Global Piecewise Carleman Embedding

• Global Piecewise Carleman Embedding is a method that partitions domains to apply local Carleman estimates, ensuring control, observability, and approximation in settings with irregular coefficients.
• It integrates tailored weight functions and interface matching to facilitate robust inverse problem solving, PDE control, dynamical simulation, and complex geometric embeddings.
• By overcoming discontinuities and variable coefficients, the approach enables adaptive charting and effective numerical strategies for challenging applied mathematics problems.

Global Piecewise Carleman Embedding is a methodological paradigm in analysis and applied mathematics that leverages locally constructed Carleman estimates—inequalities featuring exponential weight functions—to achieve control, observability, stability, approximation, and linearization for problems where global analytic or geometric regularity is not assured. By partitioning the domain (spatial, temporal, or state-space) into regions in which tailored Carleman weights are valid, and patching together the local results, one overcomes limitations arising from variable coefficients, discontinuities, multiple fixed points, or domain geometry. The global piecewise approach applies to diverse fields: inverse problems for PDEs and SPDEs, control theory for dispersive and parabolic equations, nonlinear dynamical system simulation, holomorphic embeddings in complex analysis, and geometric approximation in metric spaces.

1. Carleman Estimates and Piecewise Weight Construction

Carleman estimates are weighted integral inequalities for differential operators in which the weights—typically exponential functions involving large parameters—magnify solution behavior in specified regions and suppress others. The classical form is

$$\int_{\Omega} e^{2s\varphi(x,t)} [\text{energy terms}] \leq C\int_{\Omega} e^{2s\varphi(x,t)} [\text{residual/operator error}] + \text{observation/boundary terms}$$

where $\varphi$ is a carefully chosen weight function, and $s \gg 1$ the Carleman parameter.

Piecewise Carleman embedding arises when global regularity conditions—such as monotonicity of coefficients, geometric control, or directionality of advection fields—are not satisfied. In such cases, the domain $\Omega$ is partitioned into subdomains ($\Omega_i$), each admitting a locally adapted Carleman weight ($\varphi_i$), for example $\varphi_i(x,t) = |x + r_iv_i|^2 - \beta t$ where $v_i$ is aligned with the local flow direction and $r_i, \beta$ are parameters ensuring positivity of derivative terms.

Transition terms at interfaces must be treated by matching weight derivatives and transmission conditions, often formalized using directed graph representations for the flow or coefficient structure. The requirement of acyclicity in the associated graph ensures consistent assignment of parameters for proper cancellation or control of interface contributions (Cannarsa et al., 23 Jul 2025).

2. Control, Observability, and Inverse Problems

Piecewise Carleman estimates have powerful implications in control theory and inverse problems:

• Controllability: By applying a Carleman estimate to the adjoint system, observability inequalities are derived, which via duality arguments (Hilbert Uniqueness Method) yield explicit controls driving the solution to a desired state, even with discontinuous coefficients or singular potentials (Enciso et al., 2021, Baudouin et al., 2011, Zhang et al., 24 Jan 2024). The minimization of a weighted functional (often incorporating Carleman weights) is central in constructing the required control function.
• Observability: For transport and dispersive equations with variable or time-dependent coefficients, piecewise weights allow deduction of inequalities such as $$\|u(\cdot,0)\|_{L^2(\Omega)} \leq C\|u\|_{L^2(\Gamma \times (0,T))}$$, quantifying the ability to reconstruct an initial state from boundary measurements (Cannarsa et al., 2018, Cannarsa et al., 23 Jul 2025, Loyola, 12 May 2025).
• Inverse Problems and Lipschitz Stability: Carleman embedding provides strong stability estimates in inverse source or coefficient problems, e.g., estimating an unknown function $f$ via $$\|f\|_{L^2(\Omega)} \leq C\|u_t\|_{L^2(\Omega \times (0,T))}$$ using measurements and refined weight choices, crucial for robustness against noise and for numerical algorithms (Klibanov, 2012, Loyola, 12 May 2025, Cannarsa et al., 23 Jul 2025).

Singular potentials (e.g., inverse-square) and discontinuous coefficients are accommodated by formulating multiple Carleman inequalities with distinct weights and summing them for global control (Enciso et al., 2021, Loyola, 12 May 2025).

3. Piecewise Carleman Linearization and Dynamical System Simulation

In nonlinear dynamical systems, Carleman embedding linearizes a nonlinear ODE or SDE by “lifting” the state space to include all monomials up to order $P$. While standard Carleman embedding is restricted by its local convergence radius (near a single fixed point), the global piecewise approach partitions the state space into local “charts,” with a Carleman embedding constructed for each. Switching between charts as trajectories evolve enables simulation, reachability analysis, and estimation in systems with multiple fixed points or chaotic attractors (Novikau et al., 17 Oct 2025, Bhatt et al., 2019, Forets et al., 2021).

Three main algorithmic schemes have emerged:

Method	Description	Use Case
Chart Switching (PCE)	Switch Carleman representation upon exiting local region	Integrable/chaotic
Adaptive Chart (ACE)	Dynamically tune chart size and location for local convergence	High accuracy
Static Grid (GCE)	Precompute charts over regular grid for maximal computational speed	Quantum simulation

Adaptive methods excel in highly nonlinear or chaotic regions; non-adaptive, grid-based methods trade some accuracy for algorithmic simplicity and computational speed, relevant for quantum linear system solvers (Novikau et al., 17 Oct 2025).

4. Piecewise Holomorphic and Geometric Embedding

In complex geometry, piecewise Carleman approximation theory underpins global embedding theorems for families of holomorphic domains. The parametric Carleman theorem generalizes classical results to parameterized families, allowing construction of continuous families of proper holomorphic embeddings depending on auxiliary parameters (such as deformation moduli of domains) via approximation by entire functions (Salvo et al., 2022). The piecewise aspect is realized by patching together approximations on discs and attached Lipschitz curves, with precise control over errors and continuity with respect to the parameters.

In metric geometry, global piecewise affine Carleman embedding refers to quantitative decomposition of bi-Lipschitz homeomorphisms into compositions of nearly isometric (small distortion) affine (or almost affine) mappings. Corona-type decompositions provide local affine or almost affine approximations, subsequently patched to achieve global factorization and quantitative piecewise affine approximation, up to exceptional sets of controlled small measure (David et al., 9 Sep 2024).

5. Applications to Stochastic and Nonlinear PDEs

Recent advances extend the piecewise embedding methodology to stochastic PDEs (SPDEs), where improved Carleman estimates and weighted energy inequalities are “embedded” via piecewise-(in time) defined weights to obtain null controllability, even for equations with low regularity (Sobolev negative order sources) (Zhang et al., 24 Jan 2024). By selecting weights that vary on different temporal intervals—decaying near boundaries in forward equations and concentrating near terminal time in backward equations—one applies duality and fixed-point arguments in exponentially weighted Banach spaces to extend linear controllability results to semi-linear equations.

Such techniques generalize classical control and inverse problem theory for deterministic PDEs to randomly forced and more singular frameworks, broadening the scope for robust stabilization and estimation in applied problems.

6. Limitations, Open Problems, and Future Directions

Global piecewise Carleman embedding is subject to geometric, analytic, and combinatorial constraints:

• Weight Construction: Adequate definition and matching of weights at interfaces requires monotonicity or acyclicity conditions on coefficients (often encoded in domain decompositions and associated graphs) (Cannarsa et al., 23 Jul 2025).
• Parameter Selection: Stability, convergence, and numerical stability demand precise tuning of Carleman parameters ($s$, $\lambda$, $\beta$, etc.); tradeoffs exist between regularization strength and sensitivity to noise (Baudouin et al., 2011, Loyola, 12 May 2025).
• Numerical Implementation: Discrete versions of Carleman estimates may require further regularization (e.g., Tikhonov), and handling spurious oscillations and high-frequency artifacts remains an open issue (Baudouin et al., 2011, Zhang et al., 24 Jan 2024).
• Exceptional Sets: In geometric applications, the presence of small exceptional sets is unavoidable for global factorization, particularly in higher dimensions (David et al., 9 Sep 2024).

Future research seeks to relax geometric assumptions (e.g., optimal geometric control vs. Lions Gamma condition), extend to more general classes of PDEs (hyperbolic, nonlinear, SPDEs), and improve synchronization of piecewise local results for effective global embedding, both in theory and computational algorithms.

7. Summary and Significance

Global piecewise Carleman embedding synthesizes local analytic control or approximation into a robust global framework. By leveraging a partitioned approach—whether in PDE analysis, control theory, dynamical system simulation, complex or metric geometry, or stochastic analysis—it addresses scenarios inaccessible to single-domain Carleman techniques. Its flexibility accommodates irregular coefficients, domain discontinuities, nonlinear dynamics with multiple fixed points, and complex geometric or probabilistic structures, making it foundational for a wide array of contemporary problems in analysis and applied mathematics.