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This paper is devoted to the development and convergence analysis of greedy reconstruction algorithms based on the strategy presented in [Y. Maday and J. Salomon, Joint Proceedings of the 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference, 2009, pp. 375--379]. These procedures allow the design of a sequence of control functions that ease the identification of unknown operators in nonlinear dynamical systems. The original strategy of greedy reconstruction algorithms is based on an offline/online decomposition of the reconstruction process and an ansatz for the unknown operator obtained by an a priori chosen set of linearly independent matrices. In the previous work [S. Buchwald, G. Ciaramella, and J. Salomon, SIAM J. Control Optim., 59 (2021), pp. 4511--4537], convergence results were obtained in the case of linear identification problems. We tackle here the more general case of nonlinear systems. More precisely, we introduce a new greedy algorithm based on the linearized system. We show that the controls obtained with this new algorithm lead to the local convergence of the classical Gauss--Newton method applied to the online nonlinear identification problem. We then extend this result to the controls obtained on nonlinear systems where a local convergence result is also proved. The main convergence results are obtained for dynamical systems with linear and bilinear control structures. [ABSTRACT FROM AUTHOR]

This paper is devoted to the qualitative properties of discretized parabolic operators, such as nonnegativity and nonpositivity preservation, maximum/minimum principles and maximum norm contractivity. In the linear case, earlier papers of the authors (Faragó and Horváth in SIAM Sci Comput 28:2313–2336, 2006, IMA J Numer Anal 29:606–631, 2009) have established the connections between the above qualitative properties and have given sufficient conditions for their validity. The present paper extends the above results to nonlinear discretized parabolic operators, also motivated by the authors' recent paper (Faragó and Horváth in J Math Anal Appl 448:473–497, 2017), which has given related results on the continuous PDE level. A systematic study is presented, ranging from general discrete mesh operators to proper finite element applications. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the n-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space Hq(Rn). [ABSTRACT FROM AUTHOR]

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu-Lσ,μ[φ(u)] = f in RN×(0,T), where Lσ,μ is a general symmetric diffusion operator of Lévy type and φ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators Lσ,μ are the (fractional) Laplacians Δ and -(-Δ)α/2 for α ∈ (0,2), discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal Lévy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions, including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [F. del Teso, J. Endal, and E. R. Jakobsen, C. R. Math. Acad. Sci. Paris, 355 (2017), pp. 1154--1160]. We also present some numerical tests, but extensive testing is deferred to the companion paper [F. del Teso, J. Endal, and E. R. Jakobsen, SIAM J. Numer. Anal., 56 (2018), pp. 3611-3647] along with a more detailed discussion of the numerical methods included in our theory. [ABSTRACT FROM AUTHOR]