Decomposition Of Space Research Articles

Decomposition of Displacement Field into the Irrotational and Solenoidal Component Using Fast Fourier Transform

Purpose: A new code based on Helmholtz decomposition is presented to separate longitudinal (pressure) and transverse (shear) components of a mixed wave field. This algorithm will help isolate shear or pressure components of an elastic wave to further concentrate on each specific wave and its physical characteristics, particularly in medical imaging instrument development and image processing techniques.
 Materials and Methods: Using the combination of Fourier transform and Helmholtz decomposition, first, the mathematical basis of the work is prepared. After reaching a usable formula, this basis is embedded in the Code written in MATLAB program. Then, various test data containing shear and pressure waves were created and fed to the Code to evaluate its ability to decompose the displacements into the shear and pressure waves.
 Results: This new algorithm successfully isolated the transverse and longitudinal wavefront of the mixed wavefield. The Code demonstrated 100% accuracy for separating the shear wave and more than 99% for the pressure wave. Moreover, the background noise was kept under 0.03% in every step.
 Conclusion: The results show that using Helmholtz decomposition in Fourier space on 3D data can help decompose a displacement field into its irrotational and solenoidal components with high accuracy. A weak dependency on wave thickness and contrast was observed, but the algorithm's accuracy never fell below 99%.

A two-space-decomposition-based evolutionary algorithm for large-scale multiobjective optimization

Large-scale multiobjective optimization problems (LSMOPs) pose a great challenge to maintaining the diversity of solutions. However, existing large-scale multiobjective optimization algorithms (MOEAs) prefer to directly use environmental selection methods designed for small-scale optimization problems. These methods are not effective in solving complex LSMOPs. To address this issue, this paper proposes a two-space (decision space and objective space) decomposition (TSD)-based diversity maintenance mechanism. Its main idea is to explicitly decompose the decision space and objective space into a number of subspaces, each of which may contain some Pareto-optimal solutions. Searching for Pareto-optimal solutions in these subspaces may help maintain the diversity of solutions. To this end, a diversity design subspace (DDS) is constructed to decompose the decision space. Then, a large-scale MOEA (MOEA/TSD) is designed by using the proposed TSD-based diversity maintenance mechanism. Experimental studies validate the effectiveness of the proposed TSD mechanism. Compared with nine state-of-the-art large-scale MOEAs on 112 benchmark LSMOPs, our proposed algorithm offers considerable advantages in overall optimization performance. The source code of MOEA/TSD is available at https://github.com/yizhizhede/MOEATSD.

Accelerating heat exchanger design by combining physics-informed deep learning and transfer learning

Recently developed physics-informed deep learning is regarded as a transformative learning philosophy that has been applied in many scientific domains, but such applications are often limited to simulating relatively simple equations and well-defined physics. Here, we propose a systematic framework that can leverage the capabilities of space decomposition, physics-informed deep learning, and transfer learning to accelerate the multi-objective stochastic optimization of a heat exchanger system. In particular, this method seamlessly integrates the strengths of the modified Fourier network for capturing steep gradient variation, the point density adjustment strategy to identify the appropriate size of residual points, as well as the accelerated linear algebra to allow for kernel fusion and just-in-time compilation that enables an acceptable computational expense. The performance is verified by discovering the best-performing geometric design and the corresponding optimal operating conditions of an air cooler system under uncertainty.

Steady-State Bifurcation and Spatial Patterns of a Chemical Reaction System

This paper studies the Lengyel–Epstein chemical reaction system with nonlinear functional response and no-flux boundary conditions. We first investigate the existence of steady-state bifurcation solutions of the system. Then, the stability of bifurcation solutions is analyzed. Meanwhile, some spatial patterns induced by steady-state bifurcation are simulated numerically and depicted graphically. It is well known that the classical bifurcation theory in nonlinear dynamical systems is based on simple eigenvalues. However, this is not always the case. Sometimes, the kernel of an objective operator is of two or more dimensions. For such cases, there is no existing theory to deal with them. In this paper, by using the space decomposition technique and implicit function theorem, we analyze the bifurcation phenomenon of the system in the case of a two-dimensional kernel of some certain objective operator. The results show that the chemical reaction between activator iodide and inhibitor chlorite can proceed stably under certain conditions.

Active learning-based domain adaptive localized polynomial chaos expansion

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows the approximation of functions with strong nonlinearities, discontinuities, and/or singularities that often appear in dynamical systems. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods, as shown in numerical examples, including an engineering dynamical system exhibiting discontinuous response. The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess local variance contributions (Novák et al., 2021). The proposed approach balances both exploitation of the surrogate model and exploration of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method (Marelli et al., 2021) developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents a general framework upon which further extensions and refinements can be based and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

An automated parameter domain decomposition approach for gravitational wave surrogates using hp-greedy refinement

We introduce hp-greedy, a refinement approach for building gravitational wave (GW) surrogates as an extension of the standard reduced basis framework. Our proposal is data-driven, with a domain decomposition of the parameter space, local reduced basis, and a binary tree as the resulting structure, which are obtained in an automated way. When compared to the standard global reduced basis approach, the numerical simulations of our proposal show three salient features: (i) representations of lower dimension with no loss of accuracy, (ii) a significantly higher accuracy for a fixed maximum dimensionality of the basis, in some cases by orders of magnitude, and (iii) results that depend on the reduced basis seed choice used by the refinement algorithm. We first illustrate the key parts of our approach with a toy model and then present a more realistic use case of GWs emitted by the collision of two spinning, non-precessing black holes. We discuss performance aspects of hp-greedy, such as overfitting with respect to the depth of the tree structure, and other hyperparameter dependences. As two direct applications of the proposed hp-greedy refinement, we envision: (i) a further acceleration of statistical inference, which might be complementary to focused reduced-order quadratures, and (ii) the search of GWs through clustering and nearest neighbors.

Detection of Interaction Effects in a Nonparametric Concurrent Regression Model.

Many methods have been developed to study nonparametric function-on-function regression models. Nevertheless, there is a lack of model selection approach to the regression function as a functional function with functional covariate inputs. To study interaction effects among these functional covariates, in this article, we first construct a tensor product space of reproducing kernel Hilbert spaces and build an analysis of variance (ANOVA) decomposition of the tensor product space. We then use a model selection method with the L1 criterion to estimate the functional function with functional covariate inputs and detect interaction effects among the functional covariates. The proposed method is evaluated using simulations and stroke rehabilitation data.

ADVANCES IN UNSTRUCTURED WAVEWATCH III AND APPLICATIONS TO NEARSHORE WAVES

The spectral wave generation and propagation model WAVEWATCH III (WW3) (WW3DG 2019) is undergoing rapid community development to extend capability and nearshore modeling applicability. An option for unstructured grids and implicit solution provides WW3 with the flexibility and efficiency to resolve complex shorelines and high-gradient wave zones to drive nearshore circulation, wave setup, and wave-driven sediment transport with multi-scale spatial coverage over approximately three orders of magnitude. A hybrid approach to parallelization uses spectral partitioning for advection in geographical space and domain decomposition for spectral advection and the source term integration. The advection part of wave action equation is integrated fully implicitly, and a new convergent action limiter derived from Komen et al. (1994) and Hersbach and Janssen (1999) is applied. The model is compatible with community-based coupling infrastructure to facilitate two-way coupling with circulation models for simulating hurricane storm surge and waves. Abdolali et al. (2020) evaluated WW3 enhancements on a large-scale numerical domain employing the new parallelization algorithm and implicit solver for unstructured grids and compared to the existing parallelization algorithm, domain decomposition, and robust explicit numerical solver of WW3 on both structured and unstructured grids. These new capabilities in the wave model push the limitations of the model, including minimum resolution (10s of meters), maximum number of model grid points (~2 million), and computational scalability, as well as better accuracy in contrast to the explicit scheme. This paper describes model enhancements and validation.

Multi-Objective Optimization Algorithm With Adaptive Resource Allocation for Truck-Drone Collaborative Delivery and Pick-Up Services

To efficiently implement the truck-drone collaborative logistics system, we introduce a multi-objective truck-drone collaborative routing problem with delivery and pick-up services (MCRP-DP). A truck collaborating with a fleet of drones serves three types of customers that require delivery, pick-up, and simultaneous delivery & pick-up services, respectively. Different from most of the existing studies where the drone visits only one customer in a flight, we allow the drone to serve another customer requiring pick-up service when it completes a delivery service. Meanwhile, we simultaneously optimize three objectives: transportation costs, waiting time of vehicles (i.e., truck and drone), and service reliability. To solve MCRP-DP, we propose an objective space decomposition-based multi-objective evolutionary algorithm with adaptive resource allocation (ODEA-ARA) In ODEA-ARA, an objective space decomposition strategy is used to maintain the diversity while an adaptive resource allocation strategy is designed to improve convergence. We design an ensemble of relative improvement and relative contribution to assist the resource allocation and a variable neighborhood Pareto local search integrating 7 problem-specific neighborhood structures to improve the solution. Extensive computational experiments are carried out to evaluate the performance of ODEA-ARA. The experimental results show that ODEA-ARA outperforms its competitors. Meanwhile, several useful managerial insights are presented.

Data-Driven-Based Vector Space Decomposition Modeling of Multiphase Induction Machines

For contemporary variable-speed electric drives, the accuracy of the machine's mathematical model is critical for optimal control performance. Basically, phase variables of multiphase machines are preferably decomposed into multiple orthogonal subspaces based on vector space decomposition (VSD). In the available literature, identifying the correlation between states governed by the dynamic equations and the parameter estimate of different subspaces of multiphase IM remains scarce, especially under unbalanced conditions, where the effect of secondary subspaces sounds influential. Most available literature has relied on simple RL circuit representation to model these secondary subspaces. To this end, this paper presents an effective data-driven-based space harmonic model for n-phase IMs using sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover the IM governing equations. Moreover, the proposed approach is computationally efficient, and it precisely identifies both the electrical and mechanical dynamics of all subspaces of an IM using a single transient startup run. Additionally, the derived model can be reformulated into the standard canonical form of the induction machine model to easily extract the parameters of all subspaces based on online measurements. Eventually, the proposed modeling approach is experimentally validated using a 1.5 Hp asymmetrical six-phase induction machine.

A posteriori error analysis for a space‐time parallel discretization of parabolic partial differential equations

AbstractWe construct a space‐time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of “local” problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint‐based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our time parallel algorithm and space‐time parallel algorithm respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.

Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts

The irregular strip-packing problem, also known as nesting or marker making, is defined as the automatic computation of a non-overlapping placement of a set of non-convex polygons onto a rectangular strip of fixed width and unbounded length, such that the strip length is minimized. Nesting methods based on heuristics are a mature technology, and currently, the only practical solution to this problem. However, recent performance gains of the Mixed-Integer Programming (MIP) solvers, together with the known limitations of the heuristics methods, have encouraged the exploration of exact optimization models for nesting during the last decade. Despite the research effort, there is room to improve the efficiency of the current family of exact MIP models for nesting. In order to bridge this gap, this work introduces a new family of continuous MIP models based on a novel formulation of the NoFit-Polygon Covering Model (NFP-CM), called NFP-CM based on Vertical Slices (NFP-CM-VS). Our new family of MIP models is based on a new convex decomposition of the feasible space of relative placements between pieces into vertical slices, together with a new family of valid inequalities, symmetry breakings, and variable eliminations derived from the former convex decomposition. Our experiments show that our new NFP-CM-VS models outperform the current state-of-the-art MIP models. Ten instances are solved up to optimality within one hour for the first time, including one with 27 pieces. Finally, we provide a detailed reproducibility protocol and dataset as supplementary material to allow the exact replication of our models, experiments, and results.

STATE SPACE DECOMPOSITION AND CLASSIFICATION OF TERM STRUCTURE SHAPES IN THE TWO-FACTOR VASICEK MODEL

In this paper, we analyze the shapes of forward curves and yield curves that can be attained in the two-factor Vasicek model. We show how to partition the state space of the model, such that each partition is associated to a particular shape (normal, inverse, humped, etc.). The partitions and the corresponding shapes are determined by the winding number of a single curve with possible singularities and self-intersections, which can be constructed as the envelope of a family of lines. Building on these results, we classify possible transitions between term structure shapes, give results on attainability of shapes conditional on the level of the short rate, and propose a simple method to determine the relative frequency of different shapes of the forward curve and the yield curve.

Study on the energy conversion characteristics in the impeller of USSPAT based on velocity triangle space decomposition

With the development of micro-hydropower generation, traditional pump as turbine (PAT) is difficult to meet the needs of micro-hydropower for developing dispersed and small-flow water resources. Ultra-low specific speed pump turbine (USSPAT) has broad prospects in micro-hydropower generation in remote mountainous areas due to its advantages of larger output power at a small flow rate. In this paper, the dimensionless number ξ to measure the power output capacity of the fluid particle ξ is proposed based on the velocity triangle space decomposition. The impeller is the only energy conversion component in (USSPAT). The study of ξ in the impeller of USSPAT found that the area with the largest output power of the impeller is mainly concentrated in the area with a larger impeller radius, and the overall power output of the impeller increases with the increase of flow rate. The local high-speed jet formed by the rotor-stator interaction (RSI) between the impeller and the volute at the inlet of the region S is the fundamental reason for the NP and jet-wake phenomenon in the impeller. W* = 0.9 is the critical value of NP in region S. The jet wake formed in region A makes the critical value of the region decrease along the flow direction. Therefore, the region A is the core region of the NP inside the impeller. In addition, the secondary flow at the impeller outlet is the main reason for the NP in this region. In addition to 0.6Qd, the secondary flow intensity increases with the increase in flow rate, and WLF inhibits the secondary flow intensity in the wear ring area. However, at a small flow rate (0.6Qd), the WLF gradually enters the mainstream, which weakens the inhibition of the secondary flow in the wear ring region and promotes the secondary flow intensity in the downstream.

On Optimal Convergence Rates for Discrete Minimizers of the Gross–Pitaevskii Energy in Localized Orthogonal Decomposition Spaces

On Optimal Convergence Rates for Discrete Minimizers of the Gross–Pitaevskii Energy in Localized Orthogonal Decomposition Spaces

A framework based on symbolic regression coupled with eXtended Physics-Informed Neural Networks for gray-box learning of equations of motion from data

We propose a framework and an algorithm to uncover the unknown parts of nonlinear equations directly from data. The framework is based on eXtended Physics-Informed Neural Networks (X-PINNs), domain decomposition in space–time, but we augment the original X-PINN method by imposing flux continuity across the domain interfaces. The well-known Allen–Cahn equation is used to demonstrate the approach. The Frobenius matrix norm is used to evaluate the accuracy of the X-PINN predictions and the results show excellent performance. In addition, symbolic regression is employed to determine the closed form of the unknown part of the equation from the data, and the results confirm the accuracy of the X-PINNs based approach. To test the framework in a situation resembling real-world data, random noise is added to the datasets to mimic scenarios such as the presence of thermal noise or instrument errors. The results show that the framework is stable against significant amount of noise. As the final part, we determine the minimal amount of data required for training the neural network. For the systems studied here, the framework is able to predict the correct form of the underlying dynamical equation when at least 50% data is used for training. However, relying solely on 50% of the data is inadequate for accurately predicting the unknown coefficients. To ensure accurate predictions for the coefficients, it was necessary to train the network with a minimum of 60% of the available data.

Research on time series forecasting method of ion concentration in produced fluid

AbstractIn the research on strong alkali‐surfactant‐polymer flooding scaling prediction, the variation characteristics of ion concentration in the produced fluid are obviously consistent with those of scale component content. Consequently, studying the variation tendency of ion concentration in produced fluid is helpful for further revealing the scaling tendency. Given the periodicity and chaos characteristics of the ion concentration data of the produced fluid, an echo state network (ESN) is used to realize the relevant time series forecasting. Simultaneously, to cope with the failure of the ESN in selecting suitable reservoir parameters according to different characteristics of time series, a modified discrete particle swarm optimization algorithm based on objective space decomposition is used to optimize the reservoir parameters. The experimental results indicate that the improved ESN presents the lowest error and is the closest to the target value.

Multi-element polynomial chaos expansion based on automatic discontinuity detection for nonlinear systems

This article focuses on the stochastic modeling of nonlinear systems featuring discontinuities in their response surface. More specifically, original developments are presented to improve the efficiency of multi-element polynomial chaos expansion for such systems. First, an automated detection procedure of the discontinuities is proposed. It relies on an iterative algorithm with a polynomial annihilation edge detection method and support vector machine classification algorithms leading to the representation of the discontinuity as a B-spline curve. Based on this curve, an ad-hoc decomposition of the variable space is considered and an original approach for the application of polynomial chaos expansion on each subdomain yields a robust and versatile way to compute the response surface of the system of interest. The proposed methodology is detailed and applied to several nonlinear academic systems such as a circular discontinuity and the Duffing oscillator including one or two discontinuities in its response surface. Through these applications, it is evidenced that, compared to the classical polynomial chaos and multi-element polynomial chaos expansions, the proposed methodology yields an approximation of the discontinuous responses that is both more accurate and less computationally expensive. The influence of the main parameters of the proposed methodology is also analyzed in details. This parametric analysis underlines the robustness of the methodology and highlights the key parameters in terms of computational cost and accuracy of the discontinuities. The proposed methodology is finally applied to an industrial application, it allows to efficiently compute the surface response of an industrial compressor blade undergoing structural contacts.

Detection of incipient faults in nine-phase machines: Impact of the star winding configuration

Two approaches are commonly used for modelling and control of nine-phase (triple star) fault-tolerant machines with symmetrical and asymmetrical star winding configurations: The Vector Space Decomposition (VSD) and the decentralized d-q modelling. In this paper, it will be shown how the VSD approach used for machines with an Asymmetrical Star Winding Configuration (ASWC), unlike the decentralized d-q one used for machines with a Symmetrical Star Winding Configuration (SSWC), can be helpful for non-intrusive Fault Detection and Diagnosis (FDD) purposes. Supporting MATLAB/Simulink simulations are discussed.

An improved multi-objective framework for the Rich arc routing problem

This article introduces a Rich Arc Routing Problem (RARP). Required arcs are partitioned into given clusters, and each cluster shall be serviced for prescribed times during the planning horizon. There is a time limit that restricts the visits that can be performed in a single day. Each vehicle has to stop at one of intermediate facilities before the end of the specific hours per day. Accordingly, several days may be necessary to complete all tasks, and paths traveled by different vehicles on the same day shall not overlap. The RARP consists of finding a set of routes with (i) the minimum total distance, (ii) the minimum number of vehicles, (iii) the minimum pure working days, and (iv) the temporal service time interval consistency of each cluster. Recently, a Local-Ideal-Points based Autonomous Space Decomposition (LIP-ASD) algorithm has been demonstrated to be a competitive multi-objective framework. However, randomly generating an initial population in each decomposed space decreases the convergence speed of LIP-ASD. In addition, retaining only one individual preserved in each sub-space limits population diversity. In response to these issues, an Improved LIP-ASD (I-LIP-ASD) for RARP is developed. The two improvements of the proposed algorithm are as follows: (i) the initial population in the current sub-space is generated by combining existing nondominated individuals, and (ii) all nondominated solutions will be output from the evolved population in each decomposed sub-space. Finally, we conduct a series of experiments over converted capacitated arc routing problem instances, as well as analyze them under a real-world case to evaluate and demonstrate the effectiveness of the proposed solution algorithm.