Regular Decomposition Research Articles

A soft intelligent allocation-based hybrid model for uncertain complex time series forecasting

The generalization ability is one of the most important and the most influential factors for electing forecasting models, managing future events, and making decisions. In the literature, numerous hybrid models have been presented in order to improve the accuracy as well as generalization ability of single forecasting approaches. The main aim of these hybrid models is often to use more different and/or more individual models in order to capture all existing patterns and structures in the data, more completely; and consequently improving the accuracy and generalization. Although, it can be generally demonstrated that increasing the number of components will not decrease the performance of hybrid models in the training, it will not necessarily improve the generalizability, especially in complex and uncertain environments. In this paper, an efficient allocation strategy is proposed in order to assign the underlying data set to its appropriateness component for increasing generalizability as well as decreasing computational costs. In this paper, a novel soft intelligent hybrid model is developed using the allocation strategy for assign different IMFs to appropriateness certain linear, certain nonlinear, uncertain linear, and uncertain nonlinear components in decomposition based forecasting problems. The main purpose of this classification is to reduce the probability of the over-fitting problem and consequently to increase the generalization ability, in additional of deceasing the computational costs. Moreover, in this paper, an optimal weighting technique is proposed to find the relative importance of each component in order to yield the most accurate final predictions. On the other hand, the main motivation of the paper, in contrast to the regular decomposition based hybrid models in which components are blindly assigned to the models, is to develop a logical process to allocate components to the most appropriate model as well as optimally weighting them. Empirical results of crude oil prices and wind power forecasting indicate that despite of better performance of traditional parallel hybrid models in the training sample, the generalization ability of the proposed model in test sample is significantly higher than those hybrid models as well as its components in all considered benchmarks. The proposed model can averagely improve 64.86%, 61.93%, and 52.00% the accuracy of single linear, single nonlinear, and traditional hybrid non-decomposition; and 41.37%, 35.16%, and 32.63% the performance of single linear, single nonlinear, and traditional hybrid decomposition based models, respectively.

On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems

We consider well-known decomposition techniques for multistage stochastic programming and a new scheme based on normal solutions for stabilizing iterates during the solution process. The given algorithms combine ideas from finite perturbation of convex programs and level bundle methods to regularize the so-called forward step of these decomposition methods. Numerical experiments on a hydrothermal scheduling problem indicate that our algorithms are competitive with the state-of-the-art approaches such as multistage regularized decomposition, nested decomposition and stochastic dual dynamic programming.

Star complementary strongly regular decompositions of strongly regular graphs

In this paper we consider strongly regular graphs G which admit a decomposition into two strongly regular graphs such that one of them, say H, is a star complement of G. Relations between the parameters of graphs paired in this way are determined. We also consider several particular types of strongly regular graph and determine whether such graphs can appear in the role of G or H.

Regular Decomposition of Large Graphs: Foundation of a Sampling Approach to Stochastic Block Model Fitting

We analyze the performance of regular decomposition, a method for compression of large and dense graphs. This method is inspired by Szemerédi’s regularity lemma (SRL), a generic structural result of large and dense graphs. In our method, stochastic block model (SBM) is used as a model in maximum likelihood fitting to find a regular structure similar to the one predicted by SRL. Another ingredient of our method is Rissanen’s minimum description length principle (MDL). We consider scaling of algorithms to extremely large size of graphs by sampling a small subgraph. We continue our previous work on the subject by proving some experimentally found claims. Our theoretical setting does not assume that the graph is generated from a SBM. The task is to find a SBM that is optimal for modeling the given graph in the sense of MDL. This assumption matches with real-life situations when no random generative model is appropriate. Our aim is to show that regular decomposition is a viable and robust method for large graphs emerging, say, in Big Data area.

Noise Removal Method for Moving Images Using 3-D and Time-Domain Total Variation Regularization Decomposition

Noise Removal Method for Moving Images Using 3-D and Time-Domain Total Variation Regularization Decomposition

Parametric Order Reduction of Nonuniform Interconnect Circuits Using Integrated Congruence Transform

A parameterized model-order reduction algorithm for nonuniform high-speed interconnect with varying design parameters is presented. The constructed reduced-order model (ROM) has a direct time-domain representation, and thus, discretization techniques are not required. The key idea of the new algorithm is to find a common-projection matrix using the integrated-congruence transform (ICT) method that can be used to construct an ROM for any value of the varying parameters. However, the calculated projection matrices using the ICT method are functions of the position along the lines, and thus, finding a common-projection matrix using regular singular-value decomposition (SVD) is not applicable. To address this issue, a Hilbert-space-based SVD is proposed to extract the common-projection matrix.

An Approached Solution of Wave Equation with Cubic Damping by Homotopy Perturbation Method (HPM), Regular Pertubation Method (RPM) and Adomian Decomposition Method (ADM)

In this study, we consider the wave equation with cubic damping with its initial conditions. Homotopy Perturbation Method (HPM), Regular Pertubation Method (RPM) and Adomian decomposition Method (ADM) are applied to this equation. Then, the solution yielding the given initial conditions is gained. Finally, the solutions obtained by each method are compared.

F4 , E6 and G2 exceptional gauge groups in the vacuum domain structure model

Using vacuum domain structure model, trivial static potentials in various representations of $F_4$, $E_6$ and $G_2$ exceptional groups are calculated by means of the unit center element. Due to the absence of the non-trivial center elements, the potential of every representation is screened at far distances. However, the linear part is observed at intermediate quark separations which is investigated by the decomposition of the exceptional group to its maximal subgroups. Comparing the group factor of the super-group with the corresponding one obtained from the non-trivial center elements of $SU(3)$ subgroup, shows that $SU(3)$ is not the direct cause of temporary confinement in any of the exceptional groups. However, the trivial potential obtained from the group decomposition to the $SU(3)$ subgroup is the same as the potential of the super-group itself. In addition, any regular or singular decomposition to the $SU(2)$ subgroup which produces the Cartan generator with the same elements as $h_1$, in any exceptional group, leads to the linear intermediate potential of the exceptional gauge groups. The other $SU(2)$ decompositions with the Cartan generator different from $h_1$, are still able to describe the linear potential if the number of $SU(2)$ non-trivial center element which emerge in the decompositions is the same. As a result, it is the center vortices quantized in terms of non-trivial center element of the $SU(2)$ subgroup which give rise to the intermediate confinement in the static potentials.

Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum(A__,B__)⊕(A1,B1)⊕…⊕(At,Bt) that is congruent to (A,B), in which (A__,B__) is a pair of nonsingular matrices and (A1,B1),…,(At,Bt) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of (A,B) under congruence over an algebraically closed field of characteristic not 2.

A strongly regular decomposition of the complete graph and its association scheme

For any positive integer m, the complete graph on 22m(2m+2) vertices is decomposed into 2m+1 commuting strongly regular graphs, which give rise to a symmetric association scheme of class 2m+2−2. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.

Regular decomposition and a framework of order reduced methods for fourth order problems

This paper is devoted to the construction of order reduced method of fourth order problems. A constructive framework is presented such that a problem on the high-regularity space can be deduced to an equivalent system on three low-regularity spaces which are connected by a regular decomposition corresponding to a decomposition of the regularity of the high order space. The generated numerical schemes based on the deduced problems can be of lower complicacy, and the framework is fit for various fourth order problems. Three fourth order problems are then discussed under the framework, including one in two dimension and two in three dimension. They are each corresponding to a regular decomposition, and thus are discretised based on the discretised analogues of the regular decomposition; optimal error estimates are given.

New order bounds in differential elimination algorithms

We present a new upper bound for the orders of derivatives in the Rosenfeld–Gröbner algorithm under weighted rankings. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used to test for membership in the given radical differential ideal. In particular, this algorithm allows us to determine whether a system of polynomial PDEs is consistent.In the case of one derivation, such a bound was given by Golubitsky et al. (2008). The only known bound in the case of several derivations was given by the authors of the present paper in 2016. The bound was achieved by associating to the algorithm antichain sequences whose lengths can be bounded using the results of León Sánchez and Ovchinnikov (2016). In the present paper, the above result by the current authors is generalized and significantly improved.

The number of regular control surfaces of toric patch

Through a rational map, a toric patch is defined associated to a lattice polygon, which is the convex of a given finite integer lattice points set A . The classical rational Bézier curves, rational triangular and tensor-product patches are special cases of toric patches. One of the geometric meanings of toric patch is that the limiting of the patch is its regular control surface, when all weights tend to infinity. In this paper, we study the number of regular decompositions of A , and the relationship between regular decompositions and the corresponding secondary polytope. What is more, we indicate that the number of regular control surfaces of toric patch associated with A is equal to the number of regular decompositions of A .

Convergence of Adaptive Mixed Finite Element Methods for the Hodge Laplacian Equation: Without Harmonic Forms

Finite element exterior calculus (FEEC) has been developed as a systematic framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. This paper is devoted to analyzing the convergence of adaptive mixed finite element methods for the Hodge Laplacian equations based on FEEC without considering harmonic forms. A residual type a posteriori error estimate is obtained by using the Hodge decomposition, the regular decomposition, and bounded commuting quasi-interpolants. An additional marking strategy is added to ensure the quasi-orthogonality, based on which the convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough.

Regularly decomposable tensors and classical spin states

A spin-$j$ state can be represented by a symmetric tensor of order $N=2j$ and dimension $4$. Here, $j$ can be a positive integer, which corresponds to a boson; $j$ can also be a positive half-integer, which corresponds to a fermion. In this paper, we introduce regularly decomposable tensors and show that a spin-$j$ state is classical if and only if its representing tensor is a regularly decomposable tensor. In the even-order case, a regularly decomposable tensor is a completely decomposable tensor but not vice versa; a completely decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the odd-order case, the first row tensor of a regularly decomposable tensor is regularly decomposable and its other row tensors are induced by the regular decomposition of its first row tensor. We also show that complete decomposability and regular decomposability are invariant under orthogonal transformations, and that the completely decomposable tensor cone and the regularly decomposable tensor cone are closed convex cones. Furthermore, in the even-order case, the completely decomposable tensor cone and the PSD tensor cone are dual to each other. The Hadamard product of two completely decomposable tensors is still a completely decomposable tensor. Since one may apply the positive semi-definite programming algorithm to detect whether a symmetric tensor is an SOS tensor or not, this gives a checkable necessary condition for classicality of a spin-$j$ state. Further research issues on regularly decomposable tensors are also raised.

Ideal regular languages and strongly connected synchronizing automata

We introduce the notion of a reset left regular decomposition of an ideal regular language, and we prove that the category formed by these decompositions with the adequate set of morphisms is equivalent to the category of strongly connected synchronizing automata. We show that every ideal regular language has at least one reset left regular decomposition. As a consequence, every ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal from which follows a reformulation of Černý's conjecture in purely language theoretic terms. Finally, we present and characterize a subclass of ideal regular languages for which a better upper bound for the reset decomposition complexity holds with respect to the general case.

Thermal Decomposition of Supersaturated Ti<sub>1-x</sub>Al<sub>x</sub>N Solid Solution Synthesized by High-Energy Milling

Supersaturated titanium-aluminum nitride (Ti1-xAlxN) is a very attractive material for a wide range of applications due to its high oxidation and wear resistance accompanied by high strength, hardness, thermal conductivity and thermal shock resistance. Currently, its applications are limited to coatings obtained by physical or chemical deposition. Bulk materials based on Ti1-xAlxN may be fabricated by powder metallurgy approach using powders synthesized by high-energy ball milling (HEBM), which composition corresponds to supersaturated Ti1-xAlxN solid solution. In the present study, thermal stability of the supersaturated Ti1-xAlxN solid solution was investigated. According to the quasi-binary TiN-AlN phase diagram, constructed using density functional theory (DFT) analysis, the concentration ranges, where decomposition takes place through spinodal decomposition or through nucleation and growth, were determined. Experimental study on thermal stability of solid Ti1-xAlxN solution powder was conducted by means of differential scanning calorimetry (DSC), Brunauer-Emmited-Teller (BET) and XRD. The results indicated that spinodal decomposition of Ti1-xAlxN starts at 800°C, while at temperature higher than 1300°C regular decomposition (nucleation and growth) is occur.

Almost Regular Decompositions of a Graph

Let k ≤ 8 be a positive integer, and let G be a graph on n vertices such that the degree of each its vertex is at least $$ \frac{k-1}{k} $$ . It is proved that the vertex set of G can be partitioned into several cliques of size at most k so that for each positive integer k 0 < k, there is at most one clique of size k 0 in this partition. Bibliography: 2 titles.

Performance comparison between state-of-the-art point-cloud based collision detection approaches on the CPU and GPU

Abstract: We present two fundamentally different approaches to detect collisions between two point clouds and compare their performance on multiple datasets. A collision between points happens if they are closer to each other than a given threshold radius. One approach utilizes the main CPU with a k-d tree datastructure to efficiently carry out fixed range searches around points in 3D while the other mainly executes on a GPU using a regular grid decomposition technique implemented in the CUDA framework. We will show how massively parallel 3D range searches on a grid based datastructure on a GPU performs similarly well as a tree based approach on the CPU with orders of magnitude less parallelization. We also show how each method scales with varying input sizes and how they perform differently well depending on the spatial structure of the input data.

Efficiency of a two-step upscaling method for permeability evaluation at Darcy and pore scales

This work presents a new subdivision method to upscale absolute permeability fields. This process, called two-step method, consists in (i) solving micro-scale equations on subdomains obtained from the full domain regular decomposition and (ii) solve a second upscaling with Darcy’s law on the permeability fields obtained in the first step. The micro-scale equations used depend on the case studied. The two-step upscaling process is validated on randomly generated Darcy-scale permeability fields by measuring the numerical error induced by upscaling. The method is then applied to real domains obtained from sandstone micro-tomographic images. The method specificities due to pore-space structure are discussed. The main advantage of the two-step upscaling method resides in the drastic reduction of computational costs (CPU time and memory usage) while maintaining a numerical error similar to that of other upscaling procedures. This new upscaling method may improve permeability predictions by the use of finer meshes or larger sample volumes.