Studies — why this design, measured

Seven research notes that justify the algorithm’s design choices with argument and measurement rather than demonstration. Where the other sections show the solver running, these pages ask why it is built this way — what the alternatives were, what each costs, and what the numbers say when two designs are run head-to-head under identical, deterministic conditions.

Read in any order; a natural path is motivation → comparison framework → the measured A/B studies.

Pages

Why this algorithm

The motivation note: six drivers behind an adaptive-greedy approach to parametric eigenvalue problems — the cost of dense parameter sampling, why eigenvalue crossings break standard sensitivity theory, the absence of a standard technique linking eigensolutions across parameter values, and the method’s place in the reduced-basis greedy lineage. Each section closes with published sources.

Method comparison (1-D)

A forward-looking comparison framework positioning the adaptive-greedy algorithm against alternative methods for the same problem — uniform dense sampling, warm-started iterative eigensolvers, predictor–corrector and subspace continuation, polynomial interpolation of the operator, classical reduced-basis sampling, and contour-integration solvers — with fixed axes (cost currency, accuracy metric, failure modes) so that future head-to-head experiments are comparable by construction.

Coverage-policy validation

Does adding an approximation-coverage branch to the certification-only refinement predicate spend the eigensolve budget better? A measured cost-accuracy frontier: the surplus indicator with a self-calibrated threshold beats the certification-only floor; the power-function indicator over-refines; a too-aggressive threshold degrades accuracy.

Runtime analysis

A per-phase cost profile of the a-posteriori verdict pipeline and an end-to-end A/B against two retired verdict modes on the same problems. The pipeline is a single-digit-to-teens share of wall-clock; total cost is driven by verdict behavior, where the surviving one-path design dominates or ties. Closes with a ranked list of optimization opportunities quantified against the measured profile.

Parallel execution

The measured half of the parallelism pair: per-level eigensolves on a bounded worker pool, with byte-identical results to the serial path, and the BLAS-oversubscription fix that turns a slowdown into a speedup at large problem sizes. Three regimes — overhead-bound, crossover, win — located by measurement.

Region-parallel speedup

The projected half of the parallelism pair: an analytical layer that turns per-region work counts from domain-decomposed runs into projected speedup, parallel efficiency, and load balance — validated in one, two, and six parameter dimensions, and paired with the measured page above.

--- title: "Studies — why this design, measured" --- Seven research notes that justify the algorithm's design choices with argument and measurement rather than demonstration. Where the other sections show the solver running, these pages ask *why it is built this way* — what the alternatives were, what each costs, and what the numbers say when two designs are run head-to-head under identical, deterministic conditions. Read in any order; a natural path is motivation → comparison framework → the measured A/B studies. ## Pages ### [Why this algorithm](why-this-algorithm.md) The motivation note: six drivers behind an adaptive-greedy approach to parametric eigenvalue problems — the cost of dense parameter sampling, why eigenvalue crossings break standard sensitivity theory, the absence of a standard technique linking eigensolutions across parameter values, and the method's place in the reduced-basis greedy lineage. Each section closes with published sources. ### [Method comparison (1-D)](method-comparison-1d.md) A forward-looking comparison framework positioning the adaptive-greedy algorithm against alternative methods for the same problem — uniform dense sampling, warm-started iterative eigensolvers, predictor–corrector and subspace continuation, polynomial interpolation of the operator, classical reduced-basis sampling, and contour-integration solvers — with fixed axes (cost currency, accuracy metric, failure modes) so that future head-to-head experiments are comparable by construction. ### [Refinement strategy A/B](strategy-comparison.md) Edge-bisection versus hierarchical parent→child refinement, measured head-to-head: accuracy per eigensolve on the reference reproduction grids, the admissibility tax of the two-sided variant, and a constructed anisotropic pocket exposing the direction edge-bisection structurally cannot reach. ### [Coverage-policy validation](coverage-policy-validation.md) Does adding an approximation-coverage branch to the certification-only refinement predicate spend the eigensolve budget better? A measured cost-accuracy frontier: the surplus indicator with a self-calibrated threshold beats the certification-only floor; the power-function indicator over-refines; a too-aggressive threshold degrades accuracy. ### [Runtime analysis](runtime-analysis.md) A per-phase cost profile of the a-posteriori verdict pipeline and an end-to-end A/B against two retired verdict modes on the same problems. The pipeline is a single-digit-to-teens share of wall-clock; total cost is driven by verdict behavior, where the surviving one-path design dominates or ties. Closes with a ranked list of optimization opportunities quantified against the measured profile. ### [Parallel execution](parallel-execution.md) The measured half of the parallelism pair: per-level eigensolves on a bounded worker pool, with byte-identical results to the serial path, and the BLAS-oversubscription fix that turns a slowdown into a speedup at large problem sizes. Three regimes — overhead-bound, crossover, win — located by measurement. ### [Region-parallel speedup](region-parallel-speedup.md) The projected half of the parallelism pair: an analytical layer that turns per-region work counts from domain-decomposed runs into projected speedup, parallel efficiency, and load balance — validated in one, two, and six parameter dimensions, and paired with the measured page above.