Reconstruction of two constant coefficients in linear anisotropic diffusion model

Mola, Gianluca; Okazawa, Noboru; Yokota, Tomomi

doi:10.1088/0266-5611/32/11/115016

Cited by 3 publications

(11 citation statements)

References 8 publications

(7 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We subsume this result in the next Lemma 2.2. For all admissible x ∈ H and all (α Such a condition is in fact implying the same conclusion of Lemma 2.2, and this can be proven exactly like in [14,Lemma 2.4]. However, in that case the solutions are forced to be classical, meanwhile the admissibility assumptions we are now proposing allow the initial datum to range in a larger set, so that here the solutions may also be of weak type.…”

Section: 2mentioning

confidence: 60%

“…are linearly independent in H. As a matter of fact, like in the case n = 2 (cf. [14]), it is not difficult to realize that such a condition is strictly necessary for proving our result. In fact, without losing of generality, suppose…”

Section: 2mentioning

confidence: 92%

“…Moving forward to the more difficult case n = 2, the first results have been provided in [10], in the special case when operator A 2 is the identity on H. Therein existence and uniqueness of a weak solution (u, α 1 , α 2 ) was analyzed, as long as its continuous dependence on the data (x, ϕ 1 , ϕ 2 ), by adapting a finite-dimensional Faedo-Galerkin approximation scheme to the inverse problem. More recently, in [14], the same results have been extended for general operators A 1 and A 2 and rely on the abstract semigroup theory. In this case, a classic solution has been constructed.…”

mentioning

confidence: 92%

“…The task of the present paper is the one to generalize the uniqueness and the Lipschitz dependence in [14] to an arbitrary number of diffusion constants, under the overdetermination delivered by the same number of energy measurements performed at the time-instant T . As, for a larger number of unknown constants, the approach proposed in [14] -involving implicit functions -seems difficult to be performed, in order to recover α 1 , • • • , α n from the additional measurements we have to use a global injectivity theorem based on the positivity of the Jacobian matrix (see Subsection 3.1), which is the key tool of our investigation. Besides, this allows to prove only a uniqueness result, whereas understanding suitable conditions on the data which allow to recover also the existence of solutions is still an open issue, although less interesting for the sake of inverse problems.…”

mentioning

confidence: 99%

“…Moreover, we shall also provide such a statement for a larger class of initial data, which do not necessarily need to belong to the intersection of the domains of operators A i as in [14]. As a consequence, here we deal with weak solutions.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Mola¹

2018

Inverse Problems &Amp; Imaging

Self Cite

View full text Add to dashboard Cite

Let (H, •, •) be a separable Hilbert space and A i : D(A i) → H (i = 1, • • • , n) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function u : [0, T ] → H and n constants α 1 , • • • , αn > 0 (diffusion coefficients) that fulfill the initial-value problem u (t) + α 1 A 1 u(t) + • • • + αnAnu(t) = 0, t ∈ (0, T), u(0) = x, and the additional conditions A 1 u(T), u(T) = ϕ 1 , • • • , Anu(T), u(T) = ϕn, where ϕ i are given positive constants. Under suitable assumptions on the operators A i and on the initial data x ∈ H, we shall prove that the solution of such a problem is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion constants in a heat equation and of the Lamé parameters in a elasticity problem on a plate.

show abstract

Section: 2mentioning

confidence: 60%

Section: 2mentioning

confidence: 92%

mentioning

confidence: 92%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Mola¹

2018

Inverse Problems &Amp; Imaging

Self Cite

View full text Add to dashboard Cite

show abstract

Solvability Analysis of Problems of Determining External Influence of Combined Type in Processes Described by Parabolic Equations

Kozhanov,

Namsaraeva

2024

J. Appl. Ind. Math.

View full text Add to dashboard Cite

Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

Favini

Mola

Romanelli

2022

DCDS-S

View full text Add to dashboard Cite

Let <inline-formula><tex-math id="M1">\begin{document}$ \left(H, \langle \cdot, \cdot \rangle \right) $\end{document}</tex-math></inline-formula> be a separable Hilbert space and <inline-formula><tex-math id="M2">\begin{document}$ A_{i}:D(A_i) \to H $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ i = 1, \cdots, n $\end{document}</tex-math></inline-formula>) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function <inline-formula><tex-math id="M4">\begin{document}$ u:[0, T] \to H $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> time-dependent diffusion coefficients <inline-formula><tex-math id="M6">\begin{document}$ \alpha_{1}, \cdots, \alpha_{n}:[s, T] \to {\mathbb{R}}_+ $\end{document}</tex-math></inline-formula> that fulfill the initial-value problem<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u'(t) + \alpha_{1}(t) A_{1}u(t) + \cdots + \alpha_{n}(t) A_{n}u(t) = 0, \quad s \leq t \leq T, \quad u(s) = x, $\end{document} </tex-math></disp-formula>and the additional conditions<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \langle A_{1} u(t), u(t)\rangle = \varphi_{1}(t), \quad \cdots \quad, \langle A_{n} u(t), u(t)\rangle = \varphi_{n}(t), \quad s \leq t \leq T. $\end{document} </tex-math></disp-formula>Under suitable assumptions on the operators <inline-formula><tex-math id="M7">\begin{document}$ A_i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ i = 1, \cdots, n $\end{document}</tex-math></inline-formula>, on the initial data <inline-formula><tex-math id="M9">\begin{document}$ x\in H $\end{document}</tex-math></inline-formula> and on the given functions <inline-formula><tex-math id="M10">\begin{document}$ \varphi_i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ i = 1, \cdots, n $\end{document}</tex-math></inline-formula>, we shall prove that the solution of such a problem exists, is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion coefficients in a heat equation and of the Lamé parameters in an elasticity problem on a plate.

show abstract

Reconstruction of two constant coefficients in linear anisotropic diffusion model

Cited by 3 publications

References 8 publications

Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Solvability Analysis of Problems of Determining External Influence of Combined Type in Processes Described by Parabolic Equations

Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

Contact Info

Product

Resources

About