Approximate Diffeomorphisms in Analysis

• Approximate diffeomorphisms are techniques where less-regular Sobolev maps are closely approximated by smooth, invertible functions while preserving key topological and energy properties.
• The methodology employs cell decomposition, p-harmonic replacements, and interface smoothing to achieve controlled approximations without increasing the p–energy.
• These approaches extend to infinite-dimensional settings, enabling efficient numerical schemes in image registration, shape analysis, and control of dynamical systems.

Approximate diffeomorphisms refer to the mathematical and algorithmic frameworks wherein elements of a diffeomorphism group—or suitable analogues—are used to approximate less regular, potentially non-smooth, or otherwise intractable geometric transformations. The term encompasses both the analytic density of diffeomorphisms within broader function spaces (such as Sobolev or rearrangement-invariant spaces) and constructions in computational, dynamical, and control-theoretic contexts where exact diffeomorphic behavior is unattainable or undesirable, but approximation suffices for theoretical or practical purposes.

1. Analytic Approximation in Sobolev Spaces

A foundational result in the theory of approximate diffeomorphisms is that, in the plane, any orientation-preserving Sobolev homeomorphism $h \in W^{1,p}(X, Y)$ with $1 < p < \infty$ can be approximated both uniformly and in the $W^{1,p}$ norm by a sequence of $C^\infty$ diffeomorphisms $h_\ell$ (Iwaniec et al., 2010). The construction is structured to guarantee that at every iteration, the topological (invertibility) property is preserved and the $p$–energy does not increase. The approximation is achieved via a finite sequence of “local modifications”:

1. Cell decomposition: The target domain is partitioned into a locally finite family of dyadic squares, whose preimages via $h$ define cells in the source.
2. p–Harmonic replacement: Within each cell (or subcell), $h$ is replaced by the unique $p$–harmonic function matching the original map on the boundary. This strictly decreases $p$–energy except when already $p$–harmonic, while yielding a $C^\infty$ diffeomorphism in the cell, provided the boundary image is convex. This leverages the $p$–harmonic version of the Radó–Kneser–Choquet theorem.
3. Smoothing across interfaces: Gluing is performed across common edges and in the neighborhoods of vertices, via local $p$–harmonic replacements in “lenses” and careful smoothing along crosscuts.
4. Quantitative control: At each step, explicit estimates guarantee that
   • $\|h_\ell - h\|_{C^0} \to 0$,
   • $\|\nabla h_\ell - \nabla h\|_{L^p} \to 0$,
   • $E[h_\ell] \leq E[h]$, and that modifications are localized.

The methodology generalizes—under certain restrictions—to $W^{1,1}$ and to planar Sobolev homeomorphisms in more general rearrangement-invariant Banach spaces, though the specific analytic and measure-theoretic conditions (e.g., the Lebesgue point property and absolute continuity of the norm) are indispensable (Hencl et al., 2015, Campbell et al., 2020). In these contexts, subdivision into good/bad squares, local affine or quadratic approximation, and gluing techniques are refined to produce both piecewise affine and ultimately smooth diffeomorphic approximants.

2. Topological and Geometric Criteria for Approximation

A central analytic question is to characterize which Sobolev maps can be approximated by diffeomorphisms. The “no-crossing” condition gives a full criterion in the plane: a Sobolev map $u \in W^{1,p}\left(\Omega; \mathbb{R}^2\right)$ is a strong $W^{1,p}$-limit of diffeomorphisms if and only if, for almost every admissible curve in the domain, its image can be made injective via an arbitrarily small perturbation (Philippis et al., 2017). Sufficient (but not necessary) conditions include connectedness properties for preimages of closed connected sets.

Key distinctions:

• The closure of diffeomorphisms in the Sobolev norm is strictly contained in the broader class of orientation-preserving homeomorphisms with certain measure-theoretic properties (the "INV" condition does not suffice).
• Counterexamples illustrate the necessity of the no-crossing property and the non-triviality of preimage connectedness.

These analytic and topological criteria are especially relevant in applications where injectivity or non-interpenetration (e.g., in nonlinear elasticity) is essential.

3. Approximation in Infinite-Dimensional and Algorithmic Settings

In infinite-dimensional contexts—such as diffeomorphism groups for shape, image, or control theories—approximate diffeomorphisms play a key role in both theoretical reachability and numerical schemes.

• Sub-Riemannian geometry and reachability: Approximate reachability results state that for a right-invariant metric on the diffeomorphism group (constructed from a Hilbert space of vector fields densely spanning the tangent space), every diffeomorphism can be approximated arbitrarily well (in the manifold topology) by horizontal flows, i.e., those generated by admissible controls (Arguillere et al., 2014). This principle allows for “approximate Moser theorems,” establishing that certain geometric structures can always be “transported” via nearly horizontal diffeomorphisms.
• Large Deformation Diffeomorphic Metric Mapping (LDDMM): Within the framework of right-invariant Riemannian metrics on the Sobolev diffeomorphism group $\mathcal{D}^m$ ($m > d/2+1$), geodesics correspond to minimal-energy deformations; efficient spectral (Fourier) discretizations yield numerical approximations whose convergence is guaranteed by preservation of higher-order Sobolev regularity along geodesics (Wirth, 9 Oct 2024).
• Deep learning architectures: Several approaches parameterize approximate diffeomorphisms via compositions of “elementary” maps (residual neural networks, control-system flows, or explicit functional bases), enforcing invertibility and regularity using structural constraints, as in (Scagliotti, 2021, Celledoni et al., 2022). Universal approximation theorems hold under suitable Lie-algebraic or functional-analytic assumptions.

These algorithmic methods are crucial for applications in optimal transport, computational anatomy, and state-transfer in infinite-dimensional systems such as controlled Schrödinger equations (Pozzoli et al., 18 Mar 2025).

4. Diffeomorphisms with Prescribed (or Exotic) Derivative

The construction of diffeomorphisms (or almost everywhere approximately differentiable homeomorphisms) with prescribed derivative—potentially irregular or changing sign—is addressed via careful partitioning and patching schemes (Goldstein et al., 2023, Goldstein et al., 2015).

• Prescribed derivative: For any measurable matrix field $T(x)$ (with positive determinant, or more generally non-vanishing), there exists a diffeomorphism whose derivative matches $T$ outside a small exceptional set; for homeomorphisms, this can be realized almost everywhere. The methodology uses Lusin-type approximation, volume-correction via the Dacorogna-Moser theorem, dyadic decomposition, and recursive gluing, ensuring measure-theoretic (approximate) differentiability everywhere except on an arbitrarily small set.
• Orientation anomalies: It is possible to construct homeomorphisms that are topologically orientation preserving but whose approximate derivative has determinant $-1$ almost everywhere, and these can be uniformly approximated by orientation and measure preserving diffeomorphisms. This underscores the subtle distinction between topological and differential orientation, with implications for the change of variable formula and applications in elasticity theory.

5. Approximate Diffeomorphisms in Dynamics and Control

Approximate diffeomorphisms are central in the paper of controllability and orbits of infinite-dimensional systems such as Hamiltonian PDEs and Liouville transport equations.

• Hamiltonian flows and density orbits: In the phase space $T^* M$, approximate controllability via Hamiltonian diffeomorphisms implies that the set of reachable densities from a given initial $\rho_0$ is exactly the set of all densities sharing the measures of all sub- and super-level sets (a measure-theoretic analogue of Moser's theorem), with permutation approximations on finite grids and Poisson bracket computations as key technical tools (Kazandjian et al., 29 Sep 2025).
• Quantum state transfers: Approximate diffeomorphism constructions enable small-time, arbitrarily precise transfer of states in bilinear Schrödinger equations, implemented via ensemble optimal control and $\Gamma$-convergence arguments, connecting infinite-dimensional geometric control theory to practical pulse-design protocols in quantum technology (Pozzoli et al., 18 Mar 2025).

6. Applications, Limitations, and Open Problems

Applications of approximate diffeomorphisms span nonlinear elasticity, fluid dynamics, medical imaging, shape analysis, control of quantum and classical infinite-dimensional systems, and machine learning (e.g., normalizing flows on diffeomorphism groups).

Notable limitations and open problems include:

• Extension to higher (than two) dimensions: in particular, strong approximation results for homeomorphisms by diffeomorphisms—valid in the plane—may fail in dimensions $n \geq 4$.
• $p = 1$ or minimal regularity: for $W^{1,1}$ maps, the construction is more delicate; in various settings, exact density fails and only partial results are available.
• Prescribed derivative in full measure and regularity: the extent to which arbitrary measurable matrix fields can be realized as the derivative of a homeomorphism (subject to global geometric compatibility), or as limits in Sobolev topologies, remains open, though progress has been made via convex integration and measure-theoretic constructions.

Approximate diffeomorphisms thus represent a robust unifying concept, linking deep analytic, geometric, dynamical, and algorithmic techniques for understanding and leveraging the structure of invertible smooth transformations under relaxed conditions.