arXiv subjects

Anastassia Doktorova

Publications and source records attributed to Anastassia Doktorova.

3 recordsLinked to original sources

arXiv2026

Convergence Analysis of Block Newton Methods for 1D Shallow Neural Network Approximation

This paper analyzes local convergence of the block Newton (BN) method introduced in [5, 6] for one-dimensional shallow neural network approximation to functions and diffusion-reaction problems. The BN method consists of the 2x2 block nonlinear Gauss-Seidel, linear Gauss-Seidel, or Jacobi method for outer iteration and the Newton method for inner iteration. The blocks are corresponding to the linear and the nonlinear parameters. Under some reasonable assumptions, we establish local convergence of the BN methods as well as the reduced BN (rBN) method for one-dimensional diffusion-reaction problems and least-squares function approximation. Unlike common optimization methods, the rBN allows for the reduction of the number of parameters during the optimization process when some neurons contribute little to the approximation or are at nearly optimal locations.

math.NA↗

arXiv2025

Efficient Shallow Ritz Method For 1D Diffusion Problems

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $O(n)$. Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. To improve efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [26].

math.NA↗

arXiv2025

Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problems

This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented in [9], we present a damped block Newton (dBN) method to achieve nearly optimal order of approximation. The dBN method optimizes the Ritz functional by alternating between the linear and non-linear parameters of the shallow ReLU neural network (NN). For diffusion-reaction problems, new difficulties arise: (1) for the linear parameters, the mass matrix is dense and even more ill-conditioned than the stiffness matrix, and (2) for the non-linear parameters, the Hessian matrix is dense and may be singular. This paper addresses these challenges, resulting in a dBN method with computational cost of ${\cal O}(n)$. The ideas presented for diffusion-reaction problems can also be applied to least-squares approximation problems. For both applications, starting with the non-linear parameters as a uniform partition, numerical experiments show that the dBN method moves the mesh points to nearly optimal locations.

math.NA↗