Demonstrations — the algorithm in action

Six pages showing the adaptive-greedy solver running on real problems.

The pages build outward from a reference case, not sequentially from each other. Any one can be read first; they all share the same reference point: the scalar diffusion eigenproblem on [0,1]^2 with a matrix-valued coefficient c(\mu) that drives a 1-D parameter sweep. That case is the anchor — the others extend it along different axes (higher-dimensional parameter space, a different PDE class, accuracy of the surrogate).

Pages

Diffusion 2D — 1D parameter sweep

The reference example for the whole method. Scalar diffusion -\nabla \cdot (c(\mu)\nabla u) = \lambda u on [0,1]^2 with a single scalar parameter \mu \in [0.4, 1]. Both Window and FirstK operating modes are run to convergence and compared honestly.

Diffusion 2D — 2D parameter sweep

The same PDE over a two-dimensional parameter box (\mu_1, \mu_2) \in [0.8, 1.05]^2. Exercises both refinement mechanisms: a crossing certified at level 0 and small-gap refinement at L \ge 1. The companion to the one-parameter sweep, with the same PDE generalised to a 2-D parameter box.

Elasticity 2D

Plane-stress linear elasticity with per-face Lamé coefficients — the only vector-valued PDE among the demonstrations. Evidence that the adaptive-greedy machinery is not specialised to scalar diffusion.

§4.4 6D campaign — quality demonstration at scale

The dissertation’s §4.4 problem run end-to-end: plane-strain elasticity on the L-shape with a six-dimensional parameter box, Window mode over the natural-frequency band. Every quality instrument the rebuild ships — ROM error gauges, certification telemetry, domain decomposition with measured parallel execution, the bounded-resource store — exercised together at dissertation scale.

§4.4 6D campaign — the correctness evidence

The companion correctness case for the 6D campaign: five evidence strands (slice views vs dense truth, an independent inertia certificate, holonomy + ν-index topological consistency, a sampled statistical audit with an exact confidence interval, and setup invariance) replacing the 1D/2D branch-figure eyeball certificate that cannot exist in six dimensions.

ROM vs dense FE sweep

The central accuracy claim: evaluating the converged surrogate by Galerkin projection at every parameter point reproduces the expensive dense finite-element sweep to a measured tolerance. Two surrogates are built from the same converged adaptive graph and compared against the dense reference on the one-parameter diffusion benchmark.

--- title: "Demonstrations — the algorithm in action" --- Six pages showing the adaptive-greedy solver running on real problems. The pages build *outward* from a reference case, not sequentially from each other. Any one can be read first; they all share the same reference point: the scalar diffusion eigenproblem on $[0,1]^2$ with a matrix-valued coefficient $c(\mu)$ that drives a 1-D parameter sweep. That case is the anchor — the others extend it along different axes (higher-dimensional parameter space, a different PDE class, accuracy of the surrogate). ## Pages ### [Diffusion 2D — 1D parameter sweep](examples/diffusion-3-3-2.qmd) The reference example for the whole method. Scalar diffusion $-\nabla \cdot (c(\mu)\nabla u) = \lambda u$ on $[0,1]^2$ with a single scalar parameter $\mu \in [0.4, 1]$. Both **Window** and **FirstK** operating modes are run to convergence and compared honestly. ### [Diffusion 2D — 2D parameter sweep](examples/diffusion-3-3-3.qmd) The same PDE over a two-dimensional parameter box $(\mu_1, \mu_2) \in [0.8, 1.05]^2$. Exercises both refinement mechanisms: a crossing certified at level 0 and small-gap refinement at $L \ge 1$. The companion to the one-parameter sweep, with the same PDE generalised to a 2-D parameter box. ### [Elasticity 2D](examples/elasticity-2d.qmd) Plane-stress linear elasticity with per-face Lamé coefficients — the only vector-valued PDE among the demonstrations. Evidence that the adaptive-greedy machinery is not specialised to scalar diffusion. ### [§4.4 6D campaign — quality demonstration at scale](examples/elasticity-4-4-campaign.qmd) The dissertation's §4.4 problem run end-to-end: plane-strain elasticity on the L-shape with a **six-dimensional** parameter box, Window mode over the natural-frequency band. Every quality instrument the rebuild ships — ROM error gauges, certification telemetry, domain decomposition with measured parallel execution, the bounded-resource store — exercised together at dissertation scale. ### [§4.4 6D campaign — the correctness evidence](examples/elasticity-4-4-evidence.qmd) The companion correctness case for the 6D campaign: five evidence strands (slice views vs dense truth, an independent inertia certificate, holonomy + ν-index topological consistency, a sampled statistical audit with an exact confidence interval, and setup invariance) replacing the 1D/2D branch-figure eyeball certificate that cannot exist in six dimensions. ### [ROM vs dense FE sweep](examples/rom-vs-dense.qmd) The central accuracy claim: evaluating the converged surrogate by Galerkin projection at every parameter point reproduces the expensive dense finite-element sweep to a measured tolerance. Two surrogates are built from the same converged adaptive graph and compared against the dense reference on the one-parameter diffusion benchmark.