The adaptive-greedy parametric eigenvalue solver

Parametric eigenvalue problems ask for the eigenvalues and eigenvectors of an operator K(\mu) that depends on a parameter \mu ranging over a box \Omega \subset \mathbb{R}^d. They arise wherever a design, control, or sensitivity question needs the spectrum as a function of \mu — material properties for a strip of plates, geometric or boundary-condition coefficients in a PDE, frequency windows in a wave problem. Solving the full eigenproblem at every \mu of interest is expensive; the practical alternative is a surrogate built from a small, carefully chosen set of high-fidelity snapshots.

This site documents a clean implementation of the adaptive-greedy strategy for choosing those snapshots: start from a coarse sparse grid over the parameter box, ask an a-posteriori verifier whether the current surrogate is accurate along each edge of the grid, and bisect the edges that fail. The process terminates when every edge is certified. Once converged, querying the surrogate at a new \mu is a small dense generalised eigenproblem in the combined basis of the two enclosing snapshots — much cheaper than a full solve.

For a single execution-order walk through the whole pipeline — greedy loop → matching → verdict → replacement → plotting, each stage grounded in the code symbol that implements it — see How it works.

Two operating modes are supported. FirstK tracks the smallest n bands across the whole box. Window tracks every band that falls inside a fixed [\lambda_{\text{lo}}, \lambda_{\text{hi}}] window — the per-snapshot band count is then ragged.

How to read this site

The pages are organised into five sections, each with a distinct purpose.

Validation — before any adaptive machinery enters, two questions must be settled: does the finite-element method converge at the expected rate, and does the fast affine assembly reproduce direct assembly to floating-point round-off? Both building blocks are shared by the surrogate and the dense reference; a flaw in either is invisible to the downstream equivalence check.

  • Spectrum samples — FE convergence rate study and wiring smoke test against the closed-form eigenvalues (n\pi)^2.
  • Affine-cache cross-check — the precomputed component-matrix assembly reproduces direct assembly to floating-point round-off.

Demonstrations — the adaptive-greedy solver running on real problems. Four pages build outward from a reference case; any one can be read first.

Mechanisms — two non-obvious phenomena and the algorithmic responses to them. Neither page is a run; both are arguments.

  • Persistent Π failures at eigenvalue crossings — why Π is persistently off-diagonal at true crossings (a topological property, not a solver artefact) and why this is the precise reason the Projection Algorithm is needed.
  • Dynamic domain decomposition — why the global basis rank grows as the sum of per-regime ranks and how dynamic parameter-domain decomposition reduces it to the maximum.

Sensitivity — three honest-caveat studies asking whether the results change when the user varies something they control.

  • Mesh independence — the same adaptive grid and ROM accuracy at three spatial mesh resolutions.
  • Tolerance perturbation — tightening t_\pi by 20% terminates a level early and degrades surrogate error ~7×. The load-bearing caveat behind every other figure on the reference problem.
  • Knob sweep — per-knob sweep across all verdict thresholds; variation is sensitivity, not a bug.

Studies — research notes that justify the design choices with argument and measurement: why this algorithm at all, how it compares to alternative methods, and head-to-head A/B measurements of refinement strategies, refinement policies, the verdict pipeline’s runtime, and parallel execution.