Finite-dimensional Truncations Research Articles

Essential Spectrum of Discrete Laplacian – Revisited

Consider the discrete Laplacian operator A acting on l2(Z). It is well known from the classical literature that the essential spectrum of A is a compact interval. In this article, we give an elementary proof for this result, using the finite-dimensional truncations An of A. We do not rely on symbol analysis or any infinite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations An and make use of the filtration techniques due to Arveson. Usage of such techniques to the discrete Schrödinger operator and to the multi-dimensional settings will be interesting future problems.

A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method

A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton–Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton–Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton–Galerkin.

Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices

We introduce a new family of tensorial field theories by coupling different fields in a non-trivial way, with a view towards the investigation of the coupling between matter and gravity in the quantum regime. As a first step, we consider the simple case with two tensors of the same rank coupled together, with Dirac like kinetic kernel. We focus especially on rank-$3$ tensors, which lead to a power counting just-renormalizable model, and interpret Feynman graphs as Ising configurations on random lattices. We investigate the renormalization group flow for this model, using two different and complementary tools for approximations, namely, the effective vertex expansion method and finite-dimensional truncations for the flowing action. Due to the complicated structure of the resulting flow equations, we divided the work into two parts. In this first part we only investigate the fundamental aspects on the construction of the model and the different ways to get tractable renormalization group equations, while their numerical analysis will be addressed in a companion paper.

Discretisation of Global Attractors for Lattice Dynamical Systems

The existence of numerical attractors for lattice dynamical systems is established, where the implicit Euler scheme is used for time discretisation. Infinite dimensional discrete lattice systems as well as their finite dimensional truncations are considered. It is shown that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model as the discretisation step size tends to zero.

Invertibility of a tridiagonal operator with an application to a non-uniform sampling problem

Let T be a tridiagonal operator on which has strict row and column dominant property except for some finite number of rows and columns. This matrix is shown to be invertible under certain conditions. This result is also extended to double infinite tridiagonal matrices. Further, a general theorem is proved for solving an operator equation using its finite-dimensional truncations, where T is a double infinite tridiagonal operator. Finally, it is also shown that these results can be applied in order to obtain a stable set of sampling for a shift-invariant space.

Preconditioners in spectral approximation

Let H be a complex separable Hilbert space, and let A be a bounded self-adjoint operator on H. Consider the orthonormal basis B={e1,e2,…} and the projection Pn of H onto the finite-dimensional subspace spanned by the first n elements of B. The finite-dimensional truncations An=PnAPn shall be regarded as a sequence of finite matrices by restricting their domains to Pn(H) for each n. Many researchers used the sequence of eigenvalues of An to obtain information about the spectrum of A. But in many situations, these An’s need not be simple enough to make the computations easier. The natural question Can we use some simpler sequence of matrices Bn instead of An? is addressed in this article. The notion of preconditioners and their convergence in the sense of eigenvalue clustering are used to study the problem. The connection between preconditioners and compact perturbations of operators is identified here. The usage of preconditioners in the spectral gap prediction problems is also discussed. The examples of Toeplitz and block Toeplitz operators are considered as an application of these results. Finally, some future possibilities are discussed.

Partially-massless higher-spin algebras and their finite-dimensional truncations

The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS$_{d+1}$ are studied. The algebras involving PM generators up to depth $2\,(\ell-1)$ are defined as the maximal symmetries of free conformal scalar field with $2\,\ell$ order wave equation in $d$ dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of $(A)dS_{d+1}$ isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of $\ell-d/2\,$, which coincides with the annihilator of the one-row $\ell$-box Young diagram representation of $\mathfrak{so}_{d+2}\,$. Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.

An Evans-function approach to spectral stability of internal solitary waves in stratified fluids

Frequently encountered in nature, internal solitary waves in stratified fluids have been investigated experimentally, theoretically, and numerically. Mathematically, these waves are exact solutions of the incompressible 2D Euler equations. Contrasting with a rich existence theory and the development of methods for their computation, their stability analysis has hardly received attention at a rigorous mathematical level.This paper proposes a new approach to the investigation of stability of internal solitary waves in a continuously stratified fluid and carries out the following four steps of this approach: (I) to formulate the eigenvalue problem as an infinite-dimensional spatial-dynamical system, (II) to introduce finite-dimensional truncations of the spatial-dynamics description, (III) to demonstrate that each truncation, of any order, permits a well-defined Evans function, (IV) to prove absence of small zeros of the Evans function in the small-amplitude limit. The latter notably implies the low-frequency spectral stability of small-amplitude waves to arbitrarily high truncation order.

Circumventing the Slater conundrum in countably infinite linear programs

Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researchers for a few decades; it has recently been called the Slater conundrum. We find a way around this hurdle.We propose a pair of primal-dual spaces with three properties: the series in the primal and dual objective functions converge; the series defined by the rows and columns of the primal constraint matrix converge; and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. Our results are illustrated using several examples, and are applied to countable-state Markov decision processes and to a problem in robust optimization.

Global and local properties of AdS 2 higher spin gravity

Two-dimensional BF theory with infinitely many higher spin fields is proposed. It is interpreted as the AdS(2) higher spin gravity model describing a consistent interaction between local fields in AdS(2) space including gravitational field, higher spin partially-massless fields, and dilaton fields. We carry out analysis of the frame-like and the metric-like formulation of the theory. Infinite-dimensional higher spin global algebras and their finite-dimensional truncations are realized in terms of o(2,1) - sp(2) Howe dual auxiliary variables.

Duality in Countably Infinite Linear Programs

Duality results on countably infinite linear programs (CILPs) are scarce. Roughly speaking, one fundamental hurdle is as follows. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual problem where the constraints cannot be expressed using the ordinary transpose of the primal doubly-infinite constraint matrix. Subspaces that do allow us to write the dual using this ordinary transpose do not admit an interior point. This has recently been called the "Slater conundrum." We find a way around this hurdle. We propose a pair of primal-dual spaces with three properties: (i) the series in the primal and dual objective functions converge; (ii) the series defined by the rows and columns of the primal constraint matrix converge; and (iii) the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. We also prove that the Lagrangean of our CILPs provides a linear support to the primal optimal value function. Finally, we establish that saddle points of the Lagrangean are equivalent to complementary feasible solutions of our CILPs. Our results are illustrated using several examples, and are also applied to countable-state Markov decision processes and to a problem in robust optimization.

RG flows of Quantum Einstein Gravity on maximally symmetric spaces

We use the Wetterich-equation to study the renormalization group flow of $f(R)$-gravity in a three-dimensional, conformally reduced setting. Building on the exact heat kernel for maximally symmetric spaces, we obtain a partial differential equation which captures the scale-dependence of $f(R)$ for positive and, for the first time, negative scalar curvature. The effects of different background topologies are studied in detail and it is shown that they affect the gravitational RG flow in a way that is not visible in finite-dimensional truncations. Thus, while featuring local background independence, the functional renormalization group equation is sensitive to the topological properties of the background. The detailed analytical and numerical analysis of the partial differential equation reveals two globally well-defined fixed functionals with at most a finite number of relevant deformations. Their properties are remarkably similar to two of the fixed points identified within the $R^2$-truncation of full Quantum Einstein Gravity. As a byproduct, we obtain a nice illustration of how the functional renormalization group realizes the "integrating out" of fluctuation modes on the three-sphere.

Energy-Preserving Integrators and the Structure of B-series

B-series are a powerful tool in the analysis of Runge–Kutta numerical integrators and some of their generalizations (“B-series methods”). A general goal is to understand what structure-preservation can be achieved with B-series and to design practical numerical methods that preserve such structures. B-series of Hamiltonian vector fields have a rich algebraic structure that arises naturally in the study of symplectic or energy-preserving B-series methods and is developed in detail here. We study the linear subspaces of energy-preserving and Hamiltonian modified vector fields which admit a B-series, their finite-dimensional truncations, and their annihilators. We characterize the manifolds of B-series that are conjugate to Hamiltonian and conjugate to energy-preserving and describe the relationships of all these spaces.

Strongly differentiable solutions of the discrete coagulation–fragmentation equation

We examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters. In contrast to previous investigations, our analysis does not involve finite-dimensional truncations of the system. Instead, we treat the problem as an infinite-dimensional differential equation, posed in an appropriate Banach space, and apply perturbation results from the theory of strongly continuous semigroups of operators. The existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates.

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.

Solution of a tridiagonal operator equation

Let H be a separable Hilbert space with an orthonormal basis { e n / n ∈ N } , T be a bounded tridiagonal operator on H and T n be its truncation on span ({ e 1, e 2, … , e n }). We study the operator equation Tx = y through its finite dimensional truncations T n x n = y n . It is shown that if { ‖ T n - 1 e n ‖ } and { ‖ T n ∗ - 1 e n ‖ } are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence { T n - 1 } . We also give sufficient conditions for the boundedness of { ‖ T n - 1 e n ‖ } and { ‖ T n ∗ - 1 e n ‖ } in terms of the entries of the matrix of T.

The null controllability of thermoelastic plates and singularity of the associated minimal energy function

We consider the null controllability problem for thermoelastic plates, defined on a two dimensional domain Ω , and subject to hinged, clamped or free boundary conditions. The uncontrolled partial differential equation system generates an analytic semigroup on the space of finite energy. Consequently, the concept of null controllability is indeed appropriate for consideration here. It is shown that all finite energy states can be driven to zero by means of just one L 2 ((0,T)×Ω) control be it either mechanical or thermal. The singularity, as T ↓0, of the associated minimal energy function is the main object studied in the paper. Singularity and blow-up rates for minimal energy function are not only of interest in their own right but are also of critical importance in Stochastic PDEs. In this paper, we establish the optimal blow-up rate O (T −5/2 ) for this function. It is shown that the rate of singularity is the same as for finite-dimensional truncations of the model. In view of sharp estimates available in the finite dimensional setting [Math. Control Signals Systems 9 (1997) 327], the singularity rates provided in this paper are optimal.

The null controllability of thermoelastic plates and singularity of the associated minimal energy function

We consider the null controllability problem for thermoelastic plates, defined on a two dimensional domain Ω, and subject to hinged, clamped or free boundary conditions. The uncontrolled partial differential equation system generates an analytic semigroup on the space of finite energy. Consequently, the concept of null controllability is indeed appropriate for consideration here. It is shown that all finite energy states can be driven to zero by means of just one L 2((0,T)×Ω) control be it either mechanical or thermal. The singularity, as T↓0, of the associated minimal energy function is the main object studied in the paper. Singularity and blow-up rates for minimal energy function are not only of interest in their own right but are also of critical importance in Stochastic PDEs. In this paper, we establish the optimal blow-up rate O(T −5/2) for this function. It is shown that the rate of singularity is the same as for finite-dimensional truncations of the model. In view of sharp estimates available in the finite dimensional setting [Math. Control Signals Systems 9 (1997) 327], the singularity rates provided in this paper are optimal.

An Isotonic Regression Problem for Infinite-Dimensional Parameters

We consider an infinite-dimensional isotonic regression problem which is an extension of the suitably revised classical isotonic regression problem. Given p-summable data, for p finite and at least one, there exists an optimal estimator to our problem. For p greater than one, this estimator is unique and is the limit in the p-norm of the sequence of unique estimators in canonical finite-dimensional truncations of our problem. However, for p equal to one, our problem, as well as the finite-dimensional truncations, admit multiple optimal estimators in general. In this case, the sequence of optimal estimator sets to the truncations converges to the optimal estimator set of the infinite problem in the sense of Kuratowski. Moreover, the selection of natural best optimal estimators to the truncations converges in the 1-norm to an optimal estimator of the infinite problem.

A new proof of the existence of equilibrium in incomplete market economies

The paper presents a hypothesis of an incomplete markets equilibrium existence theorem using finite-dimensional truncations of utilities, endowments and asset returns.