Sparse initial data indentification for parabolic pde and its finite element approximations

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.The first author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711. The third author was supported by the Advanced Grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the BERC 2014-2017 program of the Basque Government, the MTM2011-29306 and SEV-2013-0323 Grants of the MINECO, the CIMI-Toulouse Excellence Chair in PDEs, Control and Numerics and a Humboldt Award at the University of Erlangen-Nürnberg

Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

202305 bcchAccepted ManuscriptOthersNSFC; Alexander von Humboldt FoundationPublishe

Weak discrete maximum principle of finite element methods in convex polyhedra

202103 bcvcAccepted ManuscriptRGC15300519Publishe

A Posteriori Error Estimates by Recovered Gradients in Parabolic Finite Element Equations

This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at a given time. The basic principle is that the time-step error needs to be smaller than the space-discretization error. Numerical illustrations are given

Multiple heart malformations in a patient with Holt—Oram syndrome

Holt—Oram syndrome is a rare genetic disease characterized by an abnormality of the upper limb, congenital heart disease and / or conduction abnormalities. The disease is caused by the mutations in the Tbox5 gene (allocation 12q24.21), inherited in an autosomal dominant manner. Heart septal defects and isolated thenar hypoplasia are typical congenital malformations. The article describes a clinical case of a 7-month-old girl with a family history of Holt—Oram syndrome: the absence of the first metacarpal bone of the left hand and multiple heart defects (atrial septal defect, multiple defects of the ventricular septum of the Swiss cheese type, aortic valve stenosis). The authors present a detailed clinical diagnosis of Holt—Oram syndrome, as well as genetic analysis and genetic testing of the child and immediate relatives

Distributed Optimal Control of Diffusion-Convection-Reaction Equations Using Discontinuous Galerkin Methods

We discuss the symmetric interior penalty Galerkin (SIPG) method, the nonsymmetric interior penalty Galerkin (NIPG) method, and the incomplete interior penalty Galerkin (IIPG) method for the discretization of optimal control problems governed by linear diffusion-convection-reaction equations. For the SIPG discretization the discretize-then-optimize (DO) and the optimize-then-discretize (OD) approach lead to the same discrete systems and in both approaches the observed L 2 convergence for states and controls is O(hk+1) , where k is the degree of polynomials used. The situation is different for NIPG and IIPG, where the the DO and the OD approach lead to different discrete systems. For example, when standard penalization is used, the L 2 error in the controls is only O(h) independent of k. However, if superpenalization is used, the lack of adjoint consistency is reduced and the observed convergence for NIPG and IIPG is essentially equal to that of the SIPG method in the DO and OD approach