Gradient Flows for Linear Solvers
numerical-analysis
linear-systems
gradient-flows
A short note on deriving iterative solvers from continuous residual flows.
The basic idea
Consider the quadratic residual functional
\[ \Phi(x) = \frac12 \|Ax-b\|^2. \]
Its gradient flow is
\[ \dot{x}(t) = -A^\top(Ax(t)-b). \]
For symmetric positive definite systems, a preconditioned flow gives
\[ \dot{x}(t) = P(b-Ax(t)). \]
The exact finite-time update is
\[ x(t+\tau) = x(t) + (PA)^{-1}\left(I-\exp(-\tau PA)\right)P(b-Ax(t)). \]
This suggests a family of solvers indexed by the time step and by the approximation of the matrix function.
Why this is useful
Small time steps recover classical stationary iterations. Large time steps approximate projection onto a Krylov or spectral subspace. The flow viewpoint gives a clean language for stability, filtering, and acceleration.
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