Tensor Product Of Spaces Research Articles

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called proper generalized decompositions (PGD) methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated PGDs for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.

Copies of ℓ 1 in positive tensor products of Orlicz sequence spaces

Let X be a Banach lattice and ℓ ϕ be an Orlicz sequence space associated to an Orlicz function ϕ with the Δ2-condition. In this paper, we prove that (i) the positive injective tensor product of ℓ ϕ and X, contains no copy of ℓ1 if and only if both ℓ ϕ and X contain no copy of ℓ 1; and (ii) the positive projective tensor product of ℓ ϕ and X, contains no copy of ℓ 1 if and only if both ℓ ϕ and X contain no copy of ℓ 1 and each positive linear operator from ℓ ϕ to X* is compact.

Yang-Baxter $${\breve R}$$ matrix, entanglement and Yangian

We present a method to construct “X” form unitary Yang-Baxter \({\breve R}\) matrices, which act on the tensor product space \({V_{i}^{j_{1}}\otimes V_{i+1}^{j_{2}}}\) . We can obtain a set of entangled states for (2j 1 + 1) × (2j 2 + 1)-dimensional system with these Yang-Baxter \({\breve R}\) matrices. By means of Yang-Baxter approach, a 8 × 8 Yang-Baxter Hamiltonian is constructed. Yangian symmetry and Yangian generators as shift operators for this Yang-Baxter system are investigated in detail.

Characterization of the class of unitary operators by operator inequalities

Let , and be the C*-algebra of all bounded linear operators acting on a complex Hilbert space H, the set of all invertible elements in and the class of all unitary operators in , respectively. In this note, we shall show that if , then the injective norm of S ⊗ S −1 + S −1 ⊗ S in the tensor product space attains its minimal value 2 if and only if S is normal and satisfies the condition for every λ, μ in the spectrum σ(S) of S. Finally, it is shown that if then the inequality ‖SXS −1 + S −1 XS‖ ≤ 2‖X‖ holds for all X in if and only if .

Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains

A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations.

Solving stochastic systems with low-rank tensor compression

For parametrised equations, which arise, for example, in equations dependent on random parameters, the solution naturally lives in a tensor product space. The application which we have in mind is a stochastic linear elliptic partial differential equation (SPDE). Usual spatial discretisation leads to a potentially large linear system for each point in the parameter space. Approximating the parametric dependence by a Galerkin ‘ansatz’, the already large number of unknowns—for a fixed parameter value—is multiplied by the dimension of the Galerkin subspace for the parametric dependence, and thus can be very large. Therefore, we try to solve the total system approximately on a smaller submanifold which is found adaptively through compression during the solution process by alternating iteration and compression. We show that for general linearly converging stationary iterative schemes and general adaptation processes—which can be seen as a modification or perturbation of the iteration—the interlaced procedure still converges. Our proposed modification can be used for most stationary solvers for systems on tensor products. We demonstrate this on an example of a discretised SPDE with a simple preconditioned iteration. We choose a truncated singular value decomposition (SVD) for the compression and give details of the implementation, finishing with some examples.

Entanglement and Yangian in a $${V^{\otimes 3}}$$ Yang-Baxter system

In this paper, a 8 × 8 unitary Yang-Baxter matrix $${\breve{R}_{123}(\theta_{1},\theta_{2},\phi)}$$ acting on the triple tensor product space, which is a solution of the Yang-Baxter Equation for three qubits, is presented. Then quantum entanglement and the Berry phase of the Yang-Baxter system are studied. The Yangian generators, which can be viewed as the shift operators, are investigated in detail. And it is worth mentioning that the Yangian operators we constructed are independent of choice of basis.

Weakly open sets in the unit ball of the projective tensor product of Banach spaces

A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for X ⊗ ˆ π Y if the centralizer of X is infinite-dimensional and the unit sphere of Y ⁎ contains an element of numerical index one. We provide examples of classical Banach spaces satisfying the assumptions of the results. If K is any infinite compact Hausdorff topological space, then C ( K ) ⊗ ˆ π Y has the diameter two property for any nonzero Banach space Y. We also provide a result on the diameter two property for the injective tensor product.

Tensor products of operator systems

The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we examine in detail several particular examples of tensor products, including a minimal, maximal, maximal commuting, maximal injective and some asymmetric tensor products. We characterize these tensor products in terms of their universal properties and give descriptions of their positive cones. We also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system. We examine notions of nuclearity for our tensor products which, on the category of C ⁎ -algebras, reduce to the classical notion. We exhibit an operator system S which is not completely order isomorphic to a C ⁎ -algebra yet has the property that for every C ⁎ -algebra A, the minimal and maximal tensor product of S and A are equal.

Weak compactness in projective tensor products of Banach spaces

Contrary to a recent conjecture, it is shown that weakly compact subsets of the projective tensor product of Banach spaces X and Y in general are not contained in the closed absolutely convex hull of a tensor product A ⊗ B of weakly compact subsets A of X and B of Y.

Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces

Let H=X⊗Y be a tensor product of separable Hilbert spaces X and Y. We establish norm estimates for the resolvent and operator-valued functions of the operator A=∑k=0mBk⊗Sk, where Bk (k=0,…,m) are bounded operators acting in Y, and S is a self-adjoint operator acting in X. By these estimates we investigate spectrum perturbations of A. The abstract results are applied to the nonself-adjoint differential operators in Hilbert and Euclidean spaces. Our main tool is a combined use of some properties of operators on tensor products of Hilbert spaces and the recent estimates for the norm of the resolvent of a nonself-adjoint operator.

On the Dunford–Pettis property of tensor product spaces

On the Dunford–Pettis property of tensor product spaces

The Lowest Order Differentiable Finite Element on Rectangular Grids

A macro type of biquadratic $C^1$ finite elements is constructed on rectangle grids. This is a rectangular version of the $C^1$ Powell-Sabin element, a $C^1$-$P_2$ element on triangular grids. Here, each rectangle of the base grid is refined into four subrectangles. As in the case of the Powell-Sabin element, we have more constraints than the number of degrees of freedom on each macroelement. However, the extra constraints are consistent. It is shown further that the constructed finite element space is the full $C^1$-$Q_2$ space on the grid. It is also shown that the finite element space is a tensor product space of one-dimensional $C^1$-$P_2$ spaces, where the nodal basis is supported on four intervals. The B-spline function of $P_2$ is supported on three intervals. The Girault-Scott operator is extended to the element. The application and the convergence of the finite element to the biharmonic equation are presented. Numerical tests are provided.

Garding′s Inequality for Elliptic Differential Operator with Infinite Number of Variables

We formulate the elliptic differential operator with infinite number of variables and investigate that it is well defined on infinite tensor product of spaces of square integrable functions. Under suitable conditions, we prove Garding′s inequality for this operator.

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m0: Σ → X0 and m1: Σ → X1 be positive vector measures with values in the Banach Kothe function spaces X0 and X1. If 0 < α < 1, we define a new vector measure [m0, m1]α with values in the Calderon lattice interpolation space X01−gaX1α and we analyze the space of integrable functions with respect to measure [m0, m1]α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of Lp-spaces. Since each p-convex order continuous Kothe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.

A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach

The Proper Generalized Decomposition (PGD) is a methodology initially proposed for the solution of partial differential equations (PDE) defined in tensor product spaces. It consists in constructing a separated representation of the solution of a given PDE. In this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting. In particular, we introduce a generalization of Eckart and Young theorem which allows to prove the convergence of the so-called progressive PGD for a large class of linear problems defined in tensor product Hilbert spaces.

Complete continuity properties for the Fremlin projective tensor product of Orlicz sequence spaces and Banach lattices

Complete continuity properties for the Fremlin projective tensor product of Orlicz sequence spaces and Banach lattices

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗ɛY be the injective tensor product of X and Y. (a) If either X** or Y has the approximation property and each continuous linear operator from X* to Y is compact, then X ⊗ɛY has the Grothendieck property. (b) In addition, if Y has an unconditional finite dimensional decomposition, then X ⊗ɛY has the Grothendieck property if and only if each continuous linear operator from X* to Y is compact.

Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems

Uncertainty quantification and propagation in physical systems appear as a critical path for the improvement of the prediction of their response. Galerkin-type spectral stochastic methods provide a general framework for the numerical simulation of physical models driven by stochastic partial differential equations. The response is searched in a tensor product space, which is the product of deterministic and stochastic approximation spaces. The computation of the approximate solution requires the solution of a very high dimensional problem, whose calculation costs are generally prohibitive. Recently, a model reduction technique, named Generalized Spectral Decomposition method, has been proposed in order to reduce these costs. This method belongs to the family of Proper Generalized Decomposition methods. It takes part of the tensor product structure of the solution function space and allows the a priori construction of a quasi optimal separated representation of the solution, which has quite the same convergence properties as a posteriori Hilbert Karhunen-Loeve decompositions. The associated algorithms only require the solution of a few deterministic problems and a few stochastic problems on deterministic reduced basis (algebraic stochastic equations), these problems being uncoupled. However, this method does not circumvent the “curse of dimensionality” which is associated with the dramatic increase in the dimension of stochastic approximation spaces, when dealing with high stochastic dimension. In this paper, we propose a marriage between the Generalized Spectral Decomposition algorithms and a separated representation methodology, which exploits the tensor product structure of stochastic functions spaces. An efficient algorithm is proposed for the a priori construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.

Deterministic multi-level algorithms for infinite-dimensional integration on [formula omitted

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E ( g ( B ) ) , the expectation of a payoff function g , that depends on a Brownian motion B . Employing a standard series expansion of B the latter problem is equivalent to the computation of the expectation of a function of the corresponding i.i.d. sequence of random coefficients. This motivates the construction and the analysis of algorithms for numerical integration with respect to a product probability measure on the sequence space R N . The class of integrands studied in this paper is the unit ball in a reproducing kernel Hilbert space obtained by superposition of weighted tensor product spaces of functions of finitely many variables. Combining tractability results for high-dimensional integration with the multi-level technique we obtain new algorithms for infinite-dimensional integration. These deterministic multi-level algorithms use variable subspace sampling and they are superior to any deterministic algorithm based on fixed subspace sampling with respect to the respective worst case error.