Symmetric Approximation Research Articles

Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dependent tensors by symmetric or anti-symmetric Tucker tensors of low multilinear rank. The proposed symmetric or anti-symmetric low-rank integrator is different from recently proposed projector-splitting integrators for dynamical low-rank approximation, which do not preserve symmetry or anti-symmetry. However, it is shown that the (anti-)symmetric low-rank integrators retain favourable properties of the projector-splitting integrators: given low-rank time-dependent matrices and tensors are reproduced exactly, and the error behaviour is robust to the presence of small singular values, in contrast to standard integration methods applied to the differential equations of dynamical low-rank approximation. Numerical experiments illustrate the behaviour of the proposed integrators.

Convergence analysis of symmetric dual-wind discontinuous Galerkin approximation methods for the obstacle problem

This paper formulates and analyzes symmetric dual-wind discontinuous Galerkin (DG) methods for second order elliptic obstacle problem. These new methods follow from the DG differential calculus framework that defines discrete differential operators to replace the continuous differential operators when discretizing a partial differential equation (PDE). We establish optimal a priori error estimates for both linear and quadratic elements provided the exact solution is sufficiently regular. These results are also shown to hold for some non-positive penalty parameters, with the emphasis on zero penalization across all interior and boundary edges. Numerical experiments are provided to validate the theoretical results and gauge the performance of the proposed methods.

Spacing homogenization in lamellar eutectic arrays with anisotropic interphase boundaries

We analyze the effect of interphase boundary anisotropy on the dynamics of lamellar eutectic solidification fronts, in the limit that the lamellar spacing varies slowly along the envelope of the front. In the isotropic case, it is known that the spacing obeys a diffusion equation, which can be obtained theoretically by making two assumptions: (i) the lamellae always grow normal to the large-scale envelope of the front, and (ii) the Jackson-Hunt law that links lamellar spacing and front temperature remains locally valid. For anisotropic boundaries, we replace hypothesis (i) by the symmetric pattern approximation, which has recently been found to yield good predictions for lamellar growth direction in presence of interphase anisotropy. We obtain a generalized Jackson-Hunt law for tilted lamellae, and an evolution equation for the envelope of the front. The latter contains a propagative term if the initial lamellar array is tilted with respect to the direction of the temperature gradient. However, the propagation velocity of the propagative wave modes are found to be small, so that the dynamics of the front can be reasonably described by a diffusion equation with a diffusion coefficient that is modified with respect to the isotropic case.

Symmetric rank-1 approximation of symmetric high-order tensors

ABSTRACTFinding the symmetric rank-1 approximation to a given symmetric tensor is an important problem due to its wide applications and its close relationship to the Z-eigenpair of a tensor. In this paper, we propose a method based on the proximal alternating linearized minimization to directly solve the optimization problem. Global convergence of our algorithm is established. Numerical experiments show that our algorithm is very competitive in speed, accuracy and robustness compared to other state-of-the-art methods.

Symmetrized Maxwell–Garnett Approximation as an Effective Method for Studying Nanocomposites

The symmetrized Maxwell-Garnett (SMG) approximation is considered as the most optimal method of an effective medium for the description of nanocomposite structures. This approximation takes into account the microstructure of the sample, which makes it possible to calculate the metal-dielectric system. Thus, SMG describes with good accuracy the structure of the nanocomposite. Besides, this approximation is applicable for granular alloys consisting of metal components. As a result, this technique can be considered as a universal approximation to describe a wide class of nanostructured materials. At the same time, this article discusses various methods of effective environment. In these methods, the metal component of nanocomposites and the dielectric matrix are replaced by an effective medium with effective permittivity εeff. It is necessary that the particles (granules) in such structures be small in comparison with the wavelength of electromagnetic radiation incident on the sample. Based on this, the spectral dependences of the transverse Kerr effect (TKE) in magnetic nanocomposites were calculated with (CoFeZr)(Al2O3) structure as an example at different concentrations of the magnetic component. The simulation was carried out at small and large concentrations (below and above the percolation threshold). The spectral dependences were obtained taking into account the form factor of nanoparticles and the quasi-classical size effect. Besides, the authors note and discuss in this paper the contribution of various mechanisms that affect the type of spectra of the transverse Kerr effect. Using the symmetrized Maxwell-Garnett approximation, the effective values of the granule size of the nanocomposites under study were found, and the tensor of effective dielectric permittivity (TEDP) was calculated. The obtained TEDP values allowed to simulate the spectral dependences of the magneto-optical transverse Kerr effect. The authors discuss and draw conclusions about the features of the obtained spectral dependences in both the visible and infrared regions of the spectrum. In addition, the practical and fundamental importance of the obtained results is noted. The importance of effective medium methods for the study of optical, transport and magneto-optical properties of magnetic nanocomposites is shown.

On almost tight Euclidean designs for rotationally symmetric integrals

We present a characterization theorem of almost tight Euclidean $$(2e+1)$$ -designs supported by $$\lfloor e/2 \rfloor + 2$$ concentric circles in terms of roots of quasi-orthogonal polynomials. We also prove that for any $$e \ge 5$$ there exist no almost tight $$(2e + 1)$$ -designs supported by $$\lfloor e/2 \rfloor + 2$$ concentric circles for Gaussian integration. Our characterization theorem is an analogue of some previous works as such by Verlinden and Cools (Numer Math 61:395–407, 1992), Cools and Schmid (Numerical integration, IV, Oberwolfach, 1992, Birkhauser, Basel, pp 57–66, 1993) and the present authors (2010), and the nonexistence theorem provides an answer to the inverse problem for Euclidean designs, posed by Bannai et al. (Eur J Combin 31:419–422, 2010). Furthermore, the present paper includes a short review of a relationship between Euclidean designs for rotationally symmetric integrals and kernel approximation in machine learning, together with some new observations.

The Symmetrical Parabolic Approximation in College Physics

Complicated functions appearing in physics are frequently simplified by a symmetrical parabolic approximation for obtaining useful results. The symmetrical parabolic approximation is employed in many different problems in a first year college physics course. Some examples of this approximation are explored in this article. With the aid of Hamilton’s equations it is shown that the classical formula for the kinetic energy of a particle is a symmetrical parabolic approximation for the more general relativistic formula.

Gait Event Detection From Accelerometry Using the Teager-Kaiser Energy Operator.

A novel method based on the application of the Teager-Kaiser Energy Operator is presented to estimate instances of initial contact (IC) and final contact (FC) from accelerometry during gait. The performance of the proposed method was evaluated against four existing gait event detection (GED) methods under three walking conditions designed to capture the variance of gait in real-world environments. A symmetric discrete approximation of the Teager-Kaiser energy operator was used to capture simultaneous amplitude and frequency modulations of the shank acceleration signal at IC and FC during flat treadmill walking, inclined treadmill walking, and flat indoor walking. Accuracy of estimated gait events were determined relative to gait events detected using force-sensitive resistors. The performance of the proposed algorithm was assessed against four established methods by comparing mean-absolute error, sensitivity, precision, and F1-score values. The proposed method demonstrated high accuracy for GED in all walking conditions, yielding higher F1-scores (IC: >0.98, FC: >0.9) and lower mean-absolute errors (IC: <0.018s, FC: <0.039s) than other methods examined. Estimated ICs from shank-based methods tended to exhibit unimodal distributions preceding the force-sensitive resistor estimated ICs, whereas estimated gait events for waist-based methods had quasiuniform random distributions and lower accuracy. Compared with the established gait event detection methods, the proposed method yielded comparably high accuracy for IC detection, and was more accurate than all other methods examined for FC detection. The results support the use of the Teager-Kaiser Energy Operator for accurate automated GED across a range of walking conditions.

A More General Theory of Static Approximations for Conjunctive Queries

Conjunctive query (CQ) evaluation is NP-complete, but becomes tractable for fragments of bounded hypertreewidth. Approximating a hard CQ by a query from such a fragment can thus allow for an efficient approximate evaluation. While underapproximations (i.e., approximations that return correct answers only) are well-understood, the dual notion of overapproximations (i.e, approximations that return complete – but not necessarily sound – answers), and also a more general notion of approximation based on the symmetric difference of query results, are almost unexplored. In fact, the decidability of the basic problems of evaluation, identification, and existence of those approximations has been open. This article establishes a connection between overapproximations and existential pebble games that allows for studying such problems systematically. Building on this connection, it is shown that the evaluation and identification problem for overapproximations can be solved in polynomial time. While the general existence problem remains open, the problem is shown to be decidable in 2EXPTIME over the class of acyclic CQs and in PTIME for Boolean CQs over binary schemata. Additionally we propose a more liberal notion of overapproximations to remedy the known shortcoming that queries might not have an overapproximation, and study how queries can be overapproximated in the presence of tuple generating and equality generating dependencies. The techniques are then extended to symmetric difference approximations and used to provide several complexity results for the identification, existence, and evaluation problem for this type of approximations.

Exact subsystem time-dependent density-functional theory.

In this communication, we show that coupled subsystem time-dependent density functional theory (subsystem TDDFT) [J. Neugebauer, J. Chem. Phys. 126, 134116 (2007)] in combination with projection-based embedding (PbE) is an exact subsystem theory in the sense that supermolecular TDDFT excitation energies can exactly be restored. A correct handling of the kernel contribution due to the enforced orthogonality is crucial in this context, which leads to different PbE kernel contributions in the A and B matrices of the general TDDFT eigenvalue problem. Although this formalism has been proposed before [D. V. Chulhai and L. Jensen, Phys. Chem. Chem. Phys. 18, 21032 (2016)], the symmetric eigenvalue problem used in that work implicitly introduces an approximation concerning this kernel contribution. We show that our treatment numerically exactly reproduces supermolecular results for the previously investigated helium dimer and for the fluoroethane molecule as a more challenging case with a partitioning of a covalent bond. We also demonstrate that the symmetric approximation can lead to significant deviations, including a wrong ordering of electronic transitions.

Dynamical generation of low-energy couplings from quark-meson fluctuations

We extend our recent computation of the low-energy limit of the linear O(4) Quark-Meson Model. The present analysis focuses on the transformation of the resulting effective action into a nonlinearly realized effective pion action, whose higher-derivative interaction terms are parametrized by so-called low-energy couplings. Their counterparts in the linear model are determined from the Functional Renormalization Group flow of the momentum-dependent four-pion vertex, which is calculated in a fully O(4)-symmetric approximation by including also momentum-dependent {\sigma}{\pi} interactions as well as {\sigma} self-interactions. Consequently, these higher-derivative couplings are dynamically generated solely from quark and meson fluctuations, initialized at a hadronic scale. Despite our restriction to low-energy degrees of freedom, we find that the qualitative features of the fluctuation dynamics allow us to comment on the range of validity and on appropriate renormalization scales for purely pionic effective models.

Free energies of Boltzmann machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in artificial intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mechanics perspective these systems display the same (random) Gibbs measure of bi-partite spin-glasses, whose rigorous treatment is notoriously difficult.In this work, beyond providing a brief review on RBMs from both the learning and the retrieval perspectives, we aim to contribute to their analytical investigation, by considering two distinct realizations of their weights (i.e. Boolean and Gaussian) and studying the properties of their related free energies. More precisely, focusing on a RBM characterized by digital couplings, we first extend the Pastur–Shcherbina–Tirozzi method (originally developed for the Hopfield model) to prove the self-averaging property for the free energy, over its quenched expectation, in the infinite volume limit, then we explicitly calculate its simplest approximation, namely its annealed bound.Next, focusing on a RBM characterized by analogical weights, we extend Guerra’s interpolating scheme to obtain a control of the quenched free-energy under the assumption of replica symmetry (i.e. we require that the order parameters do not fluctuate in the thermodynamic limit): we get self-consistencies for the order parameters (in full agreement with the existing literature) as well as the critical line for ergodicity breaking that turns out to be the same obtained in AGS theory. As we discuss, this analogy stems from the slow-noise universality.Finally, glancing beyond replica symmetry, we analyze the fluctuations of the overlaps for a correct estimation of the (slow) noise affecting the retrieval of the signal, and by a stability analysis we recover the Aizenman–Contucci identities typical of glassy systems.

Approximations, ghosts and derived equivalences

AbstractApproximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weaklyn-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

Principal symmetric space analysis

Principal Geodesic Analysis is a statistical technique that constructs low-dimensional approximations to data on Riemannian manifolds. It provides a generalization of principal components analysis to non-Euclidean spaces. The approximating submanifolds are geodesic at a reference point such as the intrinsic mean of the data. However, they are local methods as the approximation depends on the reference point and does not take into account the curvature of the manifold. Therefore, in this paper we develop a specialization of principal geodesic analysis, Principal Symmetric Space Analysis, based on nested sequences of totally geodesic submanifolds of symmetric spaces. The examples of spheres, Grassmannians, tori, and products of two-dimensional spheres are worked out in detail. The approximating submanifolds are geometrically the simplest possible, with zero exterior curvature at all points. They can deal with significant curvature and diverse topology. We show that in many cases the distance between a point and the submanifold can be computed analytically and there is a related metric that reduces the computation of principal symmetric space approximations to linear algebra.

Nonlinear Gravitational Waves and Atmospheric Instability

This is an outline of the main results of the study of dust devils (a type of atmospheric vortices) with a focus on the mechanism of vortex generation in an unstable stratified atmosphere. In the approximation of ideal hydrodynamics, a new nonlinear model of the generation of convective motions and dust devils in an unstable stratified atmosphere has been developed. Using nonlinear equations for internal gravity waves, the model of generation of convective cell plumes has been investigated in the axially symmetric approximation. It has been shown that, in a convectively unstable atmosphere with large-scale seed vorticity, the plumes extremely rapidly generate small-scale intense vertical vortices. The structure of radial, vertical, and toroidal velocity components in these vortices has been investigated. The structure of vertical vorticity and toroidal velocity in vortex areas that are limited by radius has been examined.

New hybridized mixed methods for linear elasticity and optimal multilevel solvers

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions $$n=2,3$$ , which yields a conforming and strongly symmetric approximation for stress. Applying $$\mathcal {P}_{k+1}-\mathcal {P}_k$$ as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when $$k \ge n$$ . For the lower order case $$(n-2\le k<n)$$ , the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson’s ratio. Numerical experiments are provided to validate our theoretical results.

On the dynamics of near-extremal black holes

We analyse the dynamics of near-extremal Reissner-Nordström black holes in asymptotically four-dimensional Anti de Sitter space (AdS4). We work in the spherically symmetric approximation and study the thermodynamics and the response to a probe scalar field. We find that the behaviour of the system, at low energies and to leading order in our approximations, is well described by the Jackiw-Teitelboim (JT) model of gravity. In fact, this behaviour can be understood from symmetry considerations and arises due to the breaking of time reparametrisation invariance. The JT model has been analysed in considerable detail recently and related to the behaviour of the SYK model. Our results indicate that features in these models which arise from symmetry considerations alone are more general and present quite universally in near-extremal black holes.

Study on divergence approximation formula for pressure calculation in particle method

The moving particle semi-implicit method is a meshless particle method for incompressible fluid and has proven useful in a wide variety of engineering applications of free-surface flows. Despite its wide applicability, the moving particle semi-implicit method has the defects of spurious unphysical pressure oscillation. Three various divergence approximation formulas, including basic divergence approximation formula, difference divergence approximation formula, and symmetric divergence approximation formula are proposed in this paper. The proposed three divergence approximation formulas are then applied for discretization of source term in pressure Poisson equation. Two numerical tests, including hydrostatic pressure problem and dam-breaking problem, are carried out to assess the performance of different formulas in enhancing and stabilizing the pressure calculation. The results demonstrate that the pressure calculated by basic divergence approximation formula and difference divergence approximation formula fluctuates severely. However, application of symmetric divergence approximation formula can result in a more accurate and stabilized pressure.

\(L^p\)-approximation and generalized growth of generalized biaxially symmetric potentials on hyper sphere

The generalized order of growth and generalized type of an entire function \(F^{\alpha,\beta}\) (generalized biaxisymmetric potentials) have been obtained in terms of the sequence \(E_n^p(F^{\alpha,\beta},\Sigma_r^{\alpha,\beta})\) of best real biaxially symmetric harmonic polynomial approximation on open hyper sphere \(\Sigma_r^{\alpha,\beta}\). Moreover, the results of McCoy [8] have been extended for the cases of fast growth as well as slow growth.

Global Symmetric Approximation of Frames

We solve the problem of best approximation by Parseval frames to an arbitrary frame in a subspace of an infinite dimensional Hilbert space. We explicitly describe all the solutions and we give a criterion for uniqueness. Our proof relies on the geometric structure of the set of all Parseval frames quadratically close to a given frame. In the process we show that its connected components can be parametrized by using the notion of index of a pair of projections, and we prove existence and uniqueness results of best approximation by Parseval frames restricted to these connected components.