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Olivier Lafitte

Publications and source records attributed to Olivier Lafitte.

15 recordsLinked to original sources
arXiv2026

Analytical study of the optimal combination of binary classifiers based on classifiers-induced partitioning of the training set

This paper studies an optimal linear combination of binary classifiers based on a logical structuration of the dataset via truth tables. The given classifiers partition data into equivalence classes, allowing for a rigorous analysis of the convexified empirical risk through a multidimensional generalization of classification calibrated functions. We establish sufficient conditions for the existence and uniqueness of the (global) point of minimum of the convexified empirical risk for any list of classifiers (when the number of classifiers is large, there frequently could be no point of minimum). In the case of three classifiers, our analysis allows to list all the configurations leading to either a unique solution, infima or non-unique points of minimum. Furthermore, we derive explicit analytical formulae for optimal weights using Exponential (Boost) and Logistic (Logit) loss functions, bypassing iterative optimization. The stability of the resulting classifier and the analysis of data quality can be evaluated through the introduction of the notion of $\phi$-frontiers.

cs.LG↗
arXiv2024

Analytic expression of the DOS for a new model of 1d-potential and its random perturbation

In this article we present comparisons between the spectrum of a one-dimensional Schrödinger operator for a particular periodic potential and for its restriction to a finite number of sites. We deduce from this finite, but large, number of sites, the Integrated Density of States (IDS) associated to the Hamiltonian operator whose derivate is the DOS. The exact formula for the IDS is given and the expression of the DOS is analytical. All our calculations are done on the particular periodic Airy-potential, which is a new case for which one has an analytical expression of the DOS. It is a continuous, periodic potential, piecewise affine. As a periodic operator, the spectrum is a band spectrum.

math-ph↗
arXiv2024

When Analytic Calculus Cracks AdaBoost Code

The principle of boosting in supervised learning involves combining multiple weak classifiers to obtain a stronger classifier. AdaBoost has the reputation to be a perfect example of this approach. This study analyzes the (two classes) AdaBoost procedure implemented in scikit-learn. This paper shows that AdaBoost is an algorithm in name only, as the resulting combination of weak classifiers can be explicitly calculated using a truth table. Indeed, using a logical analysis of the training set with weak classifiers constructing a truth table, we recover, through an analytical formula, the weights of the combination of these weak classifiers obtained by the procedure. We observe that this formula does not give the point of minimum of the risk, we provide a system to compute the exact point of minimum and we check that the AdaBoost procedure in scikit-learn does not implement the algorithm described by Freund and Schapire.

cs.LG↗
arXiv2023

Error estimates for Gaussian beams at a fold caustic

In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order $O(k^{-5/6})$ where $k$ is the wave number in Helmholtz.

math.CA↗
arXiv2023

The reflection coefficient of a fractional reflector

This paper considers the question of characterizing the behavior of waves reflected by a fractional singularity of the wave speed profile, i.e., of the form \[ c(x_1, x_2, x_3) = c_0 \left(1 + \left( \frac{x_1}{\ell}\right)_{+}^α\right)^{-1/2}, \] for $α> 0$ not necessarily integer. We first focus on the case of one spatial dimension and a harmonic time dependence. We define the reflection coefficient $R$ from a limiting absorption principle. We provide an exact formula for $R$ in terms of the solution to a Volterra equation. We obtain the asymptotic limit of this coefficient in the large $\ell ω/ c_0$ regime as \[ R = \frac{Γ(α+ 1)}{(2 i)^{α+ 2}} \left( \frac{c_0}{\ell ω} \right)^α + \mbox{lower order terms.} \] The amplitude is proportional to $ω^{-α}$, and the phase rotation behavior is obtained from the $i^{-(α+2)}$ factor. The proof method does not rely on representing the solution by special functions, since $α> 0$ is general. In the multi-dimensional layered case, we obtain a similar result where the nondimensional variable $\ell ω/ c_0$ is modified to account for the angle of incidence. The asymptotic analysis now requires the waves to be non-glancing. The resulting reflection coefficient can now be interpreted as a Fourier multiplier of order $- α$. In practice, the knowledge of the dependency of both the amplitude and the phase of $R$ on $ω$ and $α$ might be able to inform the kind of signal processing needed to characterize the fractional nature of reflectors, for instance in geophysics.

math.AP↗
arXiv2022

Uniqueness of the Cauchy datum for the tempered-in-time response and conductivity operator of a plasma

We study the linear Vlasov equation with a given electric field $E \in \mathcal{S}$, where $\mathcal{S}$ is the space of Schwartz functions. The associated damped partial differential equation has a unique tempered solution, which fixes the needed Cauchy datum. This tempered solution then converges to the causal solution of the linear Vlasov equation when the damping parameter goes to zero. This result allows us to define the plasma conductivity operator $σ$, which gives the current density $j = σ(E)$ induced by the electric field $E$. We prove that $σ$ is continuous from $\mathcal{S}$ to its dual $\mathcal{S}^\prime$. We can treat rigorously the case of uniform non-magnetized non-relativistic plasma (linear Landau damping) and the case of uniform magnetized relativistic plasma (cyclotron damping). In both cases, we demonstrate that the main part of the conductivity operator is a pseudo-differential operator and we give its expression rigorously. This matches the formal results widely used in the theoretical physics community.

math.AP↗
arXiv2022

Spectral analysis of the incompressible viscous Rayleigh-Taylor system in $\mathbf{R}^3$

The linear instability study of the viscous Rayleigh-Taylor model in the neighborhood of a laminar smooth increasing density profile $ρ_0(x_3)$ amounts to the study of the following ordinary differential equation of order 4: \begin{equation}\label{MainEq} -λ^2 [ ρ_0 k^2 ϕ- (ρ_0 ϕ')'] = λμ(ϕ^{(4)} - 2k^2 ϕ" + k^4 ϕ) - gk^2 ρ_0'ϕ, \end{equation} where $λ$ is the growth rate in time, $k$ is the wave number transverse to the density profile. In the case of $ρ'_0\geq 0$ compactly supported, we provide a spectral analysis showing that in accordance with the results of \cite{HL03}, there is an infinite sequence of non trivial solutions $(λ_n, ϕ_n)$, with $λ_n\rightarrow 0$ when $n\rightarrow +\infty$ and $ϕ_n\in H^4(\mathbf{R})$. In the more general case where $ρ_0'>0$ everywhere and $ρ_0$ converges at $\pm\infty$ to finite limits $ρ_{\pm}>0$, we prove that there exist finitely non trivial solutions $(λ_n, ϕ_n)$. The line of investigation is to reduce both cases to the study of an operator on a compact set.

math.AP↗
arXiv2022

Analytic solutions and numerical method for a coupled thermo-neutronic problem

We consider in this contribution a simplified idealized one-dimensional model in a nuclear core reactor coupling the diffusion equation on the neutron flux with the enthalpy equation for the water which collects the heat produced by this idealized nuclear core. These equations are coupled through the dependency of the coefficients of the diffusion equation in terms of the enthalpy. We propose a numerical method treating globally the coupled problem for finding its unique solution. Simultaneously, we use incomplete elliptic integrals to represent analytically the density of neutrons and the enthalpy in the fluid. Both methods lead to the same solution with high accuracy. However, another quantity, generally used as a benchmark for comparing results, depends considerably on the approximation used for the coefficients of the diffusion equation.

math.NA↗
arXiv2015

Block-diagonalization of ODEs in the semiclassical limit and $C^ω$ vs. $C^\infty$ stationary phase

Motivated by issues in detonation stability, we study existence of block-diagonalizing transformations for ordinary differential semiclassical limit problems arising in the study of high-frequency eigenvalue problems. Our main results are to (i) establish existence of block-diagonalizing transformations in a neighborhood of infinity for analytic-coefficient ODE, and (ii) establish by a series of counterexample sharpness of hypotheses and conclusions on existence of block-diagonalizing transformations near a finite point. In particular, we show that, in general, bounded transformations exist only locally, answering a question posed by Wasow in the 1980's, and, under the minimal condition of spectral separation, for ODE with analytic rather than $C^\infty$ coefficients. The latter issue is connected with quantitative comparisons of $C^ω$ vs. $C^\infty$ stationary phase estimates

math.CA↗
arXiv2013

High-frequency stability of multidimensional ZND detonations

The rigorous study of spectral stability of strong detonations was begun by Erpenbeck in the 1960s. Working with the Zeldovitch-von Neumann-Döring (ZND) model, he identified two fundamental classes of detonation profiles, referred to as those of decreasing (D) and increasing (I) type, which appeared to exhibit very different behavior with respect to high-frequency perturbations. Using a combination of rigorous and non-rigorous arguments, Erpenbeck concluded that type I detonations were unstable to some oscillatory perturbations for which the (vector) frequency was of arbitrarily large magnitude, while type D detonations were stable provided the frequency magnitude was sufficiently high. For type D detonations Erpenbeck's methods did not allow him to obtain a cutoff magnitude for stability that was \emph{uniform} with respect to frequency direction. Thus, he left open the question whether the cutoff magnitude for stability might approach $+\infty$ as certain frequency directions were approached. In this paper we show by quite different methods that for type D detonations there exists a uniform cutoff magnitude for stability independent of frequency direction. By reducing the search for unstable frequencies to a bounded frequency set, the uniform cutoff obtained here is a key step toward the rigorous validation of a number of results in the computational detonation literature. The main difficulty in the analysis is to treat "turning points at infinity".

math.AP↗
arXiv2011

The Erpenbeck high frequency instability theorem for ZND detonations

The rigorous study of spectral stability for strong detonations was begun by J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-Döring (ZND) model, which assumes a finite reaction rate but ignores effects like viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function $V(τ,\eps)$ whose zeros in $\Re τ>0$ correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper [Er3] he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of $V$ exist for perturbations of sufficiently large transverse wavenumber $\eps$, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly twenty years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of $V(τ,\eps)$, the paper \cite{Er3} remains the only work we know of that presents a detailed and convincing theoretical argument for detecting them. The analysis in [Er3] points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in [Er3] and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as $\eps\to \infty$ of solutions to a linear system of ODEs in $x$, depending on $\eps$ and a complex frequency $τ$ as parameters, with turning points $x_*$ on the half-line $[0,\infty)$.

math-ph↗
arXiv2009

Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity

For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with $γ$-law gas equation of state, $γ\ge 1$, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes. In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. For the monatomic and diatomic cases $γ=5/3$ and $γ= 7/5$, with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well

math.AP↗
arXiv2008

Bifurcations in a convection problem with temperature-dependent viscosity

A convection problem with temperature-dependent viscosity in an infinite layer is presented. As described, this problem has important applications in mantle convection. The existence of a stationary bifurcation is proved together with a condition to obtain the critical parameters at which the bifurcation takes place. For a general dependence of viscosity with temperature a numerical strategy for the calculation of the critical bifurcation curves and the most unstable modes has been developed. For a exponential dependence of viscosity on temperature the numerical calculations have been done. Comparisons with the classical Rayleigh-Bénard problem with constant viscosity indicate that the critical threshold decreases as the exponential rate parameter increases.

nlin.PS↗
arXiv2007

Study of the linear ablation growth rate for the quasi isobaric model of Euler equations with thermal conductivity

In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The system of equations is linearized around a stationary solution, and each perturbed quantity is written using the normal modes method. The resulting linear system is not self-adjoint, of order 5, with coefficients depending on x and on physical parameters $α, β$. We calculate Evans function associated with this linear system, using rigorous constructions of decreasing at $\pm \infty$ solutions of systems of ODE. We prove that for $α$ small, there is no bounded solution of the linearized system.

math.AP↗