Alberto Lovison

Adaptive sampling with a Lipschitz criterion for accurate metamodeling

Lipschitz Sampling, unlike standard space lling strategies (Minimax and Maximin distance, Integrated Mean Squared Error, Eadze-Eglais, etc.) for producing good metamodels, incorporates information from output evaluation in order to estimate in some sense the local complexity of the function at hand. The complexity indicator considered is a suitable denition of local Lipschitz constant. New points are proposed to be evaluated where the product of the local Lipschitz constant by the distance from the nearest already evaluated point is maximum. Benchmarks are proposed on standard test functions in comparison with standard space lling strategies. Smaller prediction errors are obtained by Lipschitz sampling when the function considered shows sudden variations in some part of the domain and varies more slowly in other regions.

Global search perspectives for multiobjective optimization

Extending the notion of global search to multiobjective optimization is far than straightforward, mainly for the reason that one almost always has to deal with infinite Pareto optima and correspondingly infinite optimal values. Adopting Stephen Smale’s global analysis framework, we highlight the geometrical features of the set of Pareto optima and we are led to consistent notions of global convergence. We formulate then a multiobjective version of a celebrated result by Stephens and Baritompa, about the necessity of generating everywhere dense sample sequences, and describe a globally convergent algorithm in case the Lipschitz constant of the determinant of the Jacobian is known.

Automatic sizing of neural networks for function approximation

Neural networks (NN) are a very efficient and powerful function approximation tool. Inspired by the brain structure and functions, NN are usually trained with backpropagation learning algorithm. A detailed benchmark on standard functions is provided, supporting in particular the automatic choice of the number of neurons in the hidden layer.

PAINT---SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

We introduce a novel approximation method for multiobjective optimization problems called PAINT–SiCon. The method can construct consistent parametric representations of Pareto sets, especially for nonconvex problems, by interpolating between nondominated solutions of a given sampling both in the decision and objective space. The proposed method is especially advantageous in computationally expensive cases, since the parametric representation of the Pareto set can be used as an inexpensive surrogate for the original problem during the decision making process.

Extracting optimal datasets for metamodelling and perspectives for incremental samplings

Selecting the best input values for the purpose of fitting a metamodel to the response of a computer code presents several issues. Classical designs for physical experiments (DoE) have been developed to deal with noisy responses, while general space filling designs, though being usually effective for complete classes of problems, are not easily translated into adaptive incremental designs for specific problems. We discuss one-stage and incremental strategies for generating designs of experiments encountered in literature and present an extraction technique along with some benchmark on theoretical functions. We finally propose complexity indicators which could be considered for developing effective incremental samplings.

Exponential estimates for oscillatory integrals with degenerate phase functions

In this paper we give precise asymptotic expansions and estimates of the remainder R(λ) for oscillatory integrals with non Morse phase functions, having degeneracies of any order k ≥ 2. We provide an algorithm for writing down explicitly the coefficients of the asymptotic expansion analysing precisely the combinatorial behaviour of the coefficients (Gevrey type) and deriving optimal exponential decay estimates for the remainder when λ → ∞. We recapture the fundamental asymptotic expansions by Erdelyi (1956 Asymptotic Expansions (New York: Dover)). As it concerns the remainder estimates, it seems they are novel even for the classical cases. The main application of this machinery is a derivation of uniform estimates with respect to control parameters of celebrated oscillatory integrals in optics appearing in the calculations of the intensity of the light along the caustics (umbilics), see e.g. Arnold (1988 Singularities of Differentiable Maps vol II (Boston: Birkhauser Boston Inc.)), (1974 USP. Mat. Nauk. 29 11–49) and Berry and Upstill (1980 Prog. Opt. 18 257–346). Finally, we mention that as an outcome of our abstract approach we obtain refinements for Morse phase functions provided suitable symmetry and Gevrey type regularity conditions on the phase functions and amplitudes hold. As far as we know, even this asymptotic expansion for the elliptic umbilic is a novelty.

New activity pattern in human interactive dynamics

We investigate the response function of human agents as demonstrated by written correspondence, uncovering a new universal pattern for how the reactive dynamics of individuals is distributed across the set of each agents contacts. In long-term empirical data on email, we find that the set of response times considered separately for the messages to each different correspondent of a given writer, generate a family of heavy-tailed distributions, which have largely the same features for all agents, and whose characteristic times grow exponentially with the rank of each correspondent. We show this universal behavioral pattern emerges robustly by considering weighted moving averages of the priority-conditioned response-time probabilities generated by a basic prioritization model. Our findings clarify how the range of priorities in the inputs from ones environment underpin and shape the dynamics of agents embedded in a net of reactive relations. These newly revealed activity patterns constrain future models of communication and interaction networks, affecting their architecture and evolution.

On Generalizing Lipschitz Global Methods forMultiobjective Optimization

Lipschitz global methods for single-objective optimization can represent the optimal solutions with desired accuracy. In this paper, we highlight some directions on how the Lipschitz global methods can be extended as faithfully as possible to multiobjective optimization problems. In particular, we present a multiobjective version of the Pijavskiǐ-Schubert algorithm.

MICROSCOPIC STRUCTURES FROM REDUCTION OF CONTINUUM NONLINEAR PROBLEMS

We present an application of the Amann–Zehnder exact finite reduction to a class of nonlinear perturbations of elliptic elasto-static problems. We propose the existence of minmax solutions by applying Ljusternik–Schnirelmann theory to a finite dimensional variational formulation of the problem, based on a suitable spectral cut–off. As a by–product, with a choice of fit variables, we establish a variational equivalence between the above spectral finite description and a discrete mechanical model. By doing so, we decrypt the abstract information encoded in the AZ reduction and give rise to a concrete and finite description of the continuous problem.