Tutorials

A guided path from SpaceCore’s core abstractions to three complete algorithms. Every page below is a runnable Jupyter notebook in tutorials/ — read it here or open the notebook and execute it yourself. The foundations build on each other; the worked algorithms can be read in any order once the foundations are in place.

Foundations

• 1 · Backend and Context — Why the backend is an explicit Context; one routine on NumPy and JAX; check levels.

• 2 · Linear algebra: spaces, operators, conjugate gradients — Spaces with geometry, operators \(A:X\to Y\), and a conjugate-gradient solve.

• 3 · Functionals and gradient descent — Scalar objectives, metric-aware gradients, and gradient descent.

• 4 · Tree spaces: structured elements and block operators — Structured unknowns, block operators, and a block solve over a TreeSpace.

Worked examples

• 5 · Weighted Tikhonov: an inverse problem with metric adjoints — An inverse problem solved with metric adjoints and weighted geometry.

• 6 · Optimal transport with a matrix-free operator — Marginalisation as a matrix-free operator; Sinkhorn powered by its adjoint.

• 7 · Gradient descent on a manifold with non-Euclidean geometry — A custom non-Euclidean geometry and Riemannian gradient descent.

• 8 · PDHG for a conic program — A primal–dual solver for a conic program with a Jordan-cone projection.

Performance and internals

• 9 · Optimized kernels, dispatch, and operator fusion — Optimized-kernel dispatch (ADR-016), operator fusion (ADR-021), and fold-stack caching (ADR-022): running the fastest bit-exact kernel, multiplying operators together with fuse(), and reusing a fold’s materialized stack across applies — all opt-in or invisible, none touching the matrix-free contract.