In tensor completion, the goal is to fill in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on ...the manifold of tensors of fixed multilinear rank. More specifically, a variant of the nonlinear conjugate gradient method is developed. Paying particular attention to efficient implementation, our algorithm scales linearly in the size of the tensor. Examples with synthetic data demonstrate good recovery even if the vast majority of the entries are unknown. We illustrate the use of the developed algorithm for the recovery of multidimensional images and for the approximation of multivariate functions.
We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution ...to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.
Considering positive
H
(
curl
,
Ω
)
problems, we propose a substructured version of the Hiptmair-Xu preconditioner based on a new formula that expresses the inverse of Schur systems in terms of the ...inverse matrix of the global volume problem.
In this research, a Bernoulli wavelet operational matrix of fractional integration is presented. Bernoulli wavelets and their properties are employed for deriving a general procedure for forming this ...matrix. The application of the proposed operational matrix for solving the fractional delay differential equations is explained. Also, upper bound for the error of operational matrix of the fractional integration is given. This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients. Several numerical examples are solved to demonstrate the validity and applicability of the presented technique.
The satisfactory location for the sources outside the closure of the domain of the problem under consideration remains one of the major issues in the application of the method of fundamental ...solutions (MFS). In this work we investigate this issue by means of two algorithms, one based on the satisfaction of the boundary conditions and one based on the leave-one-out cross validation algorithm. By applying these algorithms to several numerical examples for the Laplace and biharmonic equations in a variety of geometries in two and three dimensions, we obtain locations of the sources which lead to highly accurate results, at a relatively low cost.
Our aim in this paper is to introduce an extragradient-type method for solving variational inequality with uniformly continuous pseudomonotone operator. The strong convergence of the iterative ...sequence generated by our method is established in real Hilbert spaces. Our method uses computationally inexpensive Armijo-type linesearch procedure to compute the stepsize under reasonable assumptions. Finally, we give numerical implementations of our results for optimal control problems governed by ordinary differential equations.
A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is ...extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator.
In this paper, we extend modulus-based matrix splitting iteration methods to horizontal linear complementarity problems. We consider both standard and accelerated methods and analyze their ...convergence. In this context, we also generalize existing results on modulus-based matrix splitting iteration methods for (non-horizontal) linear complementarity problems. Lastly, we analyze the proposed methods by numerical experiments involving both symmetric and non-symmetric matrices.
In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators to be co-coercive. This ...restrictive co-coercive assumption has been dispensed with in some methods, many of which require a product space formulation of the problem. However, it has been discovered that this product space formulation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting structure of the split variational inequality problem. In this paper, we present two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation. We prove that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous. Also, we provide several numerical experiments of the proposed methods in comparison with other related methods in the literature.
In this article, we provide a rigorous study on the fractal dimension of the graph of the mixed Riemann-Liouville fractional integral for various choices of continuous functions on a rectangular ...region. We estimate bounds for the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of the functions which belong to the class of continuous functions and the class of Hölder continuous functions. We also show that the box dimension of the graph of the mixed Riemann-Liouville fractional integral of two-dimensional continuous functions is also two. Furthermore, we give the construction of unbounded variational continuous functions. Later, we prove that the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of unbounded variational continuous functions are two. Moreover, we illustrate our results by using some examples.
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