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Non-negative and bounded-variable linear regression problems arise in a variety of applications in machine learning and signal processing. In this paper, we propose a technique to accelerate existing solvers for these problems by…
In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image-processing, graph partition and classification, nonlinear physics and more. The canonical eigenproblem we…
We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale…
Eigenvalue problems are critical to several fields of science and engineering. We present a novel unsupervised neural network for discovering eigenfunctions and eigenvalues for differential eigenvalue problems with solutions that…
Recently, deep neural networks (DNNs) have shown advantages in accelerating optimization algorithms. One approach is to unfold finite number of iterations of conventional optimization algorithms and to learn parameters in the algorithms.…
We describe a convergence acceleration technique for unconstrained optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average…
Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in…
The authors previously considered a method solving optimization problems by using a system of interconnected network of two component Bose-Einstein condensates (Byrnes, Yan, Yamamoto New J. Phys. 13, 113025 (2011)). The use of bosonic…
In this work we approach the Schr\"odinger equation in quantum wells with arbitrary potentials, using the machine learning technique. Two neural networks with different architectures are proposed and trained using a set of potentials,…
Nonlinear eigenvalue problems (NEPs) present significant challenges due to their inherent complexity and the limitations of traditional linear eigenvalue theory. This paper addresses these challenges by introducing a nonlinear…
As a first approximation beyond linearity, the nonlinear Schr\"odinger equation (NLSE) reliably describes a broad class of physical systems. Though numerical solutions of this model are well-established, these methods can be computationally…
In this work, we use an explainable convolutional neural network (NLS-Net) to solve an inverse problem of the nonlinear Schr\"odinger equation, which is widely used in fiber-optic communications. The landscape and minimizers of the…
Nonlinear eigenvalue problems with eigenvector nonlinearities (NEPv) are algebraic eigenvalue problems whose matrix depends on the eigenvector. Applications range from computational quantum mechanics to machine learning. Due to its…
We introduce and analyze a fast iterative method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. The key idea, in contrast to the standard Landweber method, is to use multiple search directions per…
In this work, we solve inverse problems of nonlinear Schr\"{o}dinger equations that can be formulated as a learning process of a special convolutional neural network. Instead of attempting to approximate functions in the inverse problems,…
The nonlinear Schr\"odinger equation (NLSE) is a fundamental model for wave dynamics in nonlinear media ranging from optical fibers to Bose-Einstein condensates. Accurately estimating its parameters, which are often strongly correlated,…
This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each…
This article deals with the numerical integration in time of nonlinear Schr\"odinger equations. The main application is the numerical simulation of rotating Bose-Einstein condensates. The authors perform a change of unknown so that the…
In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for…
This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary…