arXiv subjects

Pierre Maréchal

Publications and source records attributed to Pierre Maréchal.

7 recordsLinked to original sources

arXiv2026

A variational mollification approach to circular deconvolution

We propose a variational mollification approach to circular deconvolution based on the reconstruction of a mollified target object rather than the exact solution itself. The method is formulated as a convex variational problem whose solution admits an explicit Fourier representation. We establish the consistency of the proposed reconstruction as the target resolution increases and derive convergence rates for deterministic data perturbations. For ordinary smooth kernels, the method achieves the classical order-optimal algebraic convergence rates under Besov-Nikolskii smoothness assumptions, while for supersmooth kernels it attains the corresponding order-optimal logarithmic rates. Numerical experiments on synthetic and wind-direction data illustrate the effectiveness of the proposed approach and confirm the theoretical predictions.

math.NA↗

arXiv2026

Interpolating between Tikhonov regularization and spectral cutoff

Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper introduces an interpolating formula that defines a one-parameter family of regularizations, where Tikhonov and spectral cutoff methods are represented as limiting cases. By adjusting the interpolating parameter taking into account the specific operator equation under consideration, it is possible to mitigate the limitations associated with both Tikhonov and spectral cutoff regularizations. The proposed approach is demonstrated through numerical simulations in the fields of signal and image processing.

math.NA↗

arXiv2025

Multi-resolution deconvolution

We extend the classical deconvolution framework in Rn to the case with a pseudodifferential-like solution operator with a symbol depending on both the base and cotangent variable. Our framework enables deconvolution with spatially varying resolution while maintaining a set global stability, and it additionally allows rather general distributional convolution kernels. We provide consistency, convergence and stability results, as well as convergence rates. Finally, we include numerical examples supporting our results and demonstrating advantages of the generalized framework.

math.SP↗

arXiv2023

Regularization of the inverse Laplace transform by Mollification

In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.

math.AP↗

arXiv2014

Lagrange duality for the Morozov principle

Considering a general linear ill-posed equation, we explore the duality arising from the requirement that the discrepancy should take a given value based on the estimation of the noise level, as is notably the case when using the Morozov principle. We show that, under reasonable assumptions, the dual function is smooth, and that its maximization points out the appropriate value of Tikhonov's regularization parameter.

math.OC↗

arXiv2014

Tomography by Fourier synthesis

We consider a particular approach to the regularization of the inverse problem of computerized tomography. This approach is based on notions pertaining to Fourier synthesis. It refines previous contributions, in which the preprocessing of the data was performed according to the Fourier slice theorem. Since real models must account for the geometrical system response and possibly Compton scattering and attenuation, the Fourier slice theorem does not apply, yielding redefinition of the preprocessing. In general, the latter is not explicit, and must be performed numerically. The most natural choice of preprocessing involves the computation of unstable solutions. A proximal strategy is proposed for this step, which allows for accurate computations and preserves global stability of the reconstruction process.

math.OC↗

arXiv2009

A proximal approach to the inversion of ill-conditioned matrices

We propose a general proximal algorithm for the inversion of ill-conditioned matrices. This algorithm is based on a variational characterization of pseudo-inverses. We show that a particular instance of it (with constant regularization parameter) belongs to the class of {\sl fixed point} methods. Convergence of the algorithm is also discussed.

math.NA↗