Tensor Product Structure Research Articles

Computational generation and conformal fabrication of woven fabric structures by harmonic foliation

This paper presents a framework for computational generation and conformal fabrication of woven thin-shell structures with arbitrary topology based on the foliation theory which decomposes a surface into a group of parallel leaves. By solving graph-valued harmonic maps on the input surface, we construct two sets of harmonic foliations perpendicular to each other. The warp and weft threads are created afterward and then manually woven to reconstruct the surface. The proposed computational method guarantees the smoothness of the foliation and the orthogonality between each pair of leaves from different foliations. Moreover, it minimizes the number of singularities to theoretical lower bound and produces the tensor product structure as globally as possible. This method is ideal for the physical realization of woven surface structures on a variety of applications, including wearable electronics, sheet metal craft, architectural designs, and conformal woven composite parts in the automotive and aircraft industries. The performance of the proposed method is demonstrated through the computational generation and physical fabrication of several free-form thin-shell structures.

Sum factorization for fast integration of DPG matrices on prismatic elements

Higher order finite element (FE) methods provide significant advantages in a number of applications such as wave propagation, where high order shape functions help to mitigate pollution (dispersion) error. However, classical assembly of higher order systems is computationally burdensome, requiring the evaluation of many point quadrature schemes. When the Discontinuous Petrov-Galerkin (DPG) FE methodology is employed, the use of an enriched test space further increases the computational burden of system assembly, increasing the relevance of improved assembly techniques. Sum factorization—a technique that exploits the tensor-product structure of shape functions to accelerate numerical integration—was proposed in Ref. [10] for the assembly of DPG matrices on hexahedral elements that reduced the computational complexity from order O(p9) to O(p7) (where p denotes polynomial order). In this work we extend the concept of sum factorization to the construction of DPG matrices on prismatic elements by expressing prism shape functions as tensor products of 2D simplex and 1D interval shape functions. Unexpectedly, the resulting sum factorization routines on partially-tensorized prism shape functions achieve the same O(p7) complexity as sum factorization on fully-tensorized hexahedra shape functions (as products of 1D interval shape functions) presented in Ref. [10]. Throughout this work we adhere to the theory of exact sequence energy spaces, proposing sum factorization routines for each of the 3D FE exact sequence energy spaces—H1, H(curl), H(div), and L2. Computational results for construction of the DPG Gram matrix on a prismatic element in each exact sequence energy space are presented, corroborating the expected O(p7) complexity. Additionally, construction of the DPG system for an ultraweak Maxwell problem on a prismatic element is considered and a partially-tensorized sum factorization for hexahedral elements is proposed to improve implementational compatibility between hexahedral and prismatic elements.

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

We consider the solution of linear discrete ill-posed systems of equations with a certain tensor product structure. Two aspects of this kind of problems are investigated: They are transformed to large linear systems of equations and the conditioning of the matrix of the latter system is analyzed. Also, the distance of this matrix to symmetry and skew-symmetry is investigated. The aim of our analysis is to shed light on properties of linear discrete ill-posed problems and to study the feasibility of using Krylov subspace iterative methods in conjunction with Tikhonov regularization to solve Sylvester tensor equations with severely ill-conditioned coefficient matrices. The performance of several proposed algorithms is studied numerically. Applications include color image restoration and the solution of a 3D radiative transfer equation that is discretized by a Chebyshev collocation spectral method.

Emergent gauge symmetries and quantum operations

The algebraic approach to quantum physics emphasizes the role played by the structure of the algebra of observables and its relation to the space of states. An important feature of this point of view is that subsystems can be described by subalgebras, with partial trace being replaced by the more general notion of restriction to a subalgebra. This, in turn, has recently led to applications to the study of entanglement in systems of identical particles. In the course of those investigations on entanglement and particle identity, an emergent gauge symmetry has been found by Balachandran, de Queiroz and Vaidya. In this letter we establish a novel connection between that gauge symmetry, entropy production and quantum operations. Thus, let A be a system described by a finite dimensional observable algebra and a mixed faithful state. Using the Gelfand–Naimark–Segal (GNS) representation we construct a canonical purification of , allowing us to embed A into a larger system C. Using Tomita–Takesaki theory, we obtain a subsystem decomposition of C into subsystems A and B, without making use of any tensor product structure. We identify a group of transformations that acts as a gauge group on A while at the same time giving rise to entropy increasing quantum operations on C. We provide physical means to simulate this gauge symmetry/quantum operation duality.

Cap a la sintaxi de matrius

La sintaxi de matrius és un model formal de relacions sintàctiques en el llenguatge que va sorgir del desig de modelar les cadenes. L’objectiu d’aquest treball és explicar les idees bàsiques d’aquest model a un públic lingüístic, sense entrar en gaires detalls formals (vegeu Orús et al. 2017). L’estructura matemàtica resultant s’assembla a alguns aspectes de la mecànica quàntica i s’adapta bé per descriure les cadenes lingüístiques. En particular, les oracions es modelen naturalment com a vectors en un espai de Hilbert amb una estructura de producte tensorial, construïdes a partir de matrius 2 x 2 que pertanyen a un grup específic. Curiosament, les matrius que utilitza el sistema són extensions simples de representacions habituals de les parts principals del discurs com a objectes [± N, ± V].

Typical entanglement entropy in the presence of a center: Page curve and its variance

In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P(S_A) \mathrm{d}S_A$ that a subsystem has entanglement entropy $S_A$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, $\Delta S_A/\langle{S_{A}}\rangle\ll 1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.

Quantum measurements and contextuality.

In quantum physics, the term 'contextual' can be used in more than one way. One usage, here called 'Bell contextual' since the idea goes back to Bell, is that if A, B and C are three quantum observables, with A compatible (i.e. commuting) with B and also with C, whereas B and C are incompatible, a measurement of A might yield a different result (indicating that quantum mechanics is contextual) depending upon whether A is measured along with B (the {A, B} context) or with C (the {A, C} context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell non-contextual: the outcome of a particular A measurement when A is measured along with B would have been exactly the same if A had, instead, been measured along with C. A different definition, here called 'globally (non)contextual' refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Gauge structures: From stabilizer codes to continuum models

Stabilizer codes are a powerful method for implementing fault-tolerant quantum memory and in the case of topological codes, they form useful models for topological phases of matter. In this paper, we discuss the theory of stabilizer codes as a discrete version of a linear gauge structure, a concept we introduce here. A linear gauge structure captures all the familiar ingredients of U(1) gauge theory including a generalization of charge conservation, Maxwell equations and topological terms. Using this connection, we prove some important results for stabilizer codes which can be used to characterize their error-correction properties. However, this perspective does not depend on any particular Hamiltonian or Lagrangian, which as a consequence is agnostic to the source of the gauge redundancy. Based upon the connection to stabilizer codes, we view the source of the gauge redundancy as an ambiguity in the tensor product structure of the Hilbert space. That is, the gauge provides a set of equivalent but arbitrary ways to factorize the Hilbert space. We apply this formalism to the d=2 and 3 toric code as well as the paradigm fracton models, X-Cube and Haah’s cubic code. From this perspective, we are able to map all these models to some continuum version which captures all or some of the fractonic and topological features of their parent discrete models.

Iterative low-rank approximation solvers for the extension method for fractional diffusion

We consider the numerical method for fractional diffusion problems which is based on an extension to a mixed boundary value problem for a local operator in a higher dimensional space. We observe that, when this problem is discretized using tensor product spaces as is commonly done, the solution can be very well approximated by low-rank tensors; we provide some analysis to support this observation. This fact motivates us to apply iterative low-rank approximation algorithms in order to efficiently solve this extended problem. In particular, we employ a recently proposed greedy Tucker approximation method as well as a more classical greedy rank one update method. Throughout, all objects of interest are kept in suitable low-rank approximations, which dramatically reduces the required amount of memory compared to the full formulation of the extended problem.Our approach can be used for general, non-structured space discretizations. If the space discretization itself has tensor product structure, we can further decompose the problem in order to deal with even lower-dimensional objects. We also note that the approach can be directly applied to higher-order discretizations both in space and the extended variable.A further contribution of our work is a rank one version of a diagonal preconditioner which can mitigate the severe ill-conditioning of the extended problem arising due to mesh grading and singularity of the coefficient. This preconditioner can be realized using a simple pre- and postprocessing step and does not affect the computational effort of our algorithm.In several numerical examples, we demonstrate the convergence behavior of the proposed methods. In particular, the Tucker approximation approach requires only a few iterations in order to reach the discretization error in all tested settings.

Dimensionwise multivariate orthogonal polynomials in general probability spaces

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, encompassing hierarchically organized, measure-consistent multivariate orthogonal polynomials in dependent random variables. In contrast to the existing PDD, which is valid strictly for independent random variables, no tensor-product structure or product-type probability measure is imposed or necessary. Fundamental mathematical properties of GPDD are examined by creating dimensionwise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for requisite assumptions, and upholding mean-square convergence to the correct limit. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables with a large class of non-product-type probability measures.

Green ring of the category of weight modules over the Hopf–Ore extensions of group algebras

In this paper, we continue our study of the tensor product structure of category of weight modules over a Hopf–Ore extensions of a group algebra kG, where k is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of χ and are different. Then we describe the Green ring of the tensor category . It is shown that is isomorphic to the polynomial algebra over the group ring in one variable when , and that is isomorphic to the quotient ring of the polynomial algebra over the group ring in two variables modulo a principle ideal when . When , is isomorphic to the quotient ring of a skew group ring modulo some ideal, where is a polynomial algebra over in infinitely many variables.

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called vector-valued spectral analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for multitask machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto–Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

Flag additivity in quantum resource theories

Quantum resource theories offer a powerful framework for studying various phenomena in quantum physics. Despite considerable effort has been devoted to developing a unified framework of resource theories, there are few common properties that hold for all quantum resources. In this paper, we fill this gap by introducing the flag additivity based on the tensor product structure and the flag basis for the general quantum resources. To illustrate the usefulness of flag additivity, we show that flag additivity can be used to derive other nontrivial properties in quantum resource theories, e.g., strong monotonicity, convexity, and full additivity.

Exploring the entanglement of free spin- , spin-1 and spin-2 fields * * Supported by the NSFC (11875111)

In this study, we explore the entanglement of free spin- , spin-1, and spin-2 fields. We start with an example involving Majorana fields in 1+1 and 2+1 dimensions. Subsequently, we perform the Bogoliubov transformation and express the vacuum state with a particle pair state in the configuration space, which is used to calculate the entropy. This clearly demonstrates that the entanglement entropy originates from the particles across the boundary. Finally, we generalize this method to free spin-1 and spin-2 fields. These higher free massless spin fields have well-known complications owing to gauge redundancy. We deal with the redundancy by gauge-fixing in the light-cone gauge. We show that this gauge provides a natural tensor product structure in the Hilbert space, while surrendering explicit Lorentz invariance. We also use the Bogoliubov transformation to calculate the entropy. The area law emerges naturally by this method.

A Nested Tensor Product Model Transformation

The tensor product model transformation (TPMT) is an emerging numerical framework of the Takagi–Sugeno (T–S) fuzzy (or polytopic) system modeling for a linear matrix inequality based system control design. A nested TPMT (NTPMT) is proposed in this paper, which merges the dimensions of the tensors and performs the TPMT iteratively. The resultant fuzzy model is in a multilevel nested tensor product (TP) structure. The vertex tensor obtained by NTPMT has fewer dimension results than the original TPMT so the number of vertices or fuzzy rules, which have been the main bottleneck for further application of the TPMT in higher dimensional systems, is expected to decrease significantly. It is also proven that the NTPMT contains the hierarchical fuzzy logic, meaning that the NTPMT is capable of conducting hierarchical fuzzy modeling and reduction. Furthermore, because the inclusion of multiple TPMTs is prone to augment the conservativeness of the resultant fuzzy model, a suboptimal convex hull rectification algorithm for the TPMT is developed based on a newly defined tightness measure, and then extended to render the NTPMT as less conservative as possible. Finally, numerical simulations on two real physical systems (two- and four-parameter dimension) are verified to demonstrate the performance of the methods.

Do black holes create polyamory?

Of course not, but if one believes that information cannot be destroyed in a theory of quantum gravity, then we run into apparent contradictions with quantum theory when we consider evaporating black holes. Namely that the no-cloning theorem or the principle of entanglement monogamy is violated. Here, we show that neither violation need hold, since, in arguing that black holes lead to cloning or non-monogamy, one needs to assume a tensor product structure between two points in space-time that could instead be viewed as causally connected. In the latter case, one is violating the semi-classical causal structure of space, which is a strictly weaker implication than cloning or non-monogamy. This is because both cloning and non-monogamy also lead to a break-down of the semi-classical causal structure. We show that the lack of monogamy that can emerge in evaporating space times is one that is allowed in quantum mechanics, and is very naturally related to a lack of monogamy of correlations of outputs of measurements performed at subsequent instances of time of a single system. This is due to an interesting duality between temporal correlations and entanglement. A particular example of this is the Horowitz-Maldacena proposal, and we argue that it needn’t lead to cloning or violations of entanglement monogamy. For measurements on systems which appear to be leaving a black hole, we introduce the notion of the temporal product, and argue that it is just as natural a choice for measurements as the tensor product. For black holes, the tensor and temporal products have the same measurement statistics, but result in different type of non-monogamy of correlations, with the former being forbidden in quantum theory while the latter is allowed. In the case of the AMPS firewall experiment we find that the entanglement structure is modified, and one must have entanglement between the infalling Hawking partners and early time outgoing Hawking radiation which surprisingly tames the violation of entanglement monogamy.

On non-commutative operator graphs generated by covariant resolutions of identity

We study non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary representations of a compact group. Our main goal is searching for orthogonal projections which are anticliques (error-correcting codes) for such graphs. A special attention is paid to the covariance with respect to unitary representations of the circle group. We determine a tensor product structure in the space of representation under which the obtained anticliques are generated by entangled vectors.

New application of the isogeometric boundary representations methodology with SBFEM to seepage problems in complex domains

According to the infinite representation of the scaled boundary finite element method (SBFEM), an isogeometric scaled boundary finite element method (IGA-SBFEM) using the non-uniform rational B-splines (NURBS) is presented for the numerical solution of seepage problems in the unbounded domain. Further, the proposed method is extended to the modeling of seepage problems considering the parallel semi-infinite side-faces in the multilayered media. The proposed formulation is based on the SBFEM that only requires boundary discretization of the analyzed domain. And it advances the SBFEM by using the NURBS basis functions to exactly represent the geometry of the boundary and to approximate the solutions fields at the boundary. Thus, the proposed method fits neatly with the Boundary Representations (B-Reps) technique provided completely by CAD. The rigid tensor product structure can be reduced by one, since only the boundary information of the analyzed domain is required in the proposed approach. The method enhances the flexibility for describing complex geometry and offers many key improvements in comparison to the conventional SBFEM. Finally, the results from numerical examples are compared with the analytical solutions or reference solutions, illustrating the superiority of the proposed formulation.

A Low-Rank Inexact Newton–Krylov Method for Stochastic Eigenvalue Problems

Abstract This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high-dimensional systems with tensor product structure when discretized with the stochastic Galerkin method. Here, we exploit this inherent tensor product structure to develop a globalized low-rank inexact Newton method with which we tackle the stochastic eigenproblem. We illustrate the effectiveness of our solver with numerical experiments.

Efficient quadrature rules for subdivision surfaces in isogeometric analysis

We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions.The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of (1×n) quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration.Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull–Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including non-uniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull–Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis.