Polytope Decomposition Research Articles

LPV Model Based Sensor Fault Diagnosis and Isolation for Permanent Magnet Synchronous Generator in Wind Energy Conversion Systems

This paper deals with the current sensor fault diagnosis and isolation (FDI) problem for a permanent magnet synchronous generator (PMSG) based wind system. An observer based scheme is presented to detect and isolate both additive and multiplicative faults in current sensors, under varying torque and speed. This scheme includes a robust residual generator and a fault estimation based isolator. First, the PMSG system model is reformulated as a linear parameter varying (LPV) model by incorporating the electromechanical dynamics into the current dynamics. Then, polytopic decomposition is introduced for H ∞ design of an LPV residual generator and fault estimator in the form of linear matrix inequalities (LMIs). The proposed gain-scheduled FDI is capable of online monitoring three-phase currents and isolating multiple sensor faults by comparing the diagnosis variables with the predefined thresholds. Finally, a MATLAB/SIMULINK model of wind conversion system is established to illustrate FDI performance of the proposed method. The results show that multiple sensor faults are isolated simultaneously with varying input torque and mechanical power.

Harmonic Mixture Modeling for Efficient Nonlinear Hyperspectral Unmixing

Higher order nonlinear material mixtures provide a good model to explain the effects of physical–chemical phenomena on hyperspectral remote sensing measurements. Therefore, inverting nonlinear effects starting from the measured spectral values is a very challenging yet fundamental task to provide a thorough and reliable characterization of the materials in a scene. In this paper, this task is achieved by inverting a new model for nonlinear hyperspectral mixtures. Specifically, we show that it is possible to effectively unmix hyperspectral data by assuming a harmonic description of the higher order nonlinear combination of the endmembers. The rationale for this model is that the harmonic analysis is able to understand and quantify effects that cannot be effectively described by classic polynomial combinations. Although the model is nonlinear, unmixing is performed by solving a linear system thanks to the recently proposed polytope decomposition (POD). Experimental results show that inverting this model leads to improved performances with respect to the state of the art in terms of endmember abundance estimation both over synthetic and real datasets.

A State Polytope Decomposition Formula

AbstractWe give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.

A Novel Approach for Efficient <formula formulatype="inline"><tex Notation="TeX">$p$</tex></formula>-Linear Hyperspectral Unmixing

Airborne and spaceborne hyperspectral sensors, due to their limited spatial resolution, often record the spectral response of a mixture of materials. In order to extract the abundances of these materials, linear and nonlinear unmixing algorithms have been developed. In this paper, we focus on nonlinear mixing models that are able to model macro- and microscopic scale interactions. Although very useful, these models may be inverted only by means of optimization techniques, typically impossible to be performed in matrix form. Thereby, only nonlinear mixing models that describe macroscopic effects (e.g., two-reflections schemes) are currently considered as they have lower computational costs. On the other hand, this limitation may result in a loss in terms of description accuracy for the images. In this paper, we propose a new approach for nonlinear unmixing that aims at providing excellent reconstruction performance for arbitrary polynomial nonlinearities making use of the polytope decomposition (POD) method. Additionally, POD transforms nonlinear unmixing into a linear problem, and can be easily implemented in high-performance computing architectures. Results using synthetic and real data confirm the effectiveness and accuracy of the proposed framework. To prove its feasibility for fast computational applications, its complexity is analytically derived and compared with real data analysis.

Nonlinear Hyperspectral Unmixing Using Nonlinearity Order Estimation and Polytope Decomposition

Nonlinear hyperspectral unmixing (HSU) plays a key-role in understanding and quantifying the physical-chemical phenomena occurring over geometrically complex fields of view. Nonlinear HSU methods that do not rely on prior knowledge of the ground truth to analyze the scene are especially interesting. However, they can be affected either by overfitting or performance degradation provided by inaccurate setting of unmixing parameters. In this paper, we introduce a new nonlinear HSU architecture which aims at taking advantage of the benefit provided by the combination of polytope decomposition (POD) method together with artificial neural network (ANN)-based learning. Specifically, ANN is able to efficiently estimate the order $p$ of the nonlinearity provided by the given scene even without the thorough knowledge of the ground truth. The ANN-based learning is used to feed the POD in order to deliver accurate unmixing based on a $p$ -linear polynomial model. Experimental results over simulated and real scenes show promising performance of the proposed framework.

Decomposition of polytopes using inner parallel bodies

Let \(P\) be a polytope and \(P_{\lambda }, \lambda \le 0\) an inner parallel body of \(P\), i.e., the polytope constructed by moving the facets of \(P\) inwards at distance \(\left| \lambda \right| \). We study when \(P_{\lambda }\) is a summand of \(P\). We characterize the polytopes \(P\) satisfying that \(P_{\lambda }\) is a summand of \(P_{\mu }\) with \(\lambda <\mu \le 0\). Besides, we provide an explicit decomposition of such polytopes using the so called form bodies of their inner parallel bodies.

Matroid base polytope decomposition II: Sequences of hyperplane splits

This is a continuation of an early paper Chatelain et al. (2011) [3] about matroid base polytope decomposition. We will present sufficient conditions on a matroid M so its base polytope P(M) has a sequence of hyperplane splits. These yield to decompositions of P(M) with two or more pieces for infinitely many matroids M. We also present necessary conditions on the Euclidean representation of rank three matroids M for the existence of decompositions of P(M) into 2 or 3 pieces. Finally, we prove that P(M1⊕M2) has a sequence of hyperplane splits if either P(M1) or P(M2) also has a sequence of hyperplane splits.

Polytope expansion of Lie characters and applications

The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way. Since lattice polytopes are relatively simple, this decomposition is useful, in addition to being more economical than the decomposition into single weights. An expansion of characters into polytope sums follows from the polytope decomposition of weight systems. We study this polytope expansion here. A new, general formula is given for the polytope sums involved. The combinatorics of the polytope expansion are analyzed; we point out that they are reduced from those of the Weyl character formula (described by the Kostant partition function) in an optimal way. We also show that the weight multiplicities can be found easily from the polytope multiplicities, indicating explicitly the equivalence of the two descriptions. Finally, we demonstrate the utility of the polytope expansion by showing how polytope multiplicities can be used in the calculation of tensor product decompositions, and subalgebra branching rules.

Polytopic Decomposition of the Linear Parameter-Varying Model Based on HOSVD

For the solution of an infinite number of LMI in the parameters trajectory of LPV system, a method based on the tensor product transformation is proposed to tranform the LPV system into the convex polytopic structure. Firstly, discretizing the given LPV system within the interzone of the changing parameters and storing it into a tensor, and then using higher order singular value decomposition (HOSVD), discarding smaller and non-zero singular values and their corresponding singular vectors, reconstructing the reduced-rank tensor to obtain a finite number of LTI vertex systems. The final example shows that the convex polytopic system obtained by this method can substitute the original LPV system within the allowed error.

Localization for equivariant cohomology with varying polarization

We develop a localization theorem in equivariant cohomology that generalizes both the localization with respect to a component of the momentum map and the localization with respect to the norm square of the momentum map, in that it allows a more flexible polarization. Our proof uses noncompact cobordisms. Applying a case of our formula to symplectic toric manifolds gives the measure-theoretic version of the classical Brianchon-Gram polytope decomposition, thus answering a question posed by Shlomo Sternberg.

Matroid base polytope decomposition

Let P ( M ) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P ( M ) is a decomposition of the form P ( M ) = ⋃ i = 1 t P ( M i ) where each P ( M i ) is also a matroid base polytope for some matroid M i , and for each 1 ⩽ i ≠ j ⩽ t , the intersection P ( M i ) ∩ P ( M j ) is a face of both P ( M i ) and P ( M j ) . In this paper, we investigate hyperplane splits, that is, polytope decompositions when t = 2 . We give sufficient conditions for M so P ( M ) has a hyperplane split and characterize when P ( M 1 ⊕ M 2 ) has a hyperplane split where M 1 ⊕ M 2 denote the direct sum of matroids M 1 and M 2 . We also prove that P ( M ) has not a hyperplane split if M is binary. Finally, we show that P ( M ) has not a decomposition if its 1-skeleton is the hypercube.

On the union of fat tetrahedra in three dimensions

We show that the combinatorial complexity of the union of n “fat” tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O ( n 2+ε ), for any ε &gt; 0;the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [2003]. Our result extends, in a significant way, the result of Pach et al. [2003] for the restricted case of nearly congruent cubes . The analysis uses cuttings, combined with the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R 3 , having arbitrary side lengths, is O ( n 2+ε ), for any ε &gt; 0 (again, significantly extending the result of Pach et al. [2003]). Finally, our analysis can easily be extended to yield a nearly quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in R 3 .

Root-Based Compositions of Multivariate Polynomials: Structure, Geometric Interpretations, and Decomposition Results#

The concept of a composed product for univariate polynomials has been explored oextensively by Brawley, Brown, Carlitz, Gao, Mills et al. Starting with these fundamental ideas and using fractional power series representation of bivariate polynomials, the current authors Mills and Neuerburg [Mills, D., Neuerburg, K. M. A bivariate analogue to the composed product of polynomials. Algebra Colloquium (to appear)] generalize the univariate results by defining and investigating a bivariate composed sum, composed multiplication, and composed product (based on function composition) for certain classes of bivariate polynomials. In this, the sequel, we extend the generalizations to certain classes of multivariate polynomials, written as f(x 1,…,x n ), over an algebraically closed field of characteristic zero. In particular, we consider a composed sum and composed multiplication defined on the class of quasiordinary polynomials. We then consider what algebraic structure is imposed by these operations. Next, we consider a generalization of the geometry associated with the class of ν-quasiordinary polynomials. Finally, we utilize this geometry to show that an algorithm given by Gao and Lauder [Gao, S., Lauder, A. G. B. (2001). Decomposition of polytopes and polynomials. Discrete Comput. Geom. 26(1):89–104] to determine whether a given bivariate polynomial is absolutely irreducible over a given field can be used to ascertain whether a given trivariate polynomial that factors completely in one variable decomposes according to the composed multiplication operation.

Decomposition of Polytopes and Polynomials

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.

A new procedure for identifying the frame of the convex hull of a finite collection of points in multidimensional space

Consider a set, A of n points in m-dimensional space. The convex hull of these points is a polytope, P , in R m . The frame, F , of these points in the set of extreme points of teh polytope P with F ⊆ A . The problem of identifying the frame plays a central role in optimization theory (redundancy in linear programming and stochastic programming), economics (data envelopment analysis), computational geometry (facial decomposition of polytopes) and statistics (Gastwirth estimators). The standard approach for finding the elements of F consists of solving linear programs with m rows and n − 1 columns; one for each element of A , differing only in the right-hand side vectors. Although enhancements to reduce the total number of linear programs which must ultimately be solved as well as to reduce the number of columns in the technology matrix are known, the utility of this approach is severely limited by its laboriousness and computational demands. We introduce a new procedure also based on solving linear programs but with an important and distinguishing difference. The linear programs begin small and grow larger, but never have more columns than the number of extreme points of P . Experimental results indicate that the time to find the frame using the new procedure is between about one-third and two-thirds that of an enhanced implementation of the established method currently in use.

Cell decomposition of polytopes by bending

If a lineL crosses a polygonP, then bendingP up on both sides ofL yields a 3-polytope whose “upper” boundary projects back to a cell decomposition ofP transverse toL, and to a triangulation in the general case. We generalize this simple idea to polytopes of arbitrary dimension, and use it to answer several questions posed recently about possible decompositions of polytopes and of regions between polytopes.

Espaces metriques constitutes de classes de polytopes convexes lies aux problemes decomposition

We introduce in the space of translation classes of polytopes a metric which is related to the Minkowski decomposition of polytopes. In this metric space the connected components of topological dimension one are shown to coincide with the translation classes of indecomposable polytopes.

Addition and decomposition of convex polytopes

A new addition of convex polytopes is defined and the possibility of representing each polytope as a sum of “standard” polytopes is established