Calibration of a SIR (Susceptibles–Infected–Recovered) model with official international data for the COVID-19 pandemics provides a good example of the difficulties inherent in the solution of inverse problems. Inverse modeling is set up in a framework of discrete inverse problems, which explicitly considers the role and the relevance of data. Together with a physical vision of the model, the present work addresses numerically the issue of parameters calibration in SIR models, it discusses the uncertainties in the data provided by international authorities, how they influence the reliability of calibrated model parameters and, ultimately, of model predictions.
Abstract.We prove results of uniqueness and stability at the boundary for the inverse problem of electrical impedance tomography in the presence of possibly anisotropic conduct.ivities. We assume that the unknown conductivity has the forrn A = A(x, a(x)), where a(x) is an unknown scalar function and A(x, t) is a given matrix-valued function. We also deduce results of uniqueness in the interior among conductivities A obtained by piecewise analytic perturbations of the scalar term a.
We consider the inverse boundary value problem of determining the potential q in the equation ∆u + qu = 0 in Ω ⊂ R n , from local Cauchy data. A result of global Lipschitz stability is obtained in dimension n ≥ 3 for potentials that are piecewise linear on a given partition of Ω. No sign, nor spectrum condition on q is assumed, hence our treatment encompasses the reduced wave equation ∆u + k 2 c −2 u = 0 at fixed frequency k.
Abstract. We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body Ω ⊂ R n when the so-called Dirichlet-to-Neumann map is locally given on a non empty portion Γ of the boundary ∂Ω. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001), no. 1, 153-171, where the Dirichlet-to-Neumann map was given on all of ∂Ω instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point y ∈ Γ. Our arguments also apply when the local Neumann-to-Dirichlet map is available.
We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body Ω ⊂ R n when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion Σ of the boundary ∂Ω. We prove that anisotropic conductivities that are a-priori known to be piecewise constant matrices on a given partition of Ω with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map.
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