
Optimal error estimate of the finite element approximation of second order semilinear nonautonomous parabolic PDEs
In this work, we investigate the numerical approximation of the second o...
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Mean value methods for solving the heat equation backwards in time
We investigate an iterative mean value method for the inverse (and highl...
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A splineassisted semiparametric approach to nonparametric measurement error models
Nonparametric estimation of the probability density function of a random...
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Convergence rates for Penalised Least Squares Estimators in PDEconstrained regression problems
We consider PDE constrained nonparametric regression problems in which t...
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A new nonlinear instability for scalar fields
In this letter we introduce the nonlinear partial differential equation...
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MODNet: A Machine Learning Approach via ModelOperatorData Network for Solving PDEs
In this paper, we propose a modeloperatordata network (MODNet) for so...
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Nonparametric generalised newsvendor model
In classical newsvendor model, piecewise linear shortage and excess cos...
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Statistical deconvolution of the free FokkerPlanck equation at fixed time
We are interested in reconstructing the initial condition of a nonlinear partial differential equation (PDE), namely the FokkerPlanck equation, from the observation of a Dyson Brownian motion at a given time t>0. The FokkerPlanck equation describes the evolution of electrostatic repulsive particle systems, and can be seen as the large particle limit of correctly renormalized Dyson Brownian motions. The solution of the FokkerPlanck equation can be written as the free convolution of the initial condition and the semicircular distribution. We propose a nonparametric estimator for the initial condition obtained by performing the free deconvolution via the subordination functions method. This statistical estimator is original as it involves the resolution of a fixed point equation, and a classical deconvolution by a Cauchy distribution. This is due to the fact that, in free probability, the analogue of the Fourier transform is the Rtransform, related to the Cauchy transform. In past literature, there has been a focus on the estimation of the initial conditions of linear PDEs such as the heat equation, but to the best of our knowledge, this is the first time that the problem is tackled for a nonlinear PDE. The convergence of the estimator is proved and the integrated mean square error is computed, providing rates of convergence similar to the ones known for nonparametric deconvolution methods. Finally, a simulation study illustrates the good performances of our estimator.
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