Space Of Independent Variables Research Articles

The turbulence models review for aerodynamic, thermal and structural problems in jet engine turbines

In the processes of their processing to obtain finished products, the internal structure and the properties of various scale levels of the material that depend on it change significantly, the prediction of which using empirical methods is extremely costly. For this reason, the development of processing technologies is unthinkable without mathematical models, the central element of which is constitutive relations. A promising approach for constructing the latter is the physical approach, which makes it possible to describe the evolution of the internal structure at various scale levels. As a rule, such models of materials are multilevel and represent a whole class of constitutive models. Of particular interest are elastoplastic models, since they allow calculations with sufficiently large time steps. Most models of this type are based on the Taylor - Bishop - Hill model. Models of this type have a drawback: uncertainty in the choice of a set of active slip systems. This uncertainty is due to the fact that the dislocation shear rates along slip systems are determined from a system of equations, the dimension of which may exceed the dimension of the space of independent variables being sought. In this paper, we study the influence of a random choice of sets of active slip systems in elastoplastic models of the Taylor - Bishop - Hill type on the results of mod-eling the elastoplastic deformation of polycrystals with a face-centered cubic lat-tice. The deviation in the simulation results was considered for a random choice of sets of active slip systems. For this purpose, special metrics were introduced to show the effect of uncertainty on the macrolevel stress tensor and misorientation angles between crystallites. It is shown that with increasing strains, a random choice of sets of active slip systems leads to an almost monotonous increase in differences in the calculation of the macrolevel stress tensor and misorientation angles between grains. This fact means that the problem of uncertainty in the choice of active slip systems inherent in these models affects the simulation re-sults. The indicated discrepancy between the simulation results evidences the need to develop an algorithm to eliminate the noted uncertainty, which is the sub-ject of the forthcoming publication of the authors.

Quantum integrability: Lagrangian 1-form case

A new notion of integrability called the multi-dimensional consistency for the integrable systems with the Lagrangian 1-form structure is captured in the geometrical language for quantum level. A zero-curvature condition, which implies the multi-dimensional consistency, will be a key relation, e.g. Hamiltonian operators. Therefore, the existence of the zero-curvature condition directly leads to the path-independent feature of the mapping, e.g. multi-time evolution in the Schrödinger picture. Another important result is the formulation of the continuous multi-time propagator. With this new type of the propagator, a new perspective on summing all possible paths unavoidably arises as not only all possible paths in the space of dependent variables but also in the space of independent variables must be taken into account. The semi-classical approximation is applied to the multi-time propagator expressing in terms of the classical action and the fluctuation around it. Therefore, the extremum propagator, resulting in path independent feature on the space of independent variables, would guarantee the integrability of the system.

Independent and Related Variable Fault Detection Based on Information Concentrated Variational Auto-encoder

With the rapid development of deep learning methods, the variational auto-encoder (VAE) has been utilized for nonlinear process monitoring. However, most VAE-based methods hardly consider the inner independent and related relationship of each variable. To overcome this problem, a novel VAE named independent and related variable information concentrated variational auto-encoder (IRVIC-VAE) is proposed. To concentrate the independent and related information, a loading weight matrix regularization based on the mutual information of variables with gaussian distribution is introduced so that the variables can separate into two subspaces that contain independent and related information in latent variables. The original data space decomposed via IRVIC-VAE is orthogonal and approximate to normal distribution. For process monitoring, the independent variable space and related variable space are combined to establish two statistics according to Kullback-Leibler divergence and 2-norm. Finally, the performance and effectiveness of IRVIC-VAE are testified by Tennessee Eastman (TE) process.

基于<bold>KPCA</bold>-<bold>SVM</bold>的相关和独立变量故障检测方法

In the real industrial process, some process variables are independent of other variables, a fault detection method of related and independent variable based on kernel principal component analysis and support vector machine (KPCA-SVM) is proposed to detect these independent variables separately from related variables. Firstly, a variable division strategy based on mutual information is applied to divide the process variables into related variables and independent variables by calculating the mutual information between variables. Then, KPCA and SVM models are established in the related variable space and the independent variable space to monitor the test data. Compared with the traditional KPCA and SVM methods, the KPCA-SVM method combines the advantages of KPCA in detecting related variables and SVM methods in detecting independent variables, and improves the fault detection performance of KPCA and SVM methods. Finally, the KPCA-SVM method is applied to the Tennessee-Eastman (TE) industrial process for fault detection, and compared with KPCA, kernel entropy component analysis (KECA) and SVM methods. The results show that the proposed KPCA-SVM method has a good detection effect and improves the detection effect of multiple faults, among which the detection effect of minor fault 5 is significantly improved, which further verifies the effectiveness of the KPCA-SVM method.

An Integrated Method of Bayesian Optimization and D-Optimal Design for Chemical Experiment Optimization

The smart chemical laboratory has recently emerged as a promising trend for future chemical research, where experiment optimization is of vital importance. The traditional Bayesian optimization (BO) algorithm focuses on exploring the dependent variable space while overlooking the independent variable space. Consequently, the BO algorithm suffers from becoming stuck at local optima, which severely deteriorates the optimization performance, especially with bad-quality initial points. Herein, we propose a novel stochastic framework of Bayesian optimization with D-optimal design (BODO) by integrating BO with D-optimal design. BODO can balance the exploitation in the dependent variable space and the exploration in the independent variable space. We highlight the excellent performance of BODO even with poor initial points on the benchmark alpine2 function. Meanwhile, BODO demonstrates a better average objective function value than BO on the benchmark Summit SnAr chemical process, showing its advantage in chemical experiment optimization and potential application in future chemical experiments.

Multitime propagators and the consistency condition

The time evolution of a wave function with $N$ time variables through the Feynman picture of quantum mechanics is derived. However, these evolutions will be compatible if and only if the $N$ Lagrangians satisfy a certain relation called the consistency condition or integrability condition which could be expressed in terms of the Wilson line. This consistency condition violates if there presents the interaction. As a consequence of this consistency condition, the evolution of the wave function gives rise to a key feature called the "path-independent" property on the space of time variables. This would suggest that one must consider all possible paths not only on the space of dependent variables but also on the space of independent variables. In the view of the geometry, this consistency condition can be considered as a zero curvature condition and the multi-time evolution can be treated as a compatible parallel transport on flat space of time variables.

Tree-Structured Regression Model Using a Projection Pursuit Approach

In this paper, a new tree-structured regression model—the projection pursuit regression tree—is proposed. It combines the projection pursuit classification tree with the projection pursuit regression. The main advantage of the projection pursuit regression tree is exploring the independent variable space in each range of the dependent variable. Additionally, it retains the main properties of the projection pursuit classification tree. The projection pursuit regression tree provides several methods of assigning values to the final node, which enhances predictability. It shows better performance than CART in most cases and sometimes beats random forest with a single tree. This development makes it possible to find a better explainable model with reasonable predictability.

Singularities in one-dimensional Euler flows

In this paper, a system of one-dimensional gas dynamics equations is considered. This system is a particular case of Jacobi type systems and has a natural representation in terms of 2-forms on 0-jet space. We use this observation to find a new class of multivalued solutions for an arbitrary thermodynamic state model and discuss singularities of their projections to the space of independent variables for the case of an ideal gas. Caustics and discontinuity lines are found.

Flat Structure on the Space of Isomonodromic Deformations

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

Neural networks catching up with finite differences in solving partial differential equations in higher dimensions

Fully connected multilayer perceptrons are used for obtaining numerical solutions of partial differential equations in various dimensions. Independent variables are fed into the input layer, and the output is considered as solution's value. To train such a network one can use square of equation's residual as a cost function and minimize it with respect to weights by gradient descent. Following previously developed method, derivatives of the equation's residual along random directions in space of independent variables are also added to cost function. Similar procedure is known to produce nearly machine precision results using less than 8 grid points per dimension for 2D case. The same effect is observed here for higher dimensions: solutions are obtained on low density grids, but maintain their precision in the entire region. Boundary value problems for linear and nonlinear Poisson equations are solved inside 2, 3, 4, and 5 dimensional balls. Grids for linear cases have 40, 159, 512 and 1536 points and for nonlinear 64, 350, 1536 and 6528 points respectively. In all cases maximum error is less than $8.8\cdot10^{-6}$, and median error is less than $2.4\cdot10^{-6}$. Very weak grid requirements enable neural networks to obtain solution of 5D linear problem within 22 minutes, whereas projected solving time for finite differences on the same hardware is 50 minutes. Method is applied to second order equation, but requires little to none modifications to solve systems or higher order PDEs.

Quantum variational principle and quantum multiform structure: The case of quadratic Lagrangians

A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property holds for both continuous dynamical systems as well as for discrete ones defined in discrete space-time. Possibly the simplest example in the discrete case is that of a linear quadrilateral lattice equation, which can be viewed as a linearised version of the well-known lattice potential Korteweg-de Vries (KdV) equation. In spite of the linearity, the MDC property is non-trivial in terms of the parameters of the system. The Lagrangian aspects of such equations, and their nonlinear analogues, has led to the notion of Lagrangian multiform structures, where the Lagrangians are no longer scalar functions (or volume forms) but genuine p-forms in a multidimensional space of independent variables. The variational principle involves variations not only with respect to the field variables, but also with respect to the geometry in the space of independent variables. In this paper we consider a quantum analogue of this new variational principle by means of quantum propagators (or equivalently Feynman path integrals). In the case of quadratic Lagrangians these can be evaluated in terms of Gaussian integrals. We study also periodic reductions of the lattice leading to discrete multi-time dynamical commuting mappings, the simplest example of which is the discrete harmonic oscillator, which surprisingly reveals a rich integrable structure behind it. On the basis of this study we propose a new quantum variational principle in terms of multiform path integrals.

Integration of a Linear Equation with Differential Operator, Corresponding to the Main Diagonal in the Space of Independent Variables, and Coefficients, Constant on the Diagonal

We consider an n-th order linear equation with differential operator corresponding to the direction of the main diagonal in the space of independent variables; its coefficients. in general, are variable but constant on the diagonal. We establish conditions on variable eigenvalues which give a possibility to apply some known methods of the theory of for ODEs, when integrating the equation. On this basis, the structures of solutions to the homogeneous equation are determined. We give conditions for existence of multiperiodic solutions of the equations related to variable eigenvalues and initial functions. An integral representation of multiperiodic solution to the inhomogeneous equation is given. We also suggest the concepts of variable frequency and variable period.

Интегрирование линейного уравнения с оператором дифференцирования по главной диагонали пространства независимых переменных и постоянными на ней коэффициентами

Интегрирование линейного уравнения с оператором дифференцирования по главной диагонали пространства независимых переменных и постоянными на ней коэффициентами

A Variational Principle for Discrete Integrable Systems

For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference equations with two independent variables, MDC allows us to define an action on arbitrary 2-dimensional surfaces embedded in a higher dimensional space of independent variables, where the action is not only a functional of the field variables but also the choice of surface. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also with respect to variations of the surface in the space of independent variables. Here we derive the resulting system of generalized Euler-Lagrange equations arising from that principle. We treat the case where the equations are 2 dimensional (but which due to MDC can be consistently embedded in higher-dimensional space), and show that they can be integrated to yield relations of quadrilateral type. We also derive the extended set of Euler-Lagrange equations for 3-dimensional systems, i.e., those for equations with 3 independent variables. The emerging point of view from this study is that the variational principle can be considered as the set of equations not only encoding the equations of motion but as the defining equations for the Lagrangians themselves.

A comparison of linear approaches to filter out environmental effects in structural health monitoring

Abstract This paper discusses the possibility of using the Mahalanobis squared-distance to perform robust novelty detection in the presence of important environmental variability in a multivariate feature vector. By performing an eigenvalue decomposition of the covariance matrix used to compute that distance, it is shown that the Mahalanobis squared-distance can be written as the sum of independent terms which result from a transformation from the feature vector space to a space of independent variables. In general, especially when the size of the features vector is large, there are dominant eigenvalues and eigenvectors associated with the covariance matrix, so that a set of principal components can be defined. Because the associated eigenvalues are high, their contribution to the Mahalanobis squared-distance is low, while the contribution of the other components is high due to the low value of the associated eigenvalues. This analysis shows that the Mahalanobis distance naturally filters out the variability in the training data. This property can be used to remove the effect of the environment in damage detection, in much the same way as two other established techniques, principal component analysis and factor analysis. The three techniques are compared here using real experimental data from a wooden bridge for which the feature vector consists in eigenfrequencies and modeshapes collected under changing environmental conditions, as well as damaged conditions simulated with an added mass. The results confirm the similarity between the three techniques and the ability to filter out environmental effects, while keeping a high sensitivity to structural changes. The results also show that even after filtering out the environmental effects, the normality assumption cannot be made for the residual feature vector. An alternative is demonstrated here based on extreme value statistics which results in a much better threshold which avoids false positives in the training data, while allowing detection of all damaged cases.

Water quality assessment with emphasis in parameter optimisation using pattern recognition methods and genetic algorithm

A non-supervised (k-means) and a supervised (k-Nearest Neighbour in combination with genetic algorithm optimisation, k-NN/GA) pattern recognition algorithms were applied for evaluating and interpreting a large complex matrix of water quality (WQ) data collected during five years (2008, 2010–2013) in the Paute river basin (southern Ecuador). 21 physical, chemical and microbiological parameters collected at 80 different WQ sampling stations were examined. At first, the k-means algorithm was carried out to identify classes of sampling stations regarding their associated WQ status by considering three internal validation indexes, i.e., Silhouette coefficient, Davies-Bouldin and Caliński-Harabasz. As a result, two WQ classes were identified, representing low (C1) and high (C2) pollution. The k-NN/GA algorithm was applied on the available data to construct a classification model with the two WQ classes, previously defined by the k-means algorithm, as the dependent variables and the 21 physical, chemical and microbiological parameters being the independent ones. This algorithm led to a significant reduction of the multidimensional space of independent variables to only nine, which are likely to explain most of the structure of the two identified WQ classes. These parameters are, namely, electric conductivity, faecal coliforms, dissolved oxygen, chlorides, total hardness, nitrate, total alkalinity, biochemical oxygen demand and turbidity. Further, the land use cover of the study basin revealed a very good agreement with the WQ spatial distribution suggested by the k-means algorithm, confirming the credibility of the main results of the used WQ data mining approach.

Multi-dimensional Fokker-Planck equation analysis using the modified finite element method

The Fokker-Planck equation (FPE) is a frequently used tool for the solution of cross probability density function (PDF) of a dynamic system response excited by a vector of random processes. FEM represents a very effective solution possibility, particularly when transition processes are investigated or a more detailed solution is needed. Actual papers deal with single degree of freedom (SDOF) systems only. So the respective FPE includes two independent space variables only. Stepping over this limit into MDOF systems a number of specific problems related to a true multi-dimensionality must be overcome. Unlike earlier studies, multi-dimensional simplex elements in any arbitrary dimension should be deployed and rectangular (multi-brick) elements abandoned. Simple closed formulae of integration in multi-dimension domain have been derived. Another specific problem represents the generation of multi-dimensional finite element mesh. Assembling of system global matrices should be subjected to newly composed algorithms due to multi-dimensionality. The system matrices are quite full and no advantages following from their sparse character can be profited from, as is commonly used in conventional FEM applications in 2D/3D problems. After verification of partial algorithms, an illustrative example dealing with a 2DOF non-linear aeroelastic system in combination with random and deterministic excitations is discussed.

Related and independent variable fault detection based on KPCA and SVDD

This paper proposes a new independent and related variable monitoring based on kernel principal component analysis (KPCA) and support vector data description (SVDD) algorithm. Some process variables are considered independent from other variables and the monitoring of independent and related variables should be performed separately. First, an independent variable division strategy based on mutual information is presented. Second, SVDD and KPCA methods are adopted to monitor independent variable space and related variable space, respectively. Finally, a general statistic is built according to the monitoring results of SVDD and KPCA. The proposed KPCA–SVDD method considers the related and independent characters of variables. This method combines the advantages of KPCA in managing nonlinear related variables and those of SVDD in handling independent variables. A numerical system and the Tennessee Eastman process are used to examine the efficiency of the proposed method. Simulation results have proved the superiority of KPCA–SVDD method.

Numerical Solutions of Elliptic Partial Differential Equations by Using Finite Volume Method

Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

Symmetries in Lagrangian Field Theory

By generalizing the cosymplectic setting for time-dependent Lagrangian mechanics, we propose a geometric framework for the Lagrangian formulation of classical field theories with a Lagrangian depending on the independent variables. For that purpose we consider the first order jet bundles $J^1\pi$ of a fiber bundle $\pi:E\to {\mathbb R}^k$ where ${\mathbb R}^k$ is the space of independent variables. Generalized symmetries of the Lagrangian are introduced and the corresponding Noether Theorem is proved.