Decomposition Of Space Research Articles

A criterion space decomposition approach to generalized tri-objective tactical resource allocation

We present a tri-objective mixed-integer linear programming model of the tactical resource allocation problem with inventories, called the generalized tactical resource allocation problem (GTRAP). We propose a specialized criterion space decomposition strategy, in which the projected two-dimensional criterion space is partitioned and the corresponding sub-problems are solved in parallel by application of the quadrant shrinking method (QSM) (Boland in Eur J Oper Res 260(3):873–885, 2017) for identifying non-dominated points. To obtain an efficient implementation of the parallel variant of the QSM we suggest some modifications to reduce redundancies. Our approach is tailored for the GTRAP and is shown to have superior computational performance as compared to using the QSM without parallelization when applied to industrial instances.

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

Funnel control for a class of nonlinear infinite-dimensional systems

An adaptive funnel control method is considered for the regulation of the output for a class of nonlinear infinite-dimensional systems on real Hilbert spaces. After a decomposition of the state space and some change of variables related to the Byrnes–Isidori form, it is shown that the funnel controller presented in Berger et al. (2020) achieves the control objective under some assumptions on the nonlinear system dynamics, like well-posedness and Bounded-Input-State Bounded-Output (BISBO) stability. The theory is applied to the regulation of the temperature in a chemical plug-flow tubular reactor whose reaction kinetics are modeled by the Arrhenius nonlinearity. Furthermore a damped sine–Gordon model is shown to fit the required assumptions as well. The theoretical results are illustrated by means of numerical simulations.

Generic Carrier-Based PWM Solution for Series-End Winding PMSM Traction System With Adaptative Overmodulation Scheme

The series-end winding permanent magnet synchronous machine (SW-PMSM) drive system is a new drive topology where the phase windings are connected in series. The generic carrier-based pulsewidth modulation (CBPWM) solution of SW-PMSMs will further promote their applications and avoid the computationally complex space vector PWM in multiphase SW-PMSMs. However, the existing CBPWM solution is complex and designed for symmetrical machines. Moreover, the current approach’s lack of the overmodulation method limits its application in the traction system. This article proposes an improved generic CBPWM solution for SW-PMSMs, which utilizes the vector space decomposition concept to extend the application in both symmetrical and asymmetrical machines. First, we will introduce the concept of the virtual duty axis and divide the modulation process into two steps, where the leg voltage sequence is initialized with reference phase voltages. Then, the virtual duty cycle is utilized to ensure the feasibility of the leg voltage sequence. Furthermore, we will provide an adaptative overmodulation scheme to ensure the stable fundamental currents in the whole speed range and improve the bus voltage utilization. Finally, the experimental results in three machines verify that the proposed solution can obtain better performance with lower switching frequency and have overmodulation capability.

Towards the non-perturbative cosmological bootstrap

We study quantum field theory on a de Sitter spacetime dSd+1 background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group SO(d + 1, 1). As the first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity proper- ties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS2.

General construction and classes of explicit L1-optimal couplings

The main scope of this paper is to give some explicit classes of examples of L1-optimal couplings. Optimal transportation w.r.t. the Kantorovich metric ℓ1 (resp. the Wasserstein metric W1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (Dokl. Akad. Nauk SSSR 115 (1957) 1058–1061) and Sudakov (Proc. Steklov Inst. Math. 141 (1979) 1–178) to occur on rays induced by a decomposition of the basic space (and more generally to higher dimensional decompositions in the case of general measures) induced by the corresponding dual potentials. Several papers have given this kind of structural result and established existence and uniqueness of solutions in varying generality. Since the dual problems pose typically too strong challenges to be solved in explicit form, these structural results have so far been applied for the solution of few particular instances. First, we give a self-contained review of some basic optimal coupling results and we propose and investigate in particular some basic principles for the construction of L1-optimal couplings given by a reduction principle and some usable forms of the decomposition method. This reduction principle, together with symmetry properties of the reduced measures, gives a hint to the decomposition of the space into sectors and via the non crossing property of optimal transport leads to the choice of transportation rays. The optimality of the induced transports is then a consequence of the characterization results of optimal couplings. Then, we apply these principles to determine in explicit form L1-optimal couplings for several classes of examples of elliptical distributions. In particular, we give for the first time a general construction of L1-optimal couplings between two bivariate Gaussian distributions. We also discuss optimality of special constructions like shifts and scalings, and provide an extended class of dual functionals allowing for the closed-form computation of the ℓ1-metric or of accurate lower bounds of it in a variety of examples.

Walsh-based surrogate-assisted multi-objective combinatorial optimization: A fine-grained analysis for pseudo-boolean functions

The aim of this paper is to study surrogate-assisted algorithms for expensive multiobjective combinatorial optimization problems. Targeting pseudo-boolean domains, we provide a fine-grained analysis of an optimization framework using the Walsh basis as a core surrogate model. The considered framework uses decomposition in the objective space, and integrates three different components, namely, (i) an inner optimizer for searching promising solutions with respect to the so-constructed surrogate, (ii) a selection strategy to decide which solution is to be evaluated by the expensive objectives, and (iii) the strategy used to setup the Walsh order hyper-parameter. Based on extensive experiments using two benchmark problems, namely bi-objective NK-landscapes and unconstrained binary quadratic programming problems (UBQP), we conduct a comprehensive in-depth analysis of the combined effects of the considered components on search performance, and provide evidence on the effectiveness of the proposed search strategies. As a by-product, our work shed more light on the key challenges for designing a successful surrogate-assisted multi-objective combinatorial search process.

A dynamic load balancing model coupled with DAC and ISAT for a stochastic turbulent combustion model

Due to the composition-dependent stiffness of chemistry, simulations of reactive turbulent flows may present computational load imbalance among parallel processes when spatial decomposition is used for parallelisation, causing high CPU idle time and waste of computational resources. To increase computational efficiency, a dynamic load balancing (DLB) model is proposed to redistribute computational load among computing cores. The DLB model exploits a decomposition in the mixture fraction space with two dynamic adjusting decomposition strategies to realise load redistribution. The DLB model is suitable for the integration of chemistry on stochastic particles in hybrid Eulerian/Lagrangian turbulent combustion models in which the Eulerian field is conventionally decomposed statically in physical space in a way that balances the computational load for the solution of the Navier-Stokes equation but which does not generally lead to balanced load for the computation of the composition fields. Here it is tested using an OpenFOAM-based platform, mmcFoam, which is a comprehensive object-orientated C++ library for stochastic turbulent combustion modelling. Apart from direct integration (DI) for chemistry, the DLB model is also coupled with dynamic adaptive chemistry (DAC) and in situ adaptive tabulation (ISAT), which allows for extra speedup. The performance of the coupled models is validated and assessed for two laboratory flame conditions that exhibit different levels of computational load imbalance. Overall, the DLB model effectively balances the computational load distribution and increases the effective usage of computing power, shortening the simulation wall time required. Moreover, a strong scaling test is carried out using up to 512 cores. Although all approaches have sub-ideal scalability, the scalability of each with DLB is significantly better than without DLB. While DLB-ISAT has relatively poor scalability compared to the DI- and DAC-based methods, DLB-ISAT still ranks the fastest among the algorithms in all scaling trials.

Partitions of topological spaces and a new club-like principle

We give a new proof of the following theorem due to W. Weiss and P. Komjath: if X X is a regular topological space, with character > b > \mathfrak {b} and X → ( t o p ω + 1 ) ω 1 X \rightarrow (top\, \omega + 1)^{1}_{\omega } , then, for all α > ω 1 \alpha > \omega _1 , X → ( t o p α ) ω 1 X \rightarrow (top\, \alpha )^{1}_{\omega } , fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle ♣ F \clubsuit _{F} , and use it to prove that it is consistent with ¬ C H \neg CH that b \mathfrak {b} is the optimal bound for the character of X X . In [Proc. Amer. Math. Soc. 101 (1987), pp. 767–770], this was obtained using ♢ \diamondsuit .

Numerical Robustness Evaluation of Floating-Point Closed-Loop Control Based on Interval Analysis

Power-electronics-based systems have penetrated into several critical sectors, such as the industry, power generation, energy transmission and distribution, and transportation. In this context, the system’s control, often implemented in real-time processing units, has to meet stringent requirements in terms of safety and repeatability. Given the growing complexity of the implemented algorithms, floating-point arithmetic is being increasingly adopted for high-performance systems. This paper proposes to assess the numerical stability of the control algorithms by means of an interval analysis. The case study of an electric drive is considered, given the wide adoption of such systems and the importance they hold for the safety of the applications. In particular, two different control strategies—the resonant control and the vector space decomposition—are examined, and a sensitivity analysis based on the proposed technique highlights the different characteristics of the two with respect to numerical stability. The proposed method shows how the resonant control is more robust to variations of the controller gain coefficients with respect to the numerical stability, which could make it the preferred choice for mission-critical electric drive control.

A result about the atomic decomposition of Bloch-type space in the polydisk

<abstract><p>The aim of the paper is to obtain a interesting result about the atomic decomposition of Bloch-type space in the polydisk. The existing similar results have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces in the unit ball.</p></abstract>

MULTIPLE PERIODIC SOLUTIONS FOR SUPERQUADRATIC AND SUBQUADRATIC SECOND-ORDER HAMILTONIAN SYSTEMS

In this paper, a class of second-order Hamiltonian systems is studied. Under the assumptions of superquadratic and subquadratic for the nonlinear term, the existence of six periodic solutions and nine periodic solutions is obtained by using the variational method and space decomposition. Finally, two examples are given to verify the feasibility of the new criteria.

On the Activation Space of ReLU equipped Deep Neural Networks

Modern Deep Neural Networks are getting wider and deeper in their architecture design. However, with an increasing number of parameters the decision mechanisms becomes more opaque. Therefore, there is a need for understanding the structures arising in the hidden layers of deep neural networks.In this work, we present a new mathematical framework for describing the canonical polyhedral decomposition in the input space, and in addition, we introduce the notions of collapsing- and preserving patches, pertinent to understanding the forward map and the activation space they induce.The activation space can be seen as the output of a layer and, in the particular case of ReLU activations, we prove that this output has the structure of a polyhedral complex.

Dispersive estimates for kinetic transport equation in modulation spaces

In this paper, we obtain the dispersive estimates for the kinetic transport equation in modulation spaces. To realize this goal, we establish a new and fundamental important unit decomposition in modulation space, and the frequency decomposition is exploited as well.

Evolutionary Learning of Interpretable Decision Trees

In the last decade, reinforcement learning (RL) has been used to solve several tasks with human-level performance. However, there is a growing demand for interpretable RL, i.e., there is the need to understand how a RL agent works and the rationale of its decisions. Not only do we need interpretability to assess the safety of such agents, but also we may need it to gain insights into unknown problems. In this work, we propose a novel optimization approach to interpretable RL that builds decision trees. While techniques that optimize decision trees for RL do exist, they usually employ greedy algorithms or do not exploit the rewards given by the environment. For these reasons, these techniques may either get stuck in local optima or be inefficient. On the contrary, our approach is based on a two-level optimization scheme that combines the advantages of evolutionary algorithms with the benefits of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {Q}$ </tex-math></inline-formula> -learning. This method allows decomposing the problem into two sub-problems: the problem of finding a meaningful decomposition of the state space, and the problem of associating an action to each subspace. We test the proposed method on three well-known RL benchmarks, as well as on a pandemic control task, on which it results competitive with the state-of-the-art in both performance and interpretability.

Decompositions of moduli spaces of vector bundles and graph potentials

Abstract We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

Validity of Whitham's modulation equations for dissipative systems with a conservation law: Phase dynamics in a generalized Ginzburg-Landau system

It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.

Degeneration of a graph describing conformal structure

We consider the cell decomposition of the moduli space of real genus $2$ curves with marked point on the unique real oval. The cells are enumerated by certain graphs, whose weights describe the complex structure on the curve. We show that the collapse of an edge in a graph results in a root-like singularity of the natural map from the weights on graphs to the moduli space of curves. Bibliography: 24 titles.

The molecular decomposition of anisotropic mixed-norm Hardy spaces and their application

The molecular decomposition of anisotropic mixed-norm Hardy spaces and their application

Overmodulation of Six-Phase Cascaded-CSI Using Optimal Harmonics Injection

This paper proposes a new and straightforward method to extend the modulation range of the six-phase cascaded current source inverter (CSI). The proposed technique employs vector space decomposition (VSD) to mitigate the inverter current harmonics and extend the linear modulation region by about 8%. For motor drive applications, increasing the fundamental output component can reflect higher torque production capability for the same drive size, given that thermal limits are not exceeded. The extension can be realized by injecting optimized xy harmonics while keeping the amplitude of the resulting phase currents under the maximum value. The method utilizes a look-up table of optimized values of the injected harmonics to extend the modulation range. The output filter capacitor effects are also studied in this paper, and a selection approach is introduced. Finally, the experimental results of a C-CSI laboratory prototype are presented and discussed to verify the feasibility of the proposed modulation technique.