13: Anderson Acceleration

[philipnickel]

Description

Implement Anderson acceleration to speed up convergence of nonlinear iterations. Useful for implicit time-stepping or steady-state wave problems.

Background

Anderson acceleration improves fixed-point iterations x_{k+1} = G(x_k) by mixing previous iterates:

x_{k+1} = Σᵢ αᵢ · G(x_{k-m+i})

where coefficients α are found by least-squares to minimize residual.

Tasks

• Implement Anderson mixing algorithm
  • Store m previous iterates and residuals
  • Solve least-squares for coefficients
  • Apply mixing
• Integrate with time-stepping (for implicit schemes)
• Test on model problem
• Compare iteration counts: Picard vs Anderson

Acceptance Criteria

• Converges to same solution as Picard iteration
• Reduces iteration count by factor of 2-3x
• Robust for depth m = 3-5
• Works for various wave conditions

Parameters

Depth: m = 3-5 (number of previous iterates to use)
Regularization: small ε for least-squares stability