Tensor Research Articles

Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces. However, this nonlocality makes it challenging to design efficient solvers for such problems. In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains. We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs. In addition, we introduce both serial and parallel domain decomposition solvers. We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints.

Preconditioned TBiCOR and TCORS algorithms for solving the Sylvester tensor equation

In this paper, the preconditioned TBiCOR and TCORS methods are presented for solving the Sylvester tensor equation. A tensor Lanczos L-Biorthogonalization algorithm (TLB) is derived for solving the Sylvester tensor equation. Two improved TLB methods are presented. One is the biconjugate L-orthogonal residual algorithm in tensor form (TBiCOR), which implements the LU decomposition for the triangular coefficient matrix derived by the TLB method. The other is the conjugate L-orthogonal residual squared algorithm in tensor form (TCORS), which introduces a square operator to the residual of the TBiCOR algorithm. A preconditioner based on the nearest Kronecker product is used to accelerate the TBiCOR and TCORS algorithms, and we obtain the preconditioned TBiCOR algorithm (PTBiCOR) and preconditioned TCORS algorithm (PTCORS). The proposed algorithms are proved to be convergent within finite steps of iteration without roundoff errors. Several examples illustrate that the preconditioned TBiCOR and TCORS algorithms present excellent convergence.

Stability and convergence of Strang splitting. Part II: Tensorial Allen-Cahn equations

We consider the second-order in time Strang-splitting approximation for tensorial (e.g. vector-valued and matrix-valued) Allen-Cahn equations. Both the linear propagator and the nonlinear propagator are computed explicitly. For the vector-valued case, we prove the maximum principle and unconditional energy dissipation for a judiciously modified energy functional. The modified energy functional is close to the classical energy up to O(τ) where τ is the splitting step. For the matrix-valued case, we prove a sharp maximum principle in the matrix Frobenius norm. We show modified energy dissipation under very mild splitting step constraints. We exhibit several numerical examples to show the efficiency of the method as well as the sharpness of the results.

Реалізація прийняття рішень організаційного управління на основі онтології діяльності

The complexity of the tasks of making decisions in organizational management is stipulated by the extremely large dimension of the management space, the weak structure of the tasks being solved due to their uniqueness, the uncertainty of conditions, and the changeability of the goals (aspects) of management. The ontological approach provides a range of advantages in solving the problem of organizational management automatization. The use of the activity ontology makes it possible to significantly expand the scope of formalized methods application to organ-izational multi-aspect management. Due to the ontology, the detailing of managerial alterna-tives is increased, and the factors of the performers’ activity and the multi-aspect nature of management are taken into account. The article considers the interpretation of the process of forming managerial alternatives as a process of the threefold grouping of elements of the activi-ty ontology and discrete moments of time. The method of distributed formation of an ontologi-cal network with the grouping of its elements using expert methods, implemented in the prac-tice of designing corporate ACS, is described. The topical scientific tasks of the development of the scientific and methodological apparatus of organizational management based on ontology are formulated. These tasks include the formation of decision-making spaces that are invariant to aspects of management, the concretization of characteristics-invariants and reconfiguration of procedural regulations concerning the actual aspect of management, synthesis of alternatives based on the ontology elements. Some approaches to solving the formulated tasks are shown. The formulation of the problem of synthesizing a system of procedures with given input and output parameters is presented as a problem of tensor transformation of an initial multi-coil electrical network (library of procedures) into a network with the given parameters. A tensor equation is proposed. Its solution allows determining the values of voltages on the intermediate coils of the original electrical network which forms the structural connections for the synthe-sized procedural regulations.

A note on the solvability of a tensor equation

<p style='text-indent:20px;'>In this note, we generalize a solvability result for a tensor equation with a nonsingular leading tensor to the case with possibly a singular leading tensor.</p>

Eigenvalues of the solution of the Lyapunov tensor equation

Eigenvalues of the solution of the Lyapunov tensor equation

Weighted generalized tensor functions based on the tensor-product and their applications

There are three weighted decompositions of tensors proposed in this paper, and the corresponding definitions of the weighted generalized tensor functions are given. The Cauchy integral formula of the weighted Moore-Penrose inverse is developed for solving the tensor equations. Besides above, we give the weighted projection tensors to discuss the representations of the weighted generalized power of tensors. Finally, some special tensors are studied which can preserve the structural invariance under the tensor functions defined in this paper.

Einstein-Klein-Gordon System in Higher Dimensional

In this present work, we study the Einstein equation coupled with the nonlinear Klein-Gordon equation. We obtain Ricci tensor, scalar curvature, and Einstein equation of the Einstein-Klein-Gordon system in higher dimensional. If we put D=4, our formulations reduce to the four dimensional Einstein-Klein-Gordon system.

Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields II

The geometry of curved ‘three-dimensional’ absolute intrinsic metric space (an absolute intrinsic Riemannian metric space) \(\varnothing\hat{\mathbb{M}}^3\), which is curved (as a curved hyper-surface) toward the absolute time/absolute intrinsic time ‘dimensions’ (along the vertical), and projects a flat three-dimensional absolute proper intrinsic metric space \(\varnothing\hat{\mathbb{E}}^\prime\)ab3 and its outward manifestation namely, the flat absolute proper 3-space \(\hat{\mathbb{E}}^\prime\)ab3, both as flat hyper-surfaces along the horizontal, isolated in the first part of this paper, is subjected to graphical analysis. Two absolute intrinsic metric tensor equations, one of which is of the form of Einstein free space field equations and the other which is a tensorial statement of absolute intrinsic local Euclidean invariance (A\(\varnothing\)LEI) on \(\varnothing\hat{\mathbb{M}}^3\), are derived. Simultaneous (algebraic) solution to the equations yields the absolute intrinsic metric tensor and the absolute intrinsic Ricci tensor of absolute intrinsic Riemann geometry on the curved \(\varnothing\hat{\mathbb{M}}^3\), in terms of a derived absolute intrinsic curvature parameter. A superposition procedure that yields the resultant absolute intrinsic metric tensor and the resultant absolute intrinsic Ricci tensor, when two or a larger number of absolute intrinsic Riemannian metric spaces co-exist (or are superposed) is developed.
 The fact that a curved ‘three-dimensional’ absolute intrinsic metric space \(\varnothing\hat{\mathbb{M}}^3\) is perfectly isotropic and is consequently contracted to a ‘one-dimensional’ isotropic absolute intrinsic metric space, denoted by \(\varnothing\hatρ^3\), which is curved toward the absolute time/absolute intrinsic time ’dimensions’ (\(\hat{c}\)\(\hat{t}\)/\(\varnothing\)\(\hat{c}\)\(\varnothing\)\(\hat{t}\)) along the vertical is derived.

Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli

A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined, and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect of finite rotations is fully taken into account and analyzed. Different types of the bulk and shear moduli under different constraints are defined and connected to the effective properties of polycrystalline aggregates. Expressions for elastic energy and stress-strain relationships for small distortions with respect to a pre-stressed configuration are derived in detail. Under initial hydrostatic load, general consistency conditions for elastic moduli and compliances are derived that follow from the existence of the generalized tensorial equation of state under hydrostatic loading obtained from single crystal or polycrystal. It is shown that B moduli can be found from the expression for the Gibbs energy. However, higher-order elastic moduli defined from the Gibbs energy do not have any meaning since they do not directly participate in any known equations, like stress-strain relationships and wave propagation equation. The deviatoric projection of B can also be found from the expression for the elastic energy for isochoric small strain increments, and the missing components of B can be found from the consistency conditions. Numerous inconsistencies and errors in the known works are analyzed.

A complementary covariant approach to gravito-electromagnetism

From a previous paper where we proposed a description of general relativity within the gravito-electromagnetic limit, we propose an alternative modified gravitational theory. As in the former version, we analyze the vector and tensor equations of motion, the gravitational continuity equation, the conservation of the energy, the energy-momentum tensor, the field tensor, and the constraints concerning these fields. The Lagrangian formulation is also exhibited as an unified and simple formulation that will be useful for future investigation.

Predefined-time convergent neural networks for solving the time-varying nonsingular multi-linear tensor equations

Equation solving is one of the main subjects in various research fields. In this study, we consider tensor equations with time-varying nonsingular coefficient tensors and right-hand side vectors, which generalize the time-invariant tensor equations and also the time-varying or time-invariant linear equations studied by many researchers. Four complex-valued neural networks termed as ZNN-I, ZNN-II, WsbpPTZNN-I and WsbpPTZNN-II are proposed for solving this problem. We show that, under certain conditions, the four models converge to a solution of the original tensor equations. Moreover, we prove that WsbpPTZNN-I and WsbpPTZNN-II models have predefined-time convergence property, which means that they converge to a true solution in a prescribed time. The upper bounds of the convergence time are also given. For comparison, the gradient-based model, termed as GNN, is also introduced. Through many numerical tests, we demonstrate the validity of our theoretical analysis of these models. The simulation results also illustrate the effectiveness and reliability of our proposed neural network models and show that their convergence performance can be improved by using the predefined-time convergent WsbpPTZNN models.

Microscopic Theory for the Rheology of Jammed Soft Suspensions.

We develop a constitutive model allowing for the description of the rheology of two-dimensional soft dense suspensions above jamming. Starting from a statistical description of the particle dynamics, we derive, using a set of approximations, a nonlinear tensorial evolution equation linking the deviatoric part of the stress tensor to the strain-rate and vorticity tensors. The coefficients appearing in this equation can be expressed in terms of the packing fraction and of particle-level parameters. This constitutive equation rooted in the microscopic dynamic qualitatively reproduces a number of salient features of the rheology of jammed soft suspensions, including the presence of yield stresses for the shear component of the stress and for the normal stress difference. More complex protocols like the relaxation after a preshear are also considered, showing a smaller stress after relaxation for a stronger preshear.

On tensor tubal-Krylov subspace methods

In this paper, we will introduce some new tubal-Krylov subspace methods for solving linear tensor equations. Using the well known tensor T-product, we will in particular define the tensor tubal-global GMRES that could be seen as a generalization of the global GMRES. We also give a new tubal-version of the tensor Golub–Kahan algorithm. To this end, we first introduce some new tensor-tensor products and new related algebraic properties. The presented numerical tests compare the two methods and show the efficiency of the proposed procedures.

Improved finite-time solutions to time-varying Sylvester tensor equation via zeroing neural networks

Based on the standard method of zeroing neural network (ZNN) for time-varying matrix problems, three improved ZNN models are proposed and extended to solve the time-varying Sylvester tensor equation (TV-STE) in finite-time. These presented ZNN models, which mainly use classical sign-bi-power (S-B-P) activation function and varying parameters, are S-B-P based ZNN (S-ZNN), S-B-P function based linear varying parameter ZNN (LVP-ZNN) and S-B-P function based exponential varying parameter ZNN (EVP-ZNN) models. Due to the applications of S-B-P activation function and varying parameters, the S-ZNN, LVP-ZNN and EVP-ZNN models can accomplish finite-time convergence, i.e., the state solutions generated by these models can converge to the analytical solutions of the TV-STE problem in finite-time. Further, based on the relationship between error, design parameter and convergence property, a dynamic varying parameter adapted to the error variation is designed and applied to the S-ZNN model. Then an error-adaptive dynamic varying parameter ZNN (DVP-ZNN) model is obtained for the TV-STE, which possesses faster finite-time convergence. These four models are proved to be stable and their upper bounds of convergence time are derived and analyzed. Theoretical analyses and comparison experiments demonstrate that the proposed ZNN models can solve the TV-STE in finite-time and the convergence of the DVP-ZNN model is the best.

Some iterative approaches for Sylvester tensor equations, Part I: A tensor format of truncated Loose Simpler GMRES

In recent years, Krylov subspace methods based on the tensor format have demonstrated their superiority over classical ones for handling various kinds of tensor equations. In this paper, according to the efficient computational cost of the classical Simpler GMRES method (SGMRES), we adopt the tensor format of this method, which is called SGMRES−BTF, for solving the Sylvester tensor equations. This paper is divided into two parts describing the tensor format of two types of acceleration approaches to tackle some serious problems of the restarted methods GMRES−BTF and SGMRES−BTF, such as the “stalling” and “alternating phenomenon”. In the first part of this two-part work, the truncated Loose Simpler GMRES based on the tensor format (LSGMRES−BTF) is introduced, which is an acceleration approach that aims to construct an augmented tensor Krylov subspace by means of approximation error tensors. Moreover, a detailed study is carried out on the connection between the values of sequential and skip angles and the convergence behavior of both SGMRES−BTF and truncated LSGMRES−BTF. In the second part of this paper, an acceleration approach based on the idea of inner-outer iteration in the truncated version of the generalized conjugate residual with inner orthogonalization (GCRO) method is developed. In this method, SGMRES−BTF is applied in the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Numerical experiments show that SGMRES−BTF achieves an appropriate performance compared with GMRES−BTF. In addition, the numerical results reveal the high potential of the presented accelerating strategies to deal with Sylvester tensor equations.

Some iterative approaches for Sylvester tensor equations, Part II: A tensor format of Simpler variant of GCRO-based methods

In the second part of this two-part work, another accelerator method based on the tensor format is established for solving the Sylvester tensor equations. This acceleration approach, which is called SGCRO−BTF, is based on the idea of inner-outer iteration used in the generalized conjugate residual with inner orthogonalization (GCRO) method. In SGCRO−BTF, the Simpler GMRES method based on the tensor format (SGMRES−BTF) is applied to the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Furthermore, SGCRO−BTF seeks an approximate solution over a tensor subspace spanned by the approximation error tensors produced during the previous outer iterations of SGCRO−BTF and a tensor Krylov subspace constructed by the inner iteration. In order to reduce the computational storage in the outer iteration, the truncated version of SGCRO−BTF is presented, in which only some of the last approximation error tensors are kept and will be then added to the tensor Krylov subspace in order to obtain a new search subspace. Finally, the proposed methods are tested on a set of experiments and compared to some conventional and state-of-the-art Krylov subspace methods based on the tensor format. Experimental results indicate that the truncated version of SGCRO−BTF is particularly effective for solving the Sylvester tensor equations.

Classical equations of an electron from the special form of the Dirac equation

The equivalent system of equations corresponding to the Dirac equation is derived and the WKB approximation of this system is found. Similarly, the WKB approximation for the equivalent system of equation corresponding to the squared Dirac equation is found and it is proved that the Lorentz equation and the Bargmann-Michel-Telegdi iquations follow from the new Dirac-Pardy system. The new tensor equation with sigma matrix is derived for the verification by adequate laboratories.

(3 + 1)-formulation for gravity with torsion and non-metricity: II. The hypermomentum equation

In this article, we consider a particular case of metric-affine -gravity for , i.e. the metric-affine general relativity (MAGR). As a companion to the first article in the series, we perform the (3 + 1) decomposition to the hypermomentum equation, obtained from the minimization of the MAGR action with respect to the connection ω. Moreover, we show that the hypermomentum tensor could be entirely constructed from ten hypersurfaces variables that arise from its dilation, shear, and rotational (spin) parts. The (3 + 1) hypermomentum equations consist of one scalar equation, three vector equations, three matrix equations, and one tensor equation of order-. Together with the (3 + 1) decomposition of the traceless torsion constraint, which consists of a scalar and a vector equation, we obtain ten hypersurface equations that are the main result in this article. Finally, we consider some special cases of MAGR, namely, the zero hypermomentum, metric, and torsionless cases. For vanishing hypermomentum, we could retrieve the metric compatibility and torsionless condition in the (3 + 1) framework, forcing the affine connection to be Levi-Civita as in the standard general relativity.

The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure

AbstractThis paper discusses an application of partial tensor Golub–Kahan bidiagonalization to the solution of large‐scale linear discrete ill‐posed problems based on the t‐product formalism for third‐order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641‐658). The solution methods presented first reduce a given (large‐scale) problem to a problem of small size by application of a few steps of tensor Golub–Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third‐order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third‐order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill‐posed problems defined by a matrix operator to linear discrete ill‐posed problems defined by a third‐order tensor operator. An interlacing property of singular tubes for third‐order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t‐product approach, in comparison with solution methods that are based on matricization of the tensor equation.