Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient Shallow Ritz Method For 1D Diffusion Problems

Zhiqiang Cai, Anastassia Doktorova, Robert D. Falgout, César Herrera

TL;DR

This work develops an efficient shallow Ritz framework for 1D diffusion with non-smooth interfaces, leveraging a shallow ReLU NN that is equivalent to free-knot splines to move mesh points. It introduces a damped block Newton (dBN) method that updates linear parameters by exact inversion and nonlinear knot positions by a damped Newton step, achieving $O(n)$ per-iteration cost, and further enhances efficiency with Adaptive dBN (AdBN) by coupling with Adaptive Neuron Enhancement (ANE) to refine the mesh. Theoretical results include a bound on the stiffness matrix conditioning and a structure that yields a differentiable reduced nonlinear system, enabling reliable optimization of knots toward near-optimal convergence. Numerical experiments across exponential, non-smooth, and interface problems show that dBN and especially AdBN outperform classical solvers (e.g., BFGS) in both speed and accuracy, demonstrating the practical potential of moving-mesh Ritz-NN discretizations for elliptic problems with interfaces.

Abstract

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].

Efficient Shallow Ritz Method For 1D Diffusion Problems

TL;DR

This work develops an efficient shallow Ritz framework for 1D diffusion with non-smooth interfaces, leveraging a shallow ReLU NN that is equivalent to free-knot splines to move mesh points. It introduces a damped block Newton (dBN) method that updates linear parameters by exact inversion and nonlinear knot positions by a damped Newton step, achieving per-iteration cost, and further enhances efficiency with Adaptive dBN (AdBN) by coupling with Adaptive Neuron Enhancement (ANE) to refine the mesh. Theoretical results include a bound on the stiffness matrix conditioning and a structure that yields a differentiable reduced nonlinear system, enabling reliable optimization of knots toward near-optimal convergence. Numerical experiments across exponential, non-smooth, and interface problems show that dBN and especially AdBN outperform classical solvers (e.g., BFGS) in both speed and accuracy, demonstrating the practical potential of moving-mesh Ritz-NN discretizations for elliptic problems with interfaces.

Abstract

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 . 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].
Paper Structure (12 sections, 7 theorems, 64 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 12 sections, 7 theorems, 64 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Free-Knot Splines and Shallow Neural Networks
  3. Shallow Ritz Method
  4. Optimality Conditions
  5. A Damped Block Newton Method
  6. Adaptive dBN (AdBN) Method
  7. Numerical Experiments
  8. Exponential Solution
  9. Non-smooth Solution
  10. Interface Problem
  11. Conclusion and Discussion
  12. Enforcing the Dirichlet Boundary Condition Algebraically

Key Result

Lemma 3.1

\newlabell:error_estimate10 Let $u$ and $u_n$ be the solutions of problems energy_functional and min, respectively. Then

Figures (4)

  • Figure 1: Comparison between BFGS and dBN for approximating function \ref{['Example1eq']}
  • Figure 2: (a) initial approximation with $20$ uniform breakpoints, $e_n = 0.250$, (b) approximation after $500$ iterations, $e_n = 0.104$, (c) adaptive approximation ($n=13,16, 20$), $e_n = 0.092$.
  • Figure 3: (a) initial approximation with $22$ uniform breakpoints, $e_n = 0.300$, (b) approximation after $500$ iterations, $e_n = 0.086$, (c) adaptive approximation ($n=11, 16, 22$), $e_n = 0.063$.
  • Figure 4: For $k = 10^{6}$: (a) initial approximation with $15$ uniform breakpoints, $e_n = 0.204$ and (b) approximation after 100 iterations, $e_n =0.073$.

Theorems & Definitions (18)

  • Lemma 3.1
  • Proof 1
  • Proposition 3.2
  • Proof 2
  • Definition 3.3
  • Lemma 4.1
  • Proof 3
  • Remark 4.2
  • Lemma 4.3
  • Proof 4
  • ...and 8 more