arXiv subjects

Punit Sharma

Publications and source records attributed to Punit Sharma.

32 records · Page 2Linked to original sources

arXiv2021

Tensionless Tales: Vacua and Critical Dimensions

Recently, a careful canonical quantisation of the theory of closed bosonic tensionless strings has resulted in the discovery of three separate vacua and hence three different quantum theories that emerge from this single classical tensionless theory. In this note, we perform lightcone quantisation with the aim of determination of the critical dimension of these three inequivalent quantum theories. The satisfying conclusion of a rather long and tedious calculation is that one of vacua does not lead to any constraint on the number of dimensions, while the other two give $D=26$. This implies that all three quantum tensionless theories can be thought of as consistent sub-sectors of quantum tensile bosonic closed string theory.

hep-th↗

arXiv2021

BMS Field Theories and Weyl Anomaly

Two dimensional field theories with Bondi-Metzner-Sachs symmetry have been proposed as duals to asymptotically flat spacetimes in three dimensions. These field theories are naturally defined on null surfaces and hence are conformal cousins of Carrollian theories, where the speed of light goes to zero. In this paper, we initiate an investigation of anomalies in these field theories. Specifically, we focus on the BMS equivalent of Weyl invariance and its breakdown in these field theories and derive an expression for Weyl anomaly. Considering the transformation of partition functions under this symmetry, we derive a Carrollian Liouville action different from ones obtained in the literature earlier.

hep-th↗

arXiv2021

On the non-symmetric semidefinite Procrustes problem

In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices $X,Y \in \mathbb{R}^{n,m}$, find the matrix $A \in \mathbb{R}^{n,n}$ that minimizes the Frobenius norm of $AX-Y$ and which is such that $A+A^T$ is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where $A$ is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As for the symmetric case, we first show that the NSPSDP problem can be reduced to a smaller NSPSDP problem that always has a unique solution and where the matrix $X$ is diagonal and has full rank. Then, an efficient semi-analytical algorithm to solve the NSPSDP problem is proposed, solving the smaller and well-posed problem with a fast gradient method which guarantees a linear rate of convergence. This algorithm is also applicable to solve the complex NSPSDP problem, where $X,Y \in \mathbb{C}^{n,m}$, as we show the complex NSPSDP problem can be written as an overparametrized real NSPSDP problem. The efficiency of the proposed algorithm is illustrated on several numerical examples.

math.OC↗

arXiv2021

Characterization of the dissipative mappings and their application to perturbations of dissipative-Hamiltonian systems

In this paper, we find necessary and sufficient conditions to identify pairs of matrices $X$ and $Y$ for which there exists $Δ\in \mathbb C^{n,n}$ such that $Δ+Δ^*$ is positive semidefinite and $ΔX=Y$. Such a $Δ$ is called a dissipative mapping taking $X$ to $Y$. We also provide two different characterizations for the set of all dissipative mappings, and use them to characterize the unique dissipative mapping with minimal Frobenius norm. The minimal-norm dissipative mapping is then used to determine the distance to asymptotic instability for dissipative-Hamiltonian systems under general structure-preserving perturbations. We illustrate our results over some numerical examples and compare them with those of Mehl, Mehrmann and Sharma (Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations, SIAM J. Mat. Anal. Appl.\ 37 (4): 1625-1654, 2016).

math.OC↗

arXiv2019

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to tackle this problem. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library.

math.OC↗

arXiv2019

Approximating the nearest stable discrete-time system

In this paper, we consider the problem of stabilizing discrete-time linear systems by computing a nearby stable matrix to an unstable one. To do so, we provide a new characterization for the set of stable matrices. We show that a matrix $A$ is stable if and only if it can be written as $A=S^{-1}UBS$, where $S$ is positive definite, $U$ is orthogonal, and $B$ is a positive semidefinite contraction (that is, the singular values of $B$ are less or equal to 1). This characterization results in an equivalent non-convex optimization problem with a feasible set on which it is easy to project. We propose a very efficient fast projected gradient method to tackle the problem in variables $(S,U,B)$ and generate locally optimal solutions. We show the effectiveness of the proposed method compared to other approaches.

math.OC↗

arXiv2019

On approximating the nearest Ω-stable matrix

In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω, within the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips and disks. We refer to this problem as the nearest Ω-stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time-invariant systems. In order to achieve this goal, we parametrize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.

math.OC↗

arXiv2018

A note on approximating the nearest stable discrete-time descriptor system with fixed rank

Consider a discrete-time linear time-invariant descriptor system $Ex(k+1)=Ax(k)$ for $k \in \mathbb Z_{+}$. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system $\hat E x(k+1)= \hat A x(k)$ with $\text{rank}(\hat E)=r$. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.

math.OC↗

arXiv2017

Structured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control

Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and symmetry-structure-preserving perturbations. It is shown that these eigenvalue and eigenpair backward errors are sometimes significantly larger than the corresponding backward errors that are obtained under perturbations that ignore the special structure of the pencil.

math.NA↗

arXiv2017

Finding the nearest positive-real system

The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by $E \dot{x}=Ax+Bu$ and $y=Cx+Du$, minimize the Frobenius norm of $(Δ_E,Δ_A,Δ_B,Δ_C,Δ_D)$ such that $(E+Δ_E,A+Δ_A,B+Δ_B,C+Δ_C,D+Δ_D)$ is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices $(E,A,B,C,D)$ describing the system, and (iii) does not make any assumption on the original given system.

math.OC↗

arXiv2017

A semi-analytical approach for the positive semidefinite Procrustes problem

The positive semidefinite Procrustes (PSDP) problem is the following: given rectangular matrices $X$ and $B$, find the symmetric positive semidefinite matrix $A$ that minimizes the Frobenius norm of $AX-B$. No general procedure is known that gives an exact solution. In this paper, we present a semi-analytical approach to solve the PSDP problem. First, we characterize completely the set of optimal solutions and identify the cases when the infimum is not attained. This characterization requires the unique optimal solution of a smaller PSDP problem where $B$ is square and $X$ is diagonal with positive diagonal elements. Second, we propose a very efficient strategy to solve the PSDP problem, combining the semi-analytical approach, a new initialization strategy and the fast gradient method. We illustrate the effectiveness of the new approach, which is guaranteed to converge linearly, compared to state-of-the-art methods.

math.OC↗

arXiv2017

Computing nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$, minimize the Frobenius norm of $(Δ_E,Δ_A)$ such that $(E+Δ_E,A+Δ_A)$ is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an invertible $Q$ such that $Q^TE$ is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.

math.NA↗

arXiv2017

On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix $A$ is stable if and only if it can be written as $A = (J-R)Q$, where $J=-J^T$, $R \succeq 0$ and $Q \succ 0$ (that is, $R$ is positive semidefinite and $Q$ is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables $(J,R,Q)$: (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require $\mathcal{O}(n^3)$ operations per iteration, where $n$ is the size of $A$. We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.

math.OC↗

arXiv2010

Reconstruction of Aggregation Tree in spite of Faulty Nodes in Wireless Sensor Networks

Recent advances in wireless sensor networks (WSNs) have led to many new promissing applications. However data communication between nodes consumes a large portion of the total energy of WSNs. Consequently efficient data aggregation technique can help greatly to reduce power consumption. Data aggregation has emerged as a basic approach in WSNs in order to reduce the number of transmissions of sensor nodes over {\it aggregation tree} and hence minimizing the overall power consumption in the network. If a sensor node fails during data aggregation then the aggregation tree is disconnected. Hence the WSNs rely on in-network aggregation for efficiency but a single faulty node can severely influence the outcome by contributing an arbitrary partial aggregate value. In this paper we have presented a distributed algorithm that reconstruct the aggregation tree from the initial aggregation tree excluding the faulty sensor node. This is a synchronous model that is completed in several rounds. Our proposed scheme can handle multiple number of faulty nodes as well.

cs.DC↗