Quantitative approximation properties for the fractional heat equation

Rüland, Angkana; Salo, Mikko

doi:10.3934/mcrf.2019027

Cited by 37 publications

(44 citation statements)

References 21 publications

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“…Minimal L 2 norm regularization. Finally, as a further possible means of recovering v from (−∆) s v| W , we use a variational approach which is analogous to the one presented in [RS17b] and [FZ00].…”

Section: Constructive Unique Continuation Resultsmentioning

confidence: 99%

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Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

Ghosh

Rüland

Salo

et al. 2020

Journal of Functional Analysis

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We show global uniqueness in the fractional Calderón problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work [GSU16] considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation result that involve different regularization schemes.This means that zero is not a Dirichlet eigenvalue of (−∆) s + q, and one indeed has a unique solution u ∈ H s (R n ) for any exterior value f . This problem, which was first introduced in [GSU16], should be viewed as a fractional analogue of the classical Calderón problem, which is a well-studied inverse problem for which we refer to the survey article [Uh14] and the references therein. Due to the results of [GSU16], it is known that the Dirichlet-to-Neumann map uniquely determines the potential q, i.e. if q 1 , q 2 ∈ L ∞ (Ω) are such that zero is not a Dirichlet eigenvalue of (−∆) s + q i , i ∈ {1, 2}, thenMoreover, uniqueness holds if the measurements are made on arbitrary, possibly disjoint subsets of the exterior. In [RS17a] this has further been extended to (almost) optimal function spaces, including potentials in L n 2s (Ω). Logarithmic stability for this inverse problem was also proved 2010 Mathematics Subject Classification. 35R11, 35R30, 35J10. Key words and phrases. fractional Calderón problem, single measurement recovery and uniqueness, unique continuation from measurable sets.

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Section: Constructive Unique Continuation Resultsmentioning

confidence: 99%

“…Proof. The proof follows along the lines of Lemmas 4.1 and 4.2 in [RS17b], which is based on the variational approach from [FZ00]. For self-containedness, we repeat the argument: Firstly, we note that the functional J α is strictly convex and continuous with respect to H s (W ) convergence (of f ∈ H s (W )).…”

Section: Constructive Unique Continuation Resultsmentioning

confidence: 99%

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

Ghosh

Rüland

Salo

et al. 2020

Journal of Functional Analysis

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“…This strong approximation result was first proved in [DSV17] for the fractional Laplacian and C k norms. Related results for other equations and norms are given in [DSV19, GSU16,RS17].…”

Section: Introductionmentioning

confidence: 99%

The fractional Calderón problem: Low regularity and stability

2020

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The Calderón problem for the fractional Schrödinger equation was introduced in the work [GSU16], which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant L p or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli-Silvestre extension and a duality argument.

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“…the discussion below). Moreover, these techniques have been extended to other nonlocal problems [RS17b,CL18] in a slightly different context. Also, for positive potentials, monotonicity inversion formulas have been successfully discovered in [HL17].…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem for the fractional Schrödinger equation with drift

2020

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We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from [GRSU18], this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension n ≥ 1.

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Quantitative approximation properties for the fractional heat equation

Cited by 37 publications

References 21 publications

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

The fractional Calderón problem: Low regularity and stability

The Calderón problem for the fractional Schrödinger equation with drift

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