Implicit multilinear modeling

The paper introduces a subclass of nonlinear differential-algebraic models of interest for applications. By restricting the nonlinearities to multilinear polynomials, it is possible to use modern tensor methods. This opens the door to new approximation and complexity reduction methods for large scale systems with relevant nonlinear behavior. The modeling procedures including composition, decomposition, normalization, and multilinearization steps are shown by an example of a local energy system with a nonlinear electrolyzer, a linear buck converter and a PI controller with saturation.

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.

Local convergence of tensor methods

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.

SHORTENED MAPPINGS OF SPACES WITH AFFINE CONNECTIVITY

The long history of theory of mappings was revived thanks to the tensor methods of inquiry. The notion of affine connectivity was introduced a hundred years ago. It enabled us to look at classic geometric problems from a different angle. Following the common tradition, this paper introduces a notion of a mapping for a space of affine connectivity. Modifying the method of A. P. Norden, we found the formulae for the main tensors: deformation tensor, Riemann tensor, Ricci tensor and their first and second covariant derivatives for spaces and , which are connected by a given mapping. These formulae contain both objects of and with covariant derivatives in respect to relevant connectivities. In order to simplify the expression, we introduced the notion of shortened mapping and its particular case: a half-mapping. The connectivity that appears in the case of a half-mapping is called a medium connectivity. The above mentioned formulae can be notably simplified in the case of transition to covariant derivatives in the medium connectivity. This fact permits us to obtain characteristics (the necessary conditions) for the estimates whether an object of inner character from the space of affine connectivity is preserved under a given type of mappings. Objects of the inner character are geometric objects implied by an affine connectivity. They include Riemann tensor, Ricci tensor, Weyl tensor. Every type of mapping received its own set of differential equations in covariant derivatives, which define a deformation tensor of connectivity with a necessity. The study of these equations can proceed by a research on integrability conditions. Integrability conditions are algebraic over-defined systems. That’s why there is a constant need in introduction of additionally specialized spaces or certain objects of these spaces. Applying the method of N. S. Sinyukov and J. Mikes, in the case of certain algebraic conditions, we obtained a form of a deformation tensor for a given mapping. Let us note that the medium connectivity was selected in order to simplify the calculations. Depending on the type of a model under consideration or on the physical limitations, we can construct any other connectivity (and mappings), which would be better suited for the given conditions. This approach is particularly fruitful when applied for invariant transformations connecting pairs of spaces of affine connectivity via their deformation tensor of connectivity.