arXiv subjects

Philippe-André Luneau

Publications and source records attributed to Philippe-André Luneau.

3 recordsLinked to original sources

arXiv2026

General Framework and Error Estimates for ROM-accelerated Fixed Point Iterations

Whether it is for solving nonlinear equations, optimization problems, or autonomous dynamical systems, fixed-point-type iterations are widely used in numerical sciences. On-the-fly reduced-order modelling (ROM) enables the construction of a low-dimensional, self-correcting approximation of the solution to this system during the iterative process, while removing the need to do an offline training phase and any dependence on a precomputed reduced basis (e.g., a fixed geometry or mesh). This technique has been used in specific fields before, including fluid-structure interactions and topology optimization, but no general study of this method has been done to the knowledge of the authors. A general method for accelerating fixed point schemes will be presented. We show that when the iteration mapping is contractive, the error of the approximate solution is guaranteed to be within the user-defined tolerance using inexact fixed-point theory. This methodology is then applied to the solution of systems of PDEs with a block Gauss-Seidel scheme. Errors due to the ROM are propagated through each iteration with respect to the computational graph of the system, which allows one to estimate whether the current iteration is still within the user-defined tolerance. Some working hypotheses necessary to observe a significant speedup and the limitations of the method are explored as well. As a numerical illustration, the methodology is applied to a multiphysics lid-driven cavity flow in two and three dimensions.

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arXiv2026

Convergence Analysis of the Random Bisection Method

We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation $\mathbb{E}[c(1-c)]$ of the cut $c$. We also provide a generalization of the method for $K$ random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.

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arXiv2024

Conservative Surrogate Models for Optimization with the Active Subspace Method

We are interested in building low-dimensional surrogate models to reduce optimization costs, while having theoretical guarantees that the optimum will satisfy the constraints of the full-size model, by making conservative approximations. The surrogate model is constructed using a Gaussian process regression (GPR). To ensure conservativeness, two new approaches are proposed: the first one using bootstrapping, and the second one using concentration inequalities. Those two techniques are based on a stochastic argument and thus will only enforce conservativeness up to a user-defined probability threshold. The method has applications in the context of optimization using the active subspace method for dimensionality reduction of the objective function and the constraints, addressing recorded issues about constraint violations. The resulting algorithms are tested on a toy optimization problem in thermal design.

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