Search results for "runge approximation"

showing 3 items of 3 documents

Quantitative Approximation Properties for the Fractional Heat Equation

2017

In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal} pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain \emph{qualitative} approximation results from \cite{DSV16}. Using propagation of smallness arguments, we then provide bounds on the \emph{cost} of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss genera…

osittaisdifferentiaaliyhtälöt0209 industrial biotechnologyClass (set theory)Control and Optimizationfractional parabolic Calderón problemPseudodifferential operatorsApplied Mathematics010102 general mathematics02 engineering and technologyType (model theory)nonlocal operators [cost of approximation]01 natural sciencesinversio-ongelmatControllabilityMathematics - Analysis of PDEsweak unique continuation [Runge approximation]020901 industrial engineering & automationFOS: MathematicsApplied mathematicsHeat equationapproksimointi0101 mathematicsMathematicsAnalysis of PDEs (math.AP)
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On some partial data Calderón type problems with mixed boundary conditions

2021

In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…

osittaisdifferentiaaliyhtälötinverse problemsApplied Mathematics(fractional) Calderón problem010102 general mathematicsDegenerate energy levelsMathematical analysisBoundary (topology)Duality (optimization)Type (model theory)partial dataCarleman estimates01 natural sciencesinversio-ongelmatrunge approximationcomplex geometrical optics solutions010101 applied mathematicsBounded functionBoundary value problemUniqueness0101 mathematicsapproksimointiAnalysisMathematicsestimointiJournal of Differential Equations
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The Calderón problem for the fractional wave equation: Uniqueness and optimal stability

2021

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial di…

osittaisdifferentiaaliyhtälötApplied MathematicsnonlocalCalder´on problemfractional wave equationinversio-ongelmatComputational MathematicsperidynamicMathematics - Analysis of PDEslogarithmic stabilityFOS: Mathematicsstrong uniquenessfractional LaplacianRunge approximationAnalysisAnalysis of PDEs (math.AP)
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