Domain Partitioning Research Articles

Big problems in spatio-temporal disease mapping: Methods and software

Background and objective: Fitting spatio-temporal models for areal data is crucial in many fields such as cancer epidemiology. However, when data sets are very large, many issues arise. The main objective of this paper is to propose a general procedure to analyze high-dimensional spatio-temporal areal data, with special emphasis on mortality/incidence relative risk estimation.Methods: We present a pragmatic and simple idea that permits hierarchical spatio-temporal models to be fitted when the number of small areas is very large. Model fitting is carried out using integrated nested Laplace approximations over a partition of the spatial domain. We also use parallel and distributed strategies to speed up computations in a setting where Bayesian model fitting is generally prohibitively time-consuming or even unfeasible.Results:Using simulated and real data, we show that our method outperforms classical global models. We implement the methods and algorithms that we develop in the open-source R package bigDM where specific vignettes have been included to facilitate the use of the methodology for non-expert users.Conclusions: Our scalable methodology proposal provides reliable risk estimates when fitting Bayesian hierarchical spatio-temporal models for high-dimensional data.

Mixing Local and Nonlocal Evolution Equations

In this paper, we study the homogenization of a stochastic process and its associated evolution equations in which we mix a local part (given by a Brownian motion with a reflection on the boundary) and a nonlocal part (given by a jump process with a smooth kernel). We consider a sequence of partitions of the (fixed) spacial domain into two parts (local and nonlocal) that are mixed in such a way that they both have positive density at every point in the limit. Under adequate hypotheses on the sequence of partitions, we prove convergence of the associated densities (that are solutions to an evolution equation with coupled local and nonlocal parts in two different regions of the domain) to the unique solution to a limit evolution system in which the local part disappears and the nonlocal part survives but divided into two different components. We also obtain convergence in distributions of the processes associated with the partitions and prove that the limit process has a density pair that coincides with the limit of the densities.

Numerical and Experimental Analysis of the Oil Flow in a Planetary Gearbox

The circular layout and the kinematics of planetary gearboxes result in characteristic oil flow phenomena. The goal of this paper is to apply a new remeshing strategy, based on the finite volume method, on the numerical analysis of a planetary gearbox and its evaluation of results as well as its validation. The numerical results are compared with experimental data acquired on the underlying test rig with high-speed camera recordings. By use of a transparent housing cover, the optical access in the front region of the gearbox is enabled. Different speeds of the planet carrier and immersion depths are considered. A proper domain partitioning and a specifically suited mesh-handling strategy provide a highly efficient numerical model. The open-source software OpenFOAM® is used.

Balanced Domain Partitioning for Software Defined Networks

Software Defined Networking (SDN) has been widely adopted to enable the network programmability by decoupling the controller logic from the data plane. Domain partitioning is essential for SDN-enabled communication networks to partition the oversized network topology into controllable and administrative domains. Existing domain partitioning algorithms failed to take the connectivity, balance constraint, compactness, and traffic cohesion into consideration simultaneously which are required by SDN communications. This paper formulates the domain partitioning problem and proposed a novel balanced network partitioning (BaDoP) algorithm, which can enhance the balance of the partitioned domains, and optimize the compactness and traffic cohesion with connectivity constraint always guaranteed. BaDoP employs Bubble algorithm to find initial domain partitioning, and improves the domain partitioning iteratively by moving border nodes among the partitioned domains guided by the negative cycles or shortest paths on an auxiliary graph which is an abstract view of the current domain partitioning. The simulation results indicates that our BaDoP algorithm outperforms the previous algorithms and can achieve significant improvement for the balance of the partitioned domains.

Echo-state-network-based iterative learning control of distributed systems

The paper proposes an effective modeling and control procedure for the distributed-parameter systems using the echo-state network. The main idea is to reconstruct the spatiotemporal dynamics defined in a given multi-dimensional domain. In the investigated problem positions of both sensors and actuators are fixed allowing to delegate the complex system dynamics to echo-state network. Imposing a proper partitioning of the spatial domain, a specific topology of a neural network is used to form a reservoir capable to follow not only temporal but also spatial dynamics of the system. Based on available historical data, neural network model is initially trained and then used to derive the control law in the framework of iterative learning control. The echo-state network can be retrained after a particular control iterate in order to reduce model uncertainty and to fit it to the current operating conditions as much as possible. The performance of the proposed approach is tested and evaluated on the example of the squared clamped plate control.

2D Minimum Compliance Topology Optimization Based on a Region Partitioning Strategy

This paper presents an extended sequential element rejection and admission (SERA) topology optimization method with a region partitioning strategy. Based on the partitioning of a design domain into solid regions and weak regions, the proposed optimization method sequentially implements finite element analysis (FEA) in these regions. After standard FEA in the solid regions, the boundary displacement of the weak regions is constrained using the numerical solution of the solid regions as Dirichlet boundary conditions. This treatment can alleviate the negative effect of the material interpolation model of the topology optimization method in the weak regions, such as the condition number of the structural global stiffness matrix. For optimization, in which the forward problem requires nonlinear structural analysis, a linear solver can be applied in weak regions to avoid numerical singularities caused by the over-deformed mesh. To enhance the robustness of the proposed method, the nonmanifold point and island are identified and handled separately. The performance of the proposed method is verified by three 2D minimum compliance examples.

СОПОСТАВЛЕНИЕ КОМПЬЮТЕРНЫХ РЕАЛИЗАЦИЙ РАВНОВЕСНОЙ И КОНТИНУАЛЬНОЙ МЕТОДИК МОДЕЛИРОВАНИЯ МНОГОФАЗНЫХ СРЕД НА ПРИМЕРЕ ОДНОМЕРНОГО НЕСТАЦИОНАРНОГО ТЕЧЕНИЯ ГАЗОВЗВЕСИ

The work is devoted to the problems of mathematical modeling of hydrodynamic processes in inhomogeneous media. The purpose of the study is to compare the techniques for modeling the flows of inhomogeneous media. The continuum and equilibrium methods of modeling the dynamics of a gas suspension – a gas with dispersed inclusions – were compared. Comparisons were made between calculations carried out by an equilibrium mathematical model of the dynamics of gas suspensions with numerical implementations of continual mathematical models of the dynamics of gas suspensions with an ideal and viscous heat-conducting carrier medium. Various parameters of the simulated processes obtained by various mathe¬matical models were compared. Materials and methods. The paper presents two methods for modeling the dynamics of multiphase media. In the equilibrium method, an inhomogeneous medium is described as an ideal gas with the calculation of the correction coefficients of gas-dynamic functions, in the continuum modeling method for particles, the concept of “average density” is introduced; thus, for both the carrier phase – gas and the dispersed phase, a system of equations is solved consisting of the density conservation equation, for of the dispersed phase, the “average density” conservation equation, the momentum conservation equations of the mixture components, and the energy conservation equations of the gas and dispersed phase. The systems of equations of motion of single-phase and two-phase media were integrated by a numerical finite-difference method of the second order of accuracy. To suppress numerical oscillations at each time step, the method of nonlinear correction of the grid function was used. The software package for mo¬deling the dynamics of a gas suspension consisted of several components: a subroutine for setting the boundary conditions, a subroutine for the formation of a finite-difference partition of a physical domain, and the main program for calculating the dynamics of an inhomogeneous medium. Results. The discontinuity decay process in a homogeneous gas and in a gas suspension was modeled. Numerical calculations of pressure discontinuity decay in a homogeneous gas and numerical calculations of discontinuity decay in a gas suspension, obtained by the equilibrium modeling technique, were compared with analytical solutions known from the literature. It is determined that the highest propagation velocity of a disturbance in a gas suspension is observed in an equilibrium model, a lower propagation velocity in a continuum model with an ideal medium, and the lowest velocity is observed in a gas suspension with a viscous heat-conducting medium. It was found that for flows with different initial pressure discontinuity intensities, the presence of a dispersed phase in the mixture has a different effect. Conclusions. The regularities obtained demonstrate the features of each of the methods for modeling the dynamics of gases, which can be used in the development of computer models of the dynamics of multiphase media.

Метод триангуляции для приближенного решения вариационных задач нелинейной теории упругости

A variational problem for the minimum of the stored energy functional is considered in the framework of the nonlinear theory of elasticity, taking into account admissible deformations. An algorithm for solving this problem is proposed, based on the use of a polygonal partition of the computational domain by the Delaunay triangulation method. Conditions for the convergence of the method to a local minimum in the class of piecewise affine mappings are found.

Inhomogeneous flow stresses in FSW jointed aluminum alloy sheets inversely identified by FE-VFM

Inhomogeneous plastic flow stresses are identified for the friction stir welded aluminum alloy 6061-T6 sheets using a new finite element-based virtual fields method (FE-VFM). The FE-VFM is an enhanced virtual fields method developed based on the formulations in a common FE scheme. In the present study, the following novel numerical schemes are newly adopted in the FE-VFM: (1) a two-step identification process, (2) partitioning of the whole domain of interest with an interpolation of the constitutive parameter within each subdomain, and (3) the normal distribution type virtual fields. The non-uniform stress method (nUSM) is also considered for comparison. Before applying the developed method to a real experiment, the FE-VFM is verified via numerical feasibility tests using ideal deformation data of the FE simulations conducted with the artificially generated target welding materials. The feasibility tests show that the proposed FE-VFM can calculate the inhomogeneous flow stresses of the welded aluminum sheets with reliable accuracy. In the experimental applications, plastic properties measured by FE-VFM and nUSM are compared in terms of the distributions of yield stress, uniform elongation, and ultimate tensile tests. Also, identified inhomogeneous flow stresses are validated by simulating the two tensile tests loaded parallel and perpendicular to the weld line. The validation reveals that the proposed FE-VFM can give more accurate measurements over the nUSM owing to its ability to capture the mechanical properties in the narrow thermo-mechanically affected zone (TMAZ).

Human RNase P: overview of a ribonuclease of interrelated molecular networks and gene-targeting systems.

The seminal discovery of ribonuclease P (RNase P) and its catalytic RNA by Sidney Altman has not only revolutionized our understanding of life, but also opened new fields for scientific exploration and investigation. This review focuses on human RNase P and its use as a gene-targeting tool, two topics initiated in Altman's laboratory. We outline early works on human RNase P as a tRNA processing enzyme and comment on its expanding nonconventional functions in molecular networks of transcription, chromatin remodeling, homology-directed repair, and innate immunity. The important implications and insights from these discoveries on the potential use of RNase P as a gene-targeting tool are presented. This multifunctionality calls to a modified structure-function partitioning of domains in human RNase P, as well as its relative ribonucleoprotein, RNase MRP. The role of these two catalysts in innate immunity is of particular interest in molecular evolution, as this dynamic molecular network could have originated and evolved from primordial enzymes and sensors of RNA, including predecessors of these two ribonucleoproteins.

Determine the number of unknown targets in the open world from the perspective of bidirectional analysis using Gap statistic and Isolation forest

How to ascertain the number of unknown targets in open world is a crucial issue in generalized evidence theory (GET). To this, limited works have been done, and they may trap in low accuracy, multi-choice issue, complex implementation, etc. To overcome the above limitations, a new method which introduces Isolation forest and Gap statistic into K-means method to perform “bidirectional” analysis is proposed in this paper. According to the “forward” clustering result, isolation-based anomaly detection could achieve the partition of normal domain and abnormal domain, then “reverse” clustering could directly focus on the abnormal domain to pick out those novel clusters using the newly defined novelty validation standard. Further, the defective frame of discernment (FOD) could be completed since the sum of base clusters and novel clusters is the actual target numbers. The proposed method could guarantee the integrity of FOD even the features of the samples are highly-correlated. Various simulation experiments illustrate its effectiveness and wide applicability.

Domain partitioning material point method for simulating shock in polycrystalline energetic materials

Heterogeneous energetic materials (EMs) subjected to mechanical shock loading exhibit complex thermo-mechanical processes which are driven by the high temperature, pressure, and strain rate behind the shock. These lead to spatial energy localization in the microstructure, colloquially known as “hotspots” where chemistry may commence and possibly culminating in detonation. Shock-induced pore collapse is one of the dominant mechanisms by which localization occurs. In order to physically predict the shock sensitivity of energetic materials under these extreme conditions, we formulate a multiplicative crystal plasticity model with key features inferred from molecular dynamics (MD) simulations. Within the framework of thermodynamics, we incorporate the pressure dependence of both monoclinic elasticity and critical resolved shear stress into the crystal plasticity formulation. Other fundamental mechanisms, such as strain hardening and pressure-dependent melting curves, are all inferred from atomic-scale computations performed across relevant intervals of pressure and temperature. To handle the extremely large deformation and the evolving geometry of the self-contact due to pore collapse, we leverage the capabilities of the Material Point Method (MPM) to track the interface via the Lagrangian motion of material points and the Eulerian residual update to avoid the mesh distortion issue. This combination of features enables us to simulate the shock-induced pore collapse and associated hotspot evolution with a more comprehensive physical underpinning, which we apply to the monoclinic crystal β-HMX. Treating MD predictions of the pore collapse as ground truth,head-to-head validation comparisons between MD and MPM predictions are made for samples with identical sample geometry and similar boundary conditions for reverse-ballistic impact speeds ranging from 0.5kms−1 to 2.0kms−1. Comparative studies are performed to reveal the importance of incorporating a frictional contact algorithm, pressure-dependent elastic stiffness, and non-Schmid critical resolved shear stress in the mesoscale model.

Multielement polynomial chaos Kriging-based metamodelling for Bayesian inference of non-smooth systems

This paper presents a surrogate modelling technique based on domain partitioning for Bayesian parameter inference of highly nonlinear engineering models. In order to alleviate the computational burden typically involved in Bayesian inference applications, a multielement Polynomial Chaos Expansion based Kriging metamodel is proposed. The developed surrogate model combines in a piecewise function an array of local Polynomial Chaos based Kriging metamodels constructed on a finite set of non-overlapping subdomains of the stochastic input space. Therewith, the presence of non-smoothness in the response of the forward model (e.g. nonlinearities and sparseness) can be reproduced by the proposed metamodel with minimum computational costs owing to its local adaptation capabilities. The model parameter inference is conducted through a Markov chain Monte Carlo approach comprising adaptive exploration and delayed rejection. The efficiency and accuracy of the proposed approach are validated through two case studies, including an analytical benchmark and a numerical case study. The latter relates the partial differential equation governing the hydrogen diffusion phenomenon of metallic materials in Thermal Desorption Spectroscopy tests.

A novel MPI-based parallel smoothed particle hydrodynamics framework with dynamic load balancing for free surface flow

This paper presents a new Smoothed Particle Hydrodynamics (SPH) parallel framework, which is designed for free surface flows and is scalable on a High Performance Computer (HPC). The framework is accomplished by adopting a Message Passing Interface (MPI) approach with a domain partitioning strategy. A regular background grid is used to partition the entire computational domain and each subdomain is labelled using an index ordering method. Adjacent subdomains can be determined by the index list, and avoid global communications in the particle distribution process. Within the local grid, the grid is divided into an internal grid as well as an interactive grid to identify the particles for which information is to be transferred. The implementation of the dynamic loading balance strategy considers two different ways of determining loading: computation particle numbers and running time. The dynamic load balance strategy repositions neighbouring subdomains based on the local load imbalance between cores. To demonstrate the framework's capacity and distinctive properties, a variety of free surface flow benchmarks are studied. Intensive numerical experiments at various scales are used to assess the performance in detail.

An iterative constrained least squares method for continuous piecewise linear approximation

In this work, a novel iterative least squares method is proposed to approximate nonlinear functions with continuous piecewise linear (CPWL) functions, where the continuity is guaranteed by constrained least squares. A typical least-squares-based method for CPWL fitting consists of two major steps: to perform least squares for a fixed set of breakpoints (i.e. for a specific partition of function domain), and to update breakpoints to reduce the overall fitting error. The proposed method aims to improve the existing CPWL method, the one with canonical representation, by modifying both major steps. Instead of performing ordinary least squares with canonical representation, partitioned least squares and constrained least squares are successively performed to reduce the computational complexity. For the update of breakpoints, an iterative procedure is proposed, where gradient descent with momentum method is employed to improve the convergence characteristics. The advantages of proposed method are illustrated using illustrative examples.

A well-posed multilayer model for granular avalanches: Comparisons with laboratory experiments

Granular avalanches are dangerous phenomena characterized by the rapid gravity-driven motion of granular solids. The complex dynamics of these flows can be effectively modeled by a multilayer approach, which, however, requires particular attention to the derivation of the model equations in order to allow stable solutions. In this work, we use a well-posed multilayer model, in which the μ(I)-rheology is employed and a dilatancy law, depending on the inertial number I, is also taken into account, and systematically compare it with various laboratory experiments. The model, whose well-posedness is guaranteed by a physically based viscous regularization, describes the evolution of a preset number of superimposed granular layers. As the sidewall friction is relevant under most experimental conditions, the model is fitted here with suitable resistance terms. Moreover, non-trivial closures for the mass exchanges are introduced to avoid any unrealistic partitioning of the flow domain during the avalanche evolution, and, hence, guarantee a regular spatial discretization along the normal to flow direction. The velocity fields are compared with different experiments in unsteady state, and comparisons of both velocity and volume fraction profiles are provided with steady uniform flow experiments. The results confirm the good capabilities of the multilayer model and the underlying μ(I)-rheology in capturing the granular flow dynamics. The experimental volume fraction profiles are qualitatively well reproduced by the proposed dilatancy law, while an overestimation is observed only in the upper, more dilute flow region with a thickness of a few grain diameters.

Toward Deep Adaptive Hinging Hyperplanes.

The adaptive hinging hyperplane (AHH) model is a popular piecewise linear representation with a generalized tree structure and has been successfully applied in dynamic system identification. In this article, we aim to construct the deep AHH (DAHH) model to extend and generalize the networking of AHH model for high-dimensional problems. The network structure of DAHH is determined through a forward growth, in which the activity ratio is introduced to select effective neurons and no connecting weights are involved between the layers. Then, all neurons in the DAHH network can be flexibly connected to the output in a skip-layer format, and only the corresponding weights are the parameters to optimize. With such a network framework, the backpropagation algorithm can be implemented in DAHH to efficiently tackle large-scale problems and the gradient vanishing problem is not encountered in the training of DAHH. In fact, the optimization problem of DAHH can maintain convexity with convex loss in the output layer, which brings natural advantages in optimization. Different from the existing neural networks, DAHH is easier to interpret, where neurons are connected sparsely and analysis of variance (ANOVA) decomposition can be applied, facilitating to revealing the interactions between variables. A theoretical analysis toward universal approximation ability and explicit domain partitions are also derived. Numerical experiments verify the effectiveness of the proposed DAHH.

Solution of a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized type

This paper proposes an effective method for solving a nonlocal problem for a system of second-order hyperbolic equations with piecewise constant time argument of generalized type. The method is based on the introduction of functional parameters that are set as the values of the desired solution along the lines of the domain partition with respect to the time variable. With the aid of the functional parameters and new unknown functions, the considered problem is reduced to an equivalent problem for a system of hyperbolic equations with data on the interior partition lines and functional relations with respect to the introduced parameters. We have developed a two-stage procedure to approximately solve the latter problem: firstly, we solve an initial-value problem for a system of differential equations in functional parameters; then, we solve a problem for a system of hyperbolic equations in new unknown functions with data on the interior partition lines. We have obtained some conditions for the convergence of approximate solutions to the exact solution of the problem under study in terms of input data and proved that these conditions guarantee the existence of a unique solution of the equivalent problem. Finally, we have established coefficient conditions for the unique solvability of the nonlocal problem.

Local Lagrangian reduced-order modeling for the Rayleigh-Taylor instability by solution manifold decomposition

The Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophysics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of the Rayleigh-Taylor instability, and therefore, numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretizations need to be extremely fine for traditional numerical methods, and long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of the Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by physical time and by penetration distance. Numerical results are presented to examine the performance of the proposed approaches.

Об интервальном оценивании с высоким порядком минимума функции

The article addresses the problem of two-sided estimation of the minimum of a function of one real variable using interval methods. These methods are widely used for finding the ranges of functions, but their order of accuracy often does not exceed 3 for the functions of one variable and 2 for the case of many variables. The paper proposes a method for finding an interval estimate for the minimum of a function of one variable with any odd order greater than 2. The proposed technique uses so-called functional intervals, the boundaries of which are described by polynomials of even degrees obtained from the Taylor formula for the function under consideration. The article provides a derivation of interval estimates for the minimum, and also proves an estimate for the order of convergence of the width of the resulting interval with the decrease in the size of the domain of the function. An example is given in which estimating the minimum of a function reaches the theoretical order of convergence. The approach developed in this paper allows, in principle, to achieve any odd accuracy order in the interval estimation of the minimum of a function. In practice, this method is convenient to use for the third order of accuracy, since, on the one hand, we need to find an interval estimate for derivatives of high degrees, and, on the other hand, we need to find roots of polynomials of even degrees. The new approach is shown to be useful and practical, as it uses the local Taylor expansion and therefore behaves stably as the width of the function domain decreases. The new approach can be used to improve the performance of interval global optimization algorithms based on adaptive domain partitioning. An increase in the order of estimation accuracy entails the acceleration of these algorithms, as well as a decrease in the number of subintervals generated in the process of partitioning the domain.