (it requires a full rendering or simulator run), we devise an efficient sampling scheme that allows for tractable run-times and competitive performance at little overhead. We demonstrate optimizing diverse non-convex, non-differentiable black-box problems in graphics, such as visibility in rendering, discrete parameter spaces in procedural modelling or optimal control in physics-driven animation. In contrast to other derivative-free algorithms, our approach scales well to higher dimensions, which we demonstrate on problems with up to 35k interlinked variables.

Targeted materials discovery using Bayesian algorithm execution

Rapid discovery and synthesis of future materials requires intelligent data acquisition strategies to navigate large design spaces. A popular strategy is Bayesian optimization, which aims to find candidates that maximize material properties; however, materials design often requires finding specific subsets of the design space which meet more complex or specialized goals. We present a framework that captures experimental goals through straightforward user-defined filtering algorithms. These algorithms are automatically translated into one of three intelligent, parameter-free, sequential data collection strategies (SwitchBAX, InfoBAX, and MeanBAX), bypassing the time-consuming and difficult process of task-specific acquisition function design. Our framework is tailored for typical discrete search spaces involving multiple measured physical properties and short time-horizon decision making. We demonstrate this approach on datasets for TiO2 nanoparticle synthesis and magnetic materials characterization, and show that our methods are significantly more efficient than state-of-the-art approaches. Overall, our framework provides a practical solution for navigating the complexities of materials design, and helps lay groundwork for the accelerated development of advanced materials.

Fast, approximation-free molecular simulation of the SPC/Fw water model using non-reversible Markov chains

In a world made of atoms, computer simulations of molecular systems such as proteins in water play an enormous role in science. Software packages for molecular simulation have been developed for decades. They all discretize Hamilton’s equations of motion and treat long-range potentials through cutoffs or discretization of reciprocal space. This introduces severe approximations and artifacts that must be controlled algorithmically. Here, we bring to fruition a paradigm for molecular simulation that relies on modern concepts in statistics to explore the thermodynamic equilibrium with an exact and efficient non-reversible Markov process. It is free of all discretizations, approximations, and cutoffs. We explicitly demonstrate that this approach reaches a break-even point with traditional molecular simulation performed at high precision, but without any of its approximations. We stress the potential of our paradigm for crucial applications in biophysics and other fields, and as a practical approach to molecular simulation. We set out a strategy to reach our goal of rigorous molecular simulation.

Scalar curvature for metric spaces: Defining curvature for quantum gravity without coordinates

Geometrical properties of spacetime are difficult to study in nonperturbative approaches to quantum gravity like causal dynamical triangulations (CDT), where one uses simplicial manifolds to define the gravitational path integral, instead of Riemannian manifolds. In particular, in CDT one only relies on two mathematical tools, a distance measure and a volume measure. In this paper, we define a notion of scalar curvature, for metric spaces endowed with a volume measure or a random walk, without assuming nor using notions of tensor calculus. Furthermore, we directly define the Ricci scalar, without the need of defining and computing the Riemann or the Ricci tensor . For this, we make use of quantities, like the surface of a geodesic sphere, or the return probability of scalar diffusion processes, that can be computed in these metric spaces, as in a Riemannian manifold, where they receive scalar curvature contributions. Our definitions recover the classical results of scalar curvature when the sets are Riemannian manifolds. We propose two methods to compute the scalar curvature in these spaces, and we compare their features in natural implementations in discrete spaces. The defined generalized scalar curvatures are easily implemented on discrete spaces, like graphs. We present the results of our definitions on random triangulations of a 2D sphere and plane. Additionally, we show the results of our generalized scalar curvatures on the quantum geometries of 2D CDT, where we find that all our definitions indicate a flat ground state of the gravitational path integral. Published by the American Physical Society 2024

Research on Leak Detection of Low-Pressure Gas Pipelines in Buildings Based on Improved Variational Mode Decomposition and Robust Kalman Filtering.

Aiming at the complex characteristics of negative pressure waves in low-pressure pipelines inside of buildings, we proposed an estimation method of pressure fluctuation trends based on the robust Kalman filter and the improved VMD, which can be used for leakage detection. The reconstructed baseline signal can accurately describe the fluctuation trend of the negative pressure wave after the pressure drop, and quantitatively express the characteristic difference between the leakage condition and the gas usage condition. The robust Kalman filter was used to estimate the pressure fluctuations. The parameters of VMD were adaptively calculated based on the WAA and discrete scale space. The trend components contained in the IMFs were separated by a reconstruction based on the Fourier series. Based on the simulation signal, the method can accurately restore the trend component contained in the complex pressure signal. Based on the actual signals, the accuracy of small leakage detection is 96.7% and the accuracy of large leakage detection is 73.3%.

Influence of control parameters on the stabilization of an Euler-Bernoulli flexible beam

In this work, we numerically study the influence of control parameters on the stabilization of a flexible Euler-Bernoulli beam fixed at one end and subjected at the other end to a force control and a punctual moment control proportional respectively to velocity and rotation velocity. First, we analyze the displacement stabilization and the asymptotic behavior of the beam energy using a stable numerical scheme, resulting from the Crank-Nicholson algorithm for time discretization and the finite element method based on the approximation by Hermite's cubic polynomial functions, for discretization in space. Then, by means of the finite element method, we represent the spectrum of the operator associated with this beam problem and we carry out a qualitative study of thelocus of the eigenvalues according to the positive control parameters. From these studies we conclude that rotation velocity control has more effect on the stabilization of the beam compared to velocity control. Finally, this result is confirmed by a sensitivity study on the control parameters involved in the stabilization of the beam.

Aging suppression in Multistrip Multigap Resistive Plate Chambers for high counting rate experiments

A long term operation of Multi-Strip Multi-Gap Resistive Plate Chambers (MSMGRPC) with gas mixtures based on C2H2F4 and SF6 leads to aging effects observed as depositions on the surface of the resistive electrodes, especially in the region of the spacers used for defining the gas gaps between the resistive electrodes. The observed higher noise rate and dark current has a negative impact on the performance of the chamber and leads to an increase of the data volume in a free running readout mode operation. MSMGRPC prototypes designed with a direct gas flow through the gas gaps and minimization of the number of spacers in the active area were developed as mitigation solution. Three new prototypes of different granularities were assembled using fishing line as spacers and investigated for aging effects. Although a significant reduction in the dark current and dark counting rate was evidenced, noise rate localized around the fishing line spacers, even though reduced, still remains. In this paper, a new generation of direct flow chambers based on discrete spacers is presented. The results of their aging investigations show that, even at lower gas flows, the aging effects become negligible.

Gradient-based optimization for quantum architecture search

Quantum Architecture Search (QAS) has shown significant promise in designing quantum circuits for Variational Quantum Algorithms (VQAs). However, existing QAS algorithms primarily explore circuit architectures within a discrete space, which is inherently inefficient. In this paper, we propose a Gradient-based Optimization for Quantum Architecture Search (GQAS), which leverages a circuit encoder, decoder, and predictor. Initially, the encoder embeds circuit architectures into a continuous latent representation. Subsequently, a predictor utilizes this continuous latent representation as input and outputs an estimated performance for the given architecture. The latent representation is then optimized through gradient descent within the continuous latent space based on the predicted performance. The optimized latent representation is finally mapped back to a discrete architecture via the decoder. To enhance the quality of the latent representation, we pre-train the encoder on a substantial dataset of circuit architectures using Self-Supervised Learning (SSL). Our simulation results on the Variational Quantum Eigensolver (VQE) indicate that our method outperforms the current Differentiable Quantum Architecture Search (DQAS).

Intelligent decision-making for a “Three-Variable” frequency-hopping pattern based on OC-CDRL

The frequency hopping pattern of the existing frequency hopping communication system is not designed according to the electromagnetic interference environment, resulting in blind anti-jamming. Therefore, to address this problem, a “three-variable” frequency-hopping pattern is proposed, where the frequency, hopping rate, and instantaneous bandwidth of the frequency-hopping signal vary randomly based on the background electromagnetic interference. The decision-making problem of the “three-variable” frequency-hopping pattern is modeled as a Markov decision process (MDP) by constructing the state-action-reward tuple. The designed frequency varies continuously within a small frequency band selected from a pseudo-random sequence to alleviate the problem of dimension explosion in decision-making. At the same time, discrete values for the hopping rate and instantaneous bandwidth are designed. To solve this MDP problem efficiently, a combined deep reinforcement learning algorithm (OC-CDRL) based on optimistic exploration and conservative estimation is proposed, which combines the features of TD3 and D3QN algorithms and designs the corresponding states, actions, and rewards to deal with continuous and discrete action spaces, respectively. To address the problem that the D3QN algorithm tends to fall into local optimal solutions, an optimistic exploration strategy (OES) for action selection is proposed to improve the degree of exploration. Moreover, the loss function is improved by conservatively estimating state–action pairs outside the experience replay buffer, reducing the overestimation of the optimistic action-value function and increasing the stability and convergence of the algorithm. Comparative simulation results of the algorithms in different electromagnetic interference environments show that the OC-CDRL algorithm effectively avoids most regions with higher interference and has better adaptability and anti-jamming capability.

Microscopic simulation of bicycle traffic flow incorporating cyclists’ heterogeneous dynamics and non-lane-based movement strategies

Cycling as a mode of transport is on an upward trend as a low-emission alternative to driving in urbanized areas nowadays. With the increasing number of cyclists, it is of great importance to assess the capacity of cycling infrastructure in practice. Simulation models are useful tools to investigate bicycle flow performance considering cyclists’ distinct moving behaviors. However, existing bicycle simulation models are restricted by either space discretization, lane-based setup, adaptation from models for car traffic, or complicated calibration requirement in a force-based environment. In addition, cyclists’ decision-making ability in the operational-level cycling behavior are not well-captured in these models. This paper proposes a comprehensible microscopic bicycle simulation model which includes a detailed decision-making process and the ability to simulate continuous-space lateral movement. The model consists of three levels, maneuver decision, movement planning, and physical acceleration. It is able to simulate bicycle flow dynamics in undersaturated traffic conditions on an exclusive bike path. As we do not intend to show the empirical validity of the proposed model, the simulation experiment aims at verifying the model and exploring bicycle flow performance in various scenarios by estimating the fundamental diagrams (FDs). The effect of different path widths on bicycle flow capacity is first explored. Other behavioral factors, including desired speed heterogeneity, overtaking incentive, and safety region size perceived by cyclists, which can potentially influence the shape of the FD are also tested. The model can be further extended to simulate relatively complex cycling behavior with cooperative and anticipative strategies and investigate bicycle flow characteristics in congested traffic conditions.

Neural Solving Uninterpreted Predicates with Abstract Gradient Descent

Uninterpreted predicate solving is a fundamental problem in formal verification, including loop invariant and Constrained Horn Clauses predicate solving. Existing approaches have been mostly in symbolic ways. While achieving sustainable progress, they still suffer from inefficiency and seem unable to leverage the ever-increasing computility such as GPU. Recently, Neural Relaxation has been proposed to tackle this problem. They treat the uninterpreted predicate-solving task as an optimization problem by relaxing the discrete search process into a learning process of neural networks. However, two bottlenecks keep them from being valid. First, relaxed neural networks cannot match the original semantics rigorously; second, the neural networks are difficult to train to reach global optimization. Therefore, this paper presents a novel discrete neural architecture with the Abstract Gradient Decent (AGD) algorithm to directly solve uninterpreted predicates in the discrete hypothesis space. The abstract gradient is for discrete neurons whose calculation rules are designed in an abstract domain. Our approach conforms to the original semantics, and the proposed AGD algorithm can achieve global optimization satisfactorily. We implement Dasp in the Boxes Abstract Domain to solve uninterpreted predicates in the QF-NIA SMT theory. In the experiments, Dasp has outperformed 7 state-of-the-art tools across three predicate synthesis tasks.

Phase field modeling and numerical algorithm for two-phase dielectric fluid flows

We develop a method for modeling and simulating a class of two-phase flows consisting of two immiscible incompressible dielectric fluids and their interactions with imposed external electric fields in two and three dimensions. We first present a thermodynamically-consistent and reduction-consistent phase field model for two-phase dielectric fluids. The model honors the conservation laws and thermodynamic principles, and has the property that, if only one fluid component is present in the system, the two-phase formulation will exactly reduce to that of the corresponding single-phase system. In particular, this model accommodates an equilibrium solution that is compatible with the zero-velocity requirement based on physics. This property leads to a simpler method for simulating the equilibrium state of two-phase dielectric systems. We further present an efficient numerical algorithm, together with a spectral-element (for two dimensions) or a hybrid Fourier-spectral/spectral-element (for three dimensions) discretization in space, for simulating this class of problems. This algorithm computes different dynamic variables successively in an un-coupled fashion, and involves only coefficient matrices that are time-independent in the resultant linear algebraic systems upon discretization, even when the physical properties (e.g. permittivity, density, viscosity) of the two dielectric fluids are different. This property is crucial and enables us to employ fast Fourier transforms for three-dimensional problems. Ample numerical simulations of two-phase dielectric flows under imposed voltage are presented to demonstrate the performance of the method herein and to compare the simulation results with theoretical models and experimental data.

Актуализация подходов к исследованию неоднородности экономического пространства

The article distinguishes the categories of “heterogeneity of economic space” and “interregional social differentiation”, as well as the ways of their assessment. On the basis of the theories of new economic geography and center-periphery theory, the nature of spatial heterogeneity is revealed. The heterogeneity of Russia’s economic space is analyzed through the prism of two key properties of space, i.e., density and connectivity. It is shown that the spatial heterogeneity in terms of population density is growing against the background of decreasing differentiation of space in terms of economic density, measured as GRP per 1 km 2 and volume of investment in fixed capital per 1 km 2 , which can be explained by the fact that the economy is in a phase of recession. A positive correlation between population density and highway density is revealed, which can be explained by the discreteness of space. Based on empirical data, the authors conclude that communicative compression of economic space is characteristic of regions with high population density and economic density and regions with extremely low population density and transport connectivity. In the first case, it is evidence of high business activity in the territory, while in the second case it is a way to increase the availability of services. Based on the comparison of population density and economic density of space, four types of territories were identified: territories of spatial compression; territories with growing population density against the background of high economic density of space; and territories characterized by high population density with relatively low density of economic space; territories with relatively low population density and a trend for its further reduction against the background of high economic density of space. The priorities of spatial development are defined for each type of territory.

Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations

We construct a positivity-preserving Lie–Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate 1/4 in time and rate 1/2 in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.

Adaptive Space Search-based Molecular Evolution Optimization Algorithm.

In drug development process, a significant portion of budget and research time are dedicated to the lead compound optimization procedure in order to identify potential drugs. This procedure focuses on enhancing the pharmacological and bioactive properties of compounds by optimizing their local substructures. However, due to the vast and discrete chemical structure space and the unpredictable element combinations within this space, the optimization process is inherently complex. Various structure enumeration-based combinatorial optimization methods have shown certain advantages. However, they still have limitations. Those methods fail to consider the differences between molecules and struggle to explore the unknown outer search space. In this study, we propose an adaptive space search-based molecular evolution optimization algorithm (ASSMOEA). It consists of three key modules: construction of molecule-specific search space, molecular evolutionary optimization, and adaptive expansion of molecule-specific search space. Specifically, we design a fragment similarity tree in molecule-specific search space, and apply a dynamic mutation strategy in this space to guide molecular optimization. Then we utilize an encoder-encoder structure to adaptively expand the space. Those three modules are circled iteratively to optimize molecules. Our experiments demonstrate that ASSMOEA outperforms existing methods in terms of molecular optimization. It not only enhances the efficiency of the molecular optimization process, but also exhibits a robust ability to search for correct solutions. The code is freely available on the web at https://github.com/bbbbb-b/MEOAFST. Supplementary data are available at Bioinformatics online.

Time-dependent dynamical energy analysis via convolution quadrature

Dynamical Energy Analysis was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions in complex engineering structures. In this paper we introduce the first time-dependent Dynamical Energy Analysis method. Time-domain models are important in numerous applications including sound simulation in room acoustics, predicting shock-responses in structural mechanics and modelling electromagnetic scattering from conductors. The first step is to reformulate Dynamical Energy Analysis in the time-domain by means of a convolution integral operator. We are then able to employ the Convolution Quadrature method to provide a link between the previous frequency-domain implementations of Dynamical Energy Analysis and fully time-dependent solutions by means of the Z-transform. By combining a modified multistep Convolution Quadrature approach for the time discretisation, together with Galerkin and Petrov-Galerkin methods for the space and momentum discretisations, respectively, we are able to accurately track the propagation of high-frequency transient signals through phase-space. The implementation here is detailed for finite two-dimensional spatial domains and we demonstrate the versatility of our approach by performing a range of numerical experiments for regular, non-convex and irregular geometries as well as different types of wave source.

A memory interaction quadratic interpolation whale optimization algorithm based on reverse information correction for high-dimensional feature selection

Feature selection is a key technique for data dimensionality reduction, and there are many challenges in facing the exponential expansion phenomenon of high-dimensional decision space. In order to improve the quality of feature selection, we propose a memory interaction quadratic interpolation whale optimization algorithm based on reverse information correction (RQWOA). First, in order to improve the convergence capability of the Whale Optimization Algorithm (WOA) in high-dimensional space, we propose a quadratic interpolation mechanism with memory information interaction, improve the quality of the datum points used for quadratic interpolation through the proposed memory interaction mechanism, and accelerate the convergence speed and accuracy of the algorithm with the help of the quadratic interpolation on the algorithm exploitation capability. In addition, to address the invalid probability flip in the feature selection subset when mapping to the discrete space, we propose a reverse information correction mechanism to judge and correct each dimension in the feature subset by using the reverse complementary information provided by individuals with poor fitness values in the population, so as to better remove invalid or redundant features and improve the quality of feature selection. Finally, in the 15 high-dimensional feature datasets compared with the comparison methods, the accuracy of RQWOA is improved by 7.49 % on average, and the number of features is compressed by 80.42 % on average, which indicates that the RQWOA algorithm has stronger feature identification and redundancy removal capabilities and can better solve the feature selection problem.

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

Peak response regularization for localization

Deep convolutional neural networks approaches often assume that the feature response has a Gaussian distribution with target-centered peak response, which can be used to guide the target location and classification. Nevertheless, such an assumption is implausible when there is progressive interference from other targets and/or background noise, which produces sub-peaks on the tracking response map and causes model drift. In this paper, we propose a feature response regularization approach for sub-peak response suppression and peak response enforcement and aim to handle progressive interference systematically. Our approach, referred to as Peak Response Regularization (PRR), applies simple-yet-efficient method to aggregate and align discriminative features, which convert local extremal response in discrete feature space to extremal response in continuous space, which enforces the localization and representation capability of convolutional features. Experiments on human pose detection, object detection, object tracking, and image classification demonstrate that PRR improves the performance of image tasks with a negligible computational cost.