Nonlinear Krylov Geometry: Tangent and Jet Embeddings for Closure and Convergence

Classical Krylov subspace methods such as GMRES and GCR rely on linear invariance.
Repeated actions of a fixed operator A generate a closed sequence of subspaces
that form the basis for residual minimization and guaranteed convergence.

For nonlinear systems F(x) = 0, this closure fails.
The Jacobian J(x) changes at every iterate, destroying the fixed subspace structure and the corresponding projection optimality.

1. Tangent–Krylov Embedding

To restore closure, the iteration is lifted to the tangent bundle, where each point x carries its own tangent space.
We define a moving subspace field K(x) spanned by the residual r(x) and successive Jacobian actions on it.

A transport operator moves basis vectors from one tangent space to the next, maintaining consistency between iterations.
This restores the geometric notion of closure: the Jacobian at any point maps one subspace field into the next.
Residual minimization once again becomes meaningful, and Newton–Krylov convergence holds without requiring a line search.

2. Jet–Krylov Extension

Nonlinearity can be understood as curvature.
By embedding second-order information—essentially Hessian-vector actions—the solver captures curvature directly rather than treating it as instability.
Minimizing a truncated second-order residual yields faster convergence near bifurcations and for strongly nonlinear systems.

3. Application: The Bratu Problem

Consider the Bratu system, where a Laplacian term is balanced against an exponential nonlinearity.
In its discrete form, the Tangent–Krylov method achieves grid-independent convergence.
With Jet–Krylov correction and right preconditioning by the inverse Laplacian, the solver remains stable through the Bratu turning point and tracks both the small and large solution branches using pseudo-arclength continuation.

4. Structural Insight

The higher-dimensional embedding does not make the algorithm more complicated.
It changes the space in which “linear” makes sense.
Nonlinearity becomes curvature of the bundle, not instability of the iteration.

Invariance is restored by relocation.
Linear closure lives not in the equation but in its tangent or jet geometry.

Keywords: nonlinear Krylov methods, tangent embedding, jet embedding, manifold geometry, closure problem, Bratu equation, Newton–Krylov convergence, bundle transport, curvature correction

Tags: #nonlinear-krylov #geometric-numerics #tangent-bundle #jet-space #bratu-problem