Analysis of Four-Dimensional Variational Data Assimilation Problems in Low Regularity Spaces

2023/03/01 by Paula M. Castro, Castro, Paula, Juan Carlos De los Reyes +3
Computer Science · Engineering · Mathematics · #49J20 #49K20 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2303.00847

openalex publication_date 2023/03/01 · openalex created_date 2023/03/05 · openalex updated_date 2026/08/01

Abstract

We carry out a rigorous analysis of four-dimensional variational data assimilation (4D-VAR) problems for linear and semilinear parabolic partial differential equations. Continuity of the state with respect to the spatial variable is required since pointwise observations of the state variable appear in the cost functional. Using maximal parabolic regularity tools, we prove this regularity for initial conditions with L^β-regularity guaranteed by control constraints, rather than Sobolev regularity of the controls ensured by artificial cost terms. We obtain existence of optimal controls and first order necessary optimality conditions for both the convex and nonconvex problem with spatial dimension d=2,3, as well as second order sufficient optimality conditions for the nonconvex problem for d=2.

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