Quasi Reversibility Method for Inverse Problems

Updated 27 December 2025

Quasi-reversibility method is a Tikhonov-type regularization technique that reformulates ill-posed PDE problems into stable variational formulations.
It adds a small penalization term to balance fidelity and noise, ensuring convergence and robust numerical approximations.
Widely applied to elliptic, parabolic, and hyperbolic problems, it supports reliable recovery in inverse source, data completion, and unique continuation tasks.

The quasi-reversibility method (QRM) is a principled regularization technique for numerically solving ill-posed inverse and data completion problems for partial differential equations (PDEs), notably for elliptic, parabolic, hyperbolic, and hybrid Cauchy-type and backward problems. Developed originally by Lattès and Lions, and subsequently extended in multiple directions, QRM converts unstable or non-continuously invertible operator equations into stable, well-posed variational formulations by adding a small, problem-adapted penalization or perturbation, typically of Tikhonov type. Its convergence properties, flexibility in functional/variational forms, and compatibility with standard finite element or finite difference discretizations have made QRM a foundational approach for deterministic and stochastic inverse problems in fields such as elasticity, electromagnetics, heat conduction, option pricing, and inverse scattering.

1. Core Principles and Variational Structure

At its foundation, QRM regularizes an inherently ill-posed or non-continuously invertible equation $A\,x = y$ (with $A$ not boundedly invertible) by seeking a minimizer of the penalized functional

$J_\varepsilon(x) = \|A\,x - y\|_\mathcal{Y}^2 + \varepsilon\,\|L\,x\|_\mathcal{X}^2,$

where $\varepsilon>0$ is the regularization parameter and $L$ is a suitably chosen regularizing operator, often identity or a differential operator depending on the problem context (Dardé, 2015). This yields a unique minimizer $x_\varepsilon$ satisfying a stable elliptic or elliptic-type variational problem, which approximates the true solution as $\varepsilon \to 0$ when the data are compatible. For noisy data, the parameter $\varepsilon$ must balance fidelity and robustness to instability.

In PDE settings, QRM is often expressed as a minimization of a residual-plus-penalty functional over an appropriate Sobolev space or affine set, subject to the (noisy) overdetermined Cauchy or terminal data imposed as constraints: $J_\varepsilon(u) = \big\|\mathcal{L}u - f\big\|_{Y}^2 + \varepsilon\,\|u\|_{Z}^2,$ with $\mathcal{L}$ a PDE operator, $f$ the (possibly projected or approximate) data, and $Z$ a higher-regularity Sobolev space selected to control ill-posed growth in unregularized directions. Existence and uniqueness of minimizers, stability with respect to noise, and explicit convergence rates are typically established under explicit conditional estimates, often using Carleman weights when dealing with problems with severe non-continuous dependence.

2. Regularization for Laplace and Elliptic Cauchy Problems

For classical Cauchy problems for Laplace’s equation with partial boundary data—an archetype of a Hadamard ill-posed problem—QRM is frequently realized in a mixed variational form, introducing an auxiliary field coupled to the main unknown. The formulation by Bourgeois–Chesnel prescribes, for the Laplacian in a domain $\Omega$ with Cauchy data on $\Gamma\subset\partial\Omega$ , the system (Bourgeois et al., 2019): $\begin{cases} \int_\Omega \nabla u^\varepsilon\cdot\nabla v + \nabla v\cdot\nabla\lambda^\varepsilon = 0, \ \int_\Omega \nabla u^\varepsilon\cdot\nabla\mu - \nabla\lambda^\varepsilon\cdot\nabla\mu = \langle g_1, \mu \rangle_{H^{-1/2}(\Gamma), H^{1/2}(\Gamma)}, \end{cases}$ for all test functions $v, \mu$ in suitable spaces, with boundary conditions encoding the Cauchy data (Bourgeois et al., 2019).

Uniform regularity estimates are obtained both in smooth and 2D polygonal domains. For polygonal cases (with mixed Dirichlet–Neumann corners), Grisvard-style methods fail, requiring the Kondratiev approach, which uses weighted Sobolev and spectral techniques to handle singularity structures (Bourgeois et al., 2019). The key result is a uniform estimate in Sobolev norm (exponent $s$ depends on the worst corner): $\|u^\varepsilon\|_{H^s(\Omega)} + \sqrt{\varepsilon}\|\lambda^\varepsilon\|_{H^s(\Omega)} \leq C \|f\|_{L^2(\Omega)},\quad s<1+\pi/\omega_{min},$ where $\omega_{min}$ is the smallest angle at a mixed corner (Bourgeois et al., 2019).

Error estimates demonstrate that in the compatible case,

$\|u^\varepsilon-u\|_{H^1(\Omega)}=O(\sqrt{\varepsilon}),$

and when solving the fully discrete problem (mesh size $h$ , data noise $\delta$ ), the total error satisfies

$\|u^{\delta}_{\varepsilon,h}-u\|_{H^1} \leq C\,\delta/\sqrt{\varepsilon} + C\,h^{s-1} + C\,\sqrt{\varepsilon}$

(Bourgeois et al., 2019).

3. Quasi-Reversibility for Hyperbolic, Parabolic, and Mixed Problems

QRM extends beyond elliptic PDEs to hyperbolic, parabolic, mixed and hybrid settings, including time-reversal, backward, source identification, and unique-continuation classes. A central technical innovation is the use of regularized Tikhonov-type functionals in product Sobolev spaces—typically with high-order or Carleman-weighted penalties—to stabilize severe exponential ill-posedness and derive explicit Lipschitz or logarithmic convergence rates (Le et al., 2020, Li et al., 2019, Khoa et al., 2020, Tuan et al., 2018).

In the context of inverse source problems for hyperbolic and parabolic equations, a key strategy employs a truncated Fourier (or spectrally adapted) expansion in time or angular variables, reducing the ill-posed inverse problem to a coupled system of spatial (elliptic) PDEs for the expansion coefficients. The overdetermined system is solved using QRM, with the functional

$J_\varepsilon(U) = \int_\Omega|L\,U(x) - S\,U(x)|^2\,dx + \varepsilon\,\|U\|_{H^k(\Omega)}^2,$

for a vector of coefficient fields $U$ and matrix $S$ determined by the truncated basis (Le et al., 2020). Stability analysis leverages vector-valued Carleman estimates. The convergence rate is typically

$\|U^{\varepsilon,\delta}-U^*\| \leq C(\delta^2 + \varepsilon\|U^*\|^2)$

for an appropriate Sobolev norm, yielding deterministic Lipschitz or stochastic logarithmic error as the noise and penalty vanish (Le et al., 2020, Tuan et al., 2019).

Problems exhibiting terminal or backward instabilities (backward heat conduction, inverse source in elasticity) adapt QRM with problem-dependent perturbing and stabilized operators $Q_\varepsilon, P_\varepsilon$ , constructed to annihilate high-frequency modes and yield logarithmic error contraction (Khoa et al., 2020, Tuan et al., 2019, Tuan et al., 2018). Spectral truncation, weighted energy identities, and Carleman-type exponential multipliers are systematically utilized to control the propagation of regularized solutions and to derive precise Hölder or logarithmic convergence rates.

4. Extensions: Electromagnetics, Elasticity, and Data Completion

QRM admits multiple structural and computational variants, including mixed formulations, iterated procedures, domain decomposition, and adaptation to vectorial PDE systems. In the context of time-harmonic Maxwell’s equations with incomplete boundary data, QRM is implemented as a mixed variational problem over edge element spaces, regularized by boundary-normal coercivity terms: $\delta (E_\delta, \phi)_{H(\text{curl})} + a(\phi, F_\delta) = 0,\quad a(E_\delta, \psi) - (F_\delta, \psi)_{H(\text{curl})} = \ell(\psi),$ with appropriate boundary spaces and parameter selection (Darbas et al., 2019). Robustness and convergence properties (including Céa-type finite element error estimates) are preserved under noise, with systematic parameter choice based on the “L-curve” or Morozov principles. In the elasticity context, time-dimensional reduction methods further expand the role of QRM by reducing inverse problems in space–time to a sequence of coupled spatial elasticity systems to be solved via QRM with provable consistency and stability as noise vanishes (Dang et al., 15 Jun 2025).

5. Error Analysis and Conditional Stability

Convergence and stability rates in QRM are intimately linked to the underlying conditional stability properties of the inverse problem. Where the unique continuation property or three-spheres/IAP inequalities yield logarithmic or Hölder-type conditional stability, the QRM error rates cannot improve upon these (see, e.g., (Burman et al., 30 Sep 2024)). If the underlying direct problem is unconditionally stable, QRM delivers convergence in the regularized norm at the corresponding rate. For instance, in Poisson unique continuation,

$\|u-u_h\|_{L^2(\Omega)} \leq C/[(−\log h)^\tau],$

with $\tau$ depending on the conditional stability exponent (Burman et al., 30 Sep 2024). Similarly, in parabolic and nonlinear backward problems, the rate is at best logarithmic in the noise level, with Energy–Carleman-based proofs establishing the optimal worst-case contraction (Tuan et al., 2018, Tuan et al., 2019).

6. Iterated Quasi-Reversibility and Computational Aspects

Beyond single-step Tikhonov regularization, iterated QRM generalizes the approach to a sequence of regularized minimizations: $x^M = \operatorname{argmin}_{x\in\mathcal{X}} \|A\,x - y\|_{\mathcal{Y}}^2 + \varepsilon\,\|L(x - x^{M-1})\|_{\mathcal{X}}^2,\quad x^{-1}=0,$ with convergence to the solution for compatible data as $M\to\infty$ , and regularization bias and data noise controlled explicitly according to stopping rules, notably the Morozov discrepancy principle (Dardé, 2015). This approach is especially useful in discrete contexts (finite element or finite difference), as the matrix structure is fixed and preconditioning is facilitated. Practical implementations for data completion in elliptic, parabolic, or Maxwell systems show stable reconstructions under severe noise, with robust parameter adjustment by data-driven rules.

7. Applications and Recent Developments

QRM is established in a diverse set of inverse and data completion applications:

Inverse source identification for Helmholtz, wave, and elasticity equations via spectral/truncated expansions and overdetermined elliptic/hyperbolic systems (Nguyen et al., 2022, Dang et al., 15 Jun 2025, Le et al., 2020, Nguyen, 2018).
Backward and ill-posed time-dependent PDEs (heat, reaction-diffusion, Maxwell–Cattaneo) using filter/perturbation operators, spectral energy techniques, and exponential weights to recover initial or source data with explicit error control (Khoa et al., 2020, Tuan et al., 2019, Tuan et al., 2018).
Cauchy data completion and unique continuation for Laplace, Poisson, and Maxwell equations in smooth and polygonal domains, using mixed or Tikhonov-based regularizations compatible with edge finite elements, including in high dimensions (Bourgeois et al., 2019, Burman et al., 30 Sep 2024, Darbas et al., 2019).
Coupling with modern data-driven approaches, such as the integration of QRM-processed features into convolutional neural network pipelines for financial time series and option pricing (Cao et al., 2022).

Across settings, QRM demonstrates reliable recovery of geometric and amplitude features (inclusions, sources) under heavy random noise (up to 100%), with final error controlled to a few percent in suitable Sobolev norms, and only mild smoothing or negligible distortion for moderate truncation and regularization parameters (Nguyen et al., 2022, Le et al., 2020, Dang et al., 15 Jun 2025).

References

See the following primary sources for full mathematical details, algorithms, and numerical case studies: