Search

Entanglement generation between remote qubit systems is the central tasks for quantum communication. Future quantum networks will have to be compatible with low-loss telecom bands and operate with large separation between qubit nodes. Single-click heralding schemes can be used to increase entanglement rates at the cost of needing an optically phase-synchronized architecture. In this paper we present such a phase synchronization scheme for a metropolitan quantum network, operating in the low-loss telecom L-band. To overcome various challenges such as communication delays and optical power limitations, the scheme consists of multiple tasks that are individually stabilized. We characterize each task, identify the main noise sources, motivate the design choices and describe the synchronization schemes. The performance of each of the tasks is quantified by a transfer-function measurement that investigates the frequency response and feedback bandwidth. Finally we investigate the resulting optical phase stability of the fully deployed system over a continuous period of 10 hours, reporting a short-term stability standard deviation of ${\sigma} \approx 30 \deg$ and a long-term stability of the average optical phase to within a few degrees. The scheme presented served as a key enabling technology for an NV-center based metropolitan quantum link. This scheme is of interest for other quantum network platforms that benefit from an extendable and telecom compatible phase synchronization solution., Comment: 12 pages, 7 figures, 1 table. supplementary materials: 9 pages, 3 figures, 2 tables

In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all $z$ on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of $R(z)$. This task is achieved via M\"{o}bius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of $R(z)$. Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles.

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials. Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to $r$ different measures, with $r \ge 1$. Moreover, they satisfy $(r+2)$--term recurrence relations. In this manuscript, without loss of generality, $r$ is considered equal to $2$. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the \texttt{Matlab} function \texttt{balance.m} does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a \texttt{Matlab} package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem., Comment: 32 pages, 1 figure, 6 tables. Includes an appendix with matlab codes

In the context of the European Erasmus+ project Teaching ASTronomy at the Educational level (TASTE), we investigated the extent to which a learning module at school and a set of activities during a planetarium visit help students to gain insight in the Apparent Motion of the Sun and Stars. Therefore, we have set up a two treatment study with a pretest posttest design. In the four participating countries (Belgium, Germany, Greece and Italy), secondary school students studied the concept of the celestial globe at school using newly designed learning materials. By using a latent class analysis, we identified different classes of student answers on the AMoSS test. We show how students evolve from one class to another between pretest and posttest. Overall the results of the pretest and posttest show that a good understanding of the different aspects of the apparent motion of celestial bodies is difficult to achieve.

The electrification of public transport vehicles offers the potential to relieve city centers of pollutant and noise emissions. Furthermore, electric buses have lower life-cycle greenhouse gas (GHG) emissions than diesel buses, particularly when operated with sustainably produced electricity. However, the heating, ventilation, and air-conditioning (HVAC) system can consume a significant amount of energy, thus limiting the achievable driving range. In this paper, we address the HVAC system in an electric city bus by analyzing the trade-off between the energy consumption and the thermal comfort of the passengers. We do this by developing a dynamic thermal model for the bus, which we simplify by considering it to be in steady state. We introduce a method that is able to quickly optimize the steady-state HVAC system inputs for a large number of samples representative of a year-round operation. A comparison between the results from the steady-state optimization approach and a dynamic simulation reveals small deviations in both the HVAC system power demand and achieved thermal comfort. Thus, the approximation of the system performance with a steady-state model is justified. We present two case studies to demonstrate the practical relevance of the approach. First, we show how the method can be used to compare different HVAC system designs based on a year-round performance evaluation. Second, we show how the method can be used to extract setpoints for online controllers that achieve close-to-optimal performance without any predictive information. In conclusion, this study shows that a steady-state analysis of the HVAC systems of an electric city bus is a valuable approach to evaluate and optimize its performance., Comment: Preprint submitted to Control Engineering Practice

The target stationary distribution problem (TSDP) is the following: given an irreducible stochastic matrix $G$ and a target stationary distribution $\hat \mu$, construct a minimum norm perturbation, $\Delta$, such that $\hat G = G+\Delta$ is also stochastic and has the prescribed target stationary distribution, $\hat \mu$. In this paper, we revisit the TSDP under a constraint on the support of $\Delta$, that is, on the set of non-zero entries of $\Delta$. This is particularly meaningful in practice since one cannot typically modify all entries of $G$. We first show how to construct a feasible solution $\hat G$ that has essentially the same support as the matrix $G$. Then we show how to compute globally optimal and sparse solutions using the component-wise $\ell_1$ norm and linear optimization. We propose an efficient implementation that relies on a column-generation approach which allows us to solve sparse problems of size up to $10^5 \times 10^5$ in a few minutes. We illustrate the proposed algorithms with several numerical experiments., Comment: 29 pages, code available from https://gitlab.com/ngillis/TSDP. In this third version, we have added clarifications, corrections and remarks suggested to us by anonymous reviewers

We investigate rank revealing factorizations of $m \times n$ polynomial matrices $P(\lambda)$ into products of three, $P(\lambda) = L(\lambda) E(\lambda) R(\lambda)$, or two, $P(\lambda) = L(\lambda) R(\lambda)$, polynomial matrices. Among all possible factorizations of these types, we focus on those for which $L(\lambda)$ and/or $R(\lambda)$ is a minimal basis, since they have favorable properties from the point of view of data compression and allow us to relate easily the degree of $P(\lambda)$ with some degree properties of the factors. We call these factorizations minimal rank factorizations. Motivated by the well-known fact that, generically, rank deficient polynomial matrices over the complex field do not have eigenvalues, we pay particular attention to the properties of the minimal rank factorizations of polynomial matrices without eigenvalues. We carefully analyze the degree properties of generic minimal rank factorizations in the set of complex $m \times n$ polynomial matrices with normal rank at most $r< \min \{m,n\}$ and degree at most $d$, and we prove that there are only $rd+1$ different classes of generic factorizations according to the degree properties of the factors and that all of them are of the form $L(\lambda) R(\lambda)$, where the degrees of the $r$ columns of $L(\lambda)$ differ at most by one, the degrees of the $r$ rows of $R(\lambda)$ differ at most by one, and, for each $i=1, \ldots, r$, the sum of the degrees of the $i$th column of $L(\lambda)$ and of the $i$th row of $R(\lambda)$ is equal to $d$. Finally, we show how these sets of polynomial matrices with generic factorizations are related to the sets of polynomial matrices with generic eigenstructures.

We study the tangential interpolation problem for a passive transfer function in standard state-space form. We derive new interpolation conditions based on the computation of a deflating subspace associated with a selection of spectral zeros of a parameterized para-Hermitian transfer function. We show that this technique improves the robustness of the low order model and that it can also be applied to non-passive systems, provided they have sufficiently many spectral zeros in the open right half plane. We analyze the accuracy needed for the computation of the deflating subspace, in order to still have a passive lower order model and we derive a novel selection procedure of spectral zeros in order to obtain low order models with a small approximation error.

We define a compact local Smith-McMillan form of a rational matrix $R(\lambda)$ as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of $R(\lambda)$. We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of $R(\lambda)$ around an arbitrary complex point $\lambda_0$, allows us to compute a compact local Smith-McMillan form of that rational matrix $R(\lambda)$ at the point $\lambda_0$, provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point $\lambda_0$. Numerical tests illustrate the promising performance of the resulting algorithm.

This paper presents an online-capable controller for the energy management system of a parallel hybrid electric vehicle based on model predictive control. Its task is to minimize the vehicle's fuel consumption along a predicted driving mission by calculating the distribution of the driver's power request between the electrical and the combustive part of the powertrain, and by choosing the driving mode, which depends on the vehicle's clutch state. The inclusion of the clutch state in a model predictive control structure is not trivial because the underlying optimization problem becomes a mixed-integer program as a consequence. Using Pontryagin's Minimum Principle and a simplified vehicle model, it is possible to prove that a drive cycle-dependent critical power request Pcrit exists, which uniquely separates the optimal driving mode. Based on this result, a learning algorithm is proposed to determine Pcrit during the operation of the vehicle. The learning algorithm is incorporated into a multi-level controller structure and the working principle of the resulting multi-level learning-based model predictive controller is analyzed in detail using two realistic driving missions. A comparison to the solution obtained by Dynamic Programming reveals that the proposed controller achieves close-to-optimal performance.

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.

In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(\lambda)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column dimension $n$. We give necessary and sufficient conditions for a matrix $G(\la)\in \F[\la]^{\ell\times n}$ to be a GCRD using the Smith normal form of the $m \times n$ compound matrix $P(\lambda)$ obtained by concatenating $P_i(\lambda)$ vertically, where $m=\sum_{i=1}^k m_i$. We also describe the complete set of degrees of freedom for the solution $G(\la)$, and we link it to the Smith form and Hermite form of $P(\la)$. We then give an algorithm for constructing a particular minimum rank solution for this problem when $\F=\C$ or $\R$, using state-space techniques. This new method works directly on the coefficient matrices of $P(\la)$, using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of $P(\la)$.

This paper considers the optimization problem in the form of $\min_{X \in \mathcal{F}_v} f(x) + \lambda \|X\|_1,$ where $f$ is smooth, $\mathcal{F}_v = \{X \in \mathbb{R}^{n \times q} : X^T X = I_q, v \in \mathrm{span}(X)\}$, and $v$ is a given positive vector. The clustering models including but not limited to the models used by $k$-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain $\mathcal{F}_v$ forms a compact embedded submanifold of $\mathbb{R}^{n \times q}$ and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of $\mathcal{F}_v$ are derived. An inexact accelerated Riemannian proximal gradient method that allows adaptive step size is proposed and its global convergence is established. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method.

The notion of root polynomials of a polynomial matrix $P(\lambda)$ was thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020]. In this paper, we extend such a systematic approach to general rational matrices $R(\lambda)$, possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix $R(\lambda)$, without any need to assume that $R(\lambda)$ has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix $R(\lambda)$ and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point $\lambda_0$. If $\lambda_0$ is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.

To estimate the corneal endothelial parameters from specular microscopy images depicting cornea guttata (Fuchs dystrophy), we propose a new deep learning methodology that includes a novel attention mechanism named feedback non-local attention (fNLA). Our approach first infers the cell edges, then selects the cells that are well detected, and finally applies a postprocessing method to correct mistakes and provide the binary segmentation from which the corneal parameters are estimated (cell density [ECD], coefficient of variation [CV], and hexagonality [HEX]). In this study, we analyzed 1203 images acquired with a Topcon SP-1P microscope, 500 of which contained guttae. Manual segmentation was performed in all images. We compared the results of different networks (UNet, ResUNeXt, DenseUNets, UNet++) and found that DenseUNets with fNLA provided the best performance, with a mean absolute error of 23.16 [cells/mm$^{2}$] in ECD, 1.28 [%] in CV, and 3.13 [%] in HEX, which was 3-6 times smaller than the error obtained by Topcon's built-in software. Our approach handled the cells affected by guttae remarkably well, detecting cell edges occluded by small guttae while discarding areas covered by large guttae. Overall, the proposed method obtained accurate estimations in extremely challenging specular images., Comment: 9 pages, 7 figures, 2 tables. Code: https://github.com/jpviguerasguillen/feedback-non-local-attention-fNLA

We analyse the recovery of different roles in a network modelled by a directed graph, based on the so-called Neighbourhood Pattern Similarity approach. Our analysis uses results from random matrix theory to show that when assuming the graph is generated as a particular Stochastic Block Model with Bernoulli probability distributions for the different blocks, then the recovery is asymptotically correct when the graph has a sufficiently large dimension. Under these assumptions there is a sufficient gap between the dominant and dominated eigenvalues of the similarity matrix, which guarantees the asymptotic correct identification of the number of different roles. We also comment on the connections with the literature on Stochastic Block Models, including the case of probabilities of order log(n)/n where n is the graph size. We provide numerical experiments to assess the effectiveness of the method when applied to practical networks of finite size.

We revisit the notion of root polynomials, thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial matrices, and show how they can efficiently be computed in the case of matrix pencils. The staircase algorithm implicitly computes so-called zero directions, as defined in [P. Van Dooren, Computation of zero directions of transfer functions, Proceedings IEEE 32nd CDC, 3132--3137, 1993]. However, zero directions generally do not provide the correct information on partial multiplicities and minimal indices. These indices are instead provided by two special cases of zero directions, namely, root polynomials and vectors of a minimal basis of the pencil. We show how to extract, starting from the block triangular pencil that the staircase algorithm computes, both a minimal basis and a maximal set of root polynomials in an efficient manner. Moreover, we argue that the accuracy of the computation of the root polynomials can be improved by making use of iterative refinement.

We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for {\em any} rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality., Comment: 29 pages

We present quadratically convergent algorithms to compute the extremal value of a real parameter for which a given rational transfer function of a linear time-invariant system is passive. This problem is formulated for both continuous-time and discrete-time systems and is linked to the problem of finding a realization of a rational transfer function such that its passivity radius is maximized. Our new methods make use of the Hybrid Expansion-Contraction algorithm, which we extend and generalize to the setting of what we call root-max problems., Comment: Revision #1

We study the backward stability of running a backward stable eigenstructure solver on a pencil $S(\lambda)$ that is a strong linearization of a rational matrix $R(\lambda)$ expressed in the form $R(\lambda)=D(\lambda)+ C(\lambda I_\ell-A)^{-1}B$, where $D(\lambda)$ is a polynomial matrix and $C(\lambda I_\ell-A)^{-1}B$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $R(\lambda)$, which are highly structured pencils. Backward stable eigenstructure solvers applied to $S(\lambda)$ will compute the exact eigenstructure of a perturbed pencil $\widehat S(\lambda):=S(\lambda)+\Delta_S(\lambda)$ and the special structure of $S(\lambda)$ will be lost. In order to link this perturbed pencil with a nearby rational matrix, we construct a strictly equivalent pencil $\widetilde S(\lambda)$ to $\widehat S(\lambda)$ that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix $\widetilde R(\lambda) = \widetilde D(\lambda)+ \widetilde C(\lambda I_\ell- \widetilde A)^{-1} \widetilde B$, where $\widetilde D(\lambda)$ is a polynomial matrix with the same degree as $D(\lambda)$. Moreover, we bound appropriate norms of $\widetilde D(\lambda)- D(\lambda)$, $\widetilde C - C$, $\widetilde A - A$ and $\widetilde B - B$ in terms of an appropriate norm of $\Delta_S(\lambda)$. These bounds may be inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny. Thus, for this scaled representation, we prove that the staircase and the $QZ$ algorithms compute the exact eigenstructure of a rational matrix $\widetilde R(\lambda)$ that can be expressed in exactly the same form as $R(\lambda)$ with the parameters defining the representation very near to those of $R(\lambda)$. This shows that this approach is backward stable in a structured sense.