• Abstract

In this study, based on an iterative method to solve nonlinear equations, a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed. Numerical comparisons of the proposed method with other methods show that the average number of iterations, number of the Einstein products, and CPU time of our method are considerably less than other methods. In some applications, partial and fractional differential equations that lead to sparse matrices are considered as prototypes. We use the iterates obtained by the method as a preconditioner, based on tensor form to solve the multilinear system $\mathcal{A}∗_N\mathcal{X}=\mathcal{B}.$ Finally, several practical numerical examples are also given to display the accuracy and efficiency of the new method. The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.

• General Summary

• Abstract

In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${\rm ReLU}^2$ activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth $\mathcal{D},$ width $\mathcal{W}$ of the ${\rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

• General Summary

• Abstract

We apply the local method of fundamental solutions (LMFS) to boundary value problems (BVPs) for the Laplace and homogeneous biharmonic equations in annuli. By appropriately choosing the collocation points, the LMFS discretization yields sparse block circulant system matrices. As a result, matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs) can be used for the solution of the systems resulting in considerable savings in both computational time and storage requirements. The accuracy of the method and its ability to solve large scale problems are demonstrated by applying it to several numerical experiments.

• General Summary