RM Research https://allthingsdynamical.github.io/blog.html Mathematical essays and reflections. quarto-1.9.38 Wed, 08 Jul 2026 00:00:00 GMT Gradient Flows for Linear Solvers Rahul Manavalan https://allthingsdynamical.github.io/posts/gradient-flows-linear-solvers/

The basic idea

Consider the quadratic residual functional

Its gradient flow is

For symmetric positive definite systems, a preconditioned flow gives

The exact finite-time update is

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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Back to top ]]> numerical-analysis linear-systems gradient-flows https://allthingsdynamical.github.io/posts/gradient-flows-linear-solvers/ Wed, 08 Jul 2026 00:00:00 GMT Nonlinear RKHS Regression as a Dissipative Flow Rahul Manavalan https://allthingsdynamical.github.io/posts/nonlinear-rkhs-regression/

Setup

Let be an unknown function in a Hilbert space . A nonlinear regression problem can be written abstractly as

The associated gradient flow is

This turns nonlinear regression into an infinite-dimensional dissipative dynamical system.

Analysis program

A natural paper structure is:

  1. well-posedness of the flow;
  2. existence of absorbing sets;
  3. existence and regularity of a global attractor;
  4. convergence to equilibria under a Łojasiewicz–Simon inequality;
  5. exponential-in-time discretization;
  6. spatial discretization by FEM or domain decomposition;
  7. numerical experiments.
Back to top ]]> rkhs pde dynamical-systems https://allthingsdynamical.github.io/posts/nonlinear-rkhs-regression/ Tue, 07 Jul 2026 00:00:00 GMT PDF Import Demo Rahul Manavalan https://allthingsdynamical.github.io/posts/pdf-import-demo/

Workflow

This template treats PDF upload as a draft-import step, not as the canonical writing format.

python scripts/pdf_to_post.py assets/papers/sample-paper.pdf

The script creates a new draft under posts/.

For math-heavy documents, keep the .tex, .qmd, or .md source whenever possible.

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