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We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier--Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretised in…

In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal…

In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for…

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a…

This paper addresses the challenge of reconstructing a 3D power spectrum map from sparse, scattered, and incomplete spectrum measurements. It proposes an integrated approach combining interpolation and block-term tensor decomposition (BTD).…

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of…

Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have…

We propose a differential geometric construction for families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant…

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In…

We study the problem of space and time efficient evaluation of a nonparametric estimator that approximates an unknown density. In the regime where consistent estimation is possible, we use a piecewise multivariate polynomial interpolation…

Based on the tensor-based large margin distribution and the nonparallel support tensor machine, we establish a novel classifier for binary classification problem in this paper, termed the Large Margin Distribution based NonParallel Support…

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying…

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based…

We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…

Black-box optimization methods play an important role in many fields of computational simulation. In particular, such methods are often used in the design and modelling of biological systems, including proteins and their complexes with…

We introduce a new tensor integration method for time-dependent PDEs that controls the tensor rank of the PDE solution via time-dependent diffeomorphic coordinate transformations. Such coordinate transformations are generated by minimizing…

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional geometrical domain in the tensor train format; secondly, to develop a new algorithm…

In this study, we present a tensor--train framework for nonintrusive operator inference aimed at learning discrete operators and using them to predict solutions of physical governing equations. Our framework comprises three approaches:…

The tensor cross interpolation (TCI) algorithm is a rank-revealing algorithm for decomposing low-rank, high-dimensional tensors into tensor trains/matrix product states (MPS). TCI learns a compact MPS representation of the entire object…