Henri Calandra

Laurent Demanet, Pierre-David Létourneau, Nicolas Boumal et al.

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.

Henri Calandra, Serge Gratton, Elisa Riccietti et al.

This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential equation. The learning problem is formulated as a least squares problem, choosing the residual of the partial differential equation as a loss function, whereas a multilevel Levenberg-Marquardt method is employed as a training method. This setting allows us to get further insight into the potential of multilevel methods. Indeed, when the least squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by any geometrical constraints and the standard interpolation and restriction operators cannot be employed any longer. A heuristic, inspired by algebraic multigrid methods, is then proposed to construct the multilevel transfer operators. Numerical experiments show encouraging results related to the efficiency of the new multilevel optimization method for the training of artificial neural networks, compared to the standard corresponding one-level procedure.

Henri Calandra, Serge Gratton, Elisa Riccietti et al.

We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q >= 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a complexity bound to reach first order stationary points is also derived. A multilevel version of the well known adaptive method based on cubic regularization (ARC, corresponding to q = 2 in our setting) has been implemented. Numerical experiments clearly highlight the relevance of the new multilevel approach leading to considerable computational savings in terms of floating point operations compared to the classical one-level strategy.