Introduction to Inverse Problems for Differential Equations

Hasanoğlu, Alemdar Hasanov; Романов, В. Г.

doi:10.1007/978-3-319-62797-7

Cited by 76 publications

(78 citation statements)

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“…On the other hand, when P −1 is unbounded, the situation is different, and the inverse problem is ill-posed [30,24]. In this case, although the kernel of P reduces to {0}, we have…”

Section: The Randomised Stability Constant For Abstract Inverse Problemsmentioning

confidence: 98%

“…which models how the quantity of interest x belonging to a space X is related to the measurements y = T (x) in the space Y . The reader is referred to the many books on inverse problems for a comprehensive exposition (see, e.g., [27,3,47,30,44,2,24,4]). Inverse problems can be ill-posed: the map T may not be injective (i.e., two different x 1 and x 2 may correspond to the same measurement T (x 1 ) = T (x 2 )) or, when injective, T −1 : ran T ⊆ Y → X may not be continuous (i.e., two different and not close x 1 and x 2 may correspond to almost identical measurements T (x 1 ) ≈ T (x 2 )).…”

Section: Introductionmentioning

confidence: 99%

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On the randomised stability constant for inverse problems

Alberti¹,

Capdeboscq²,

Privat³

2020

Mathematics in Engineering

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In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.2010 Mathematics Subject Classification. 65J22, 35R30.

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“…On the other hand, when P −1 is unbounded, the situation is different, and the inverse problem is ill-posed [30,24]. In this case, although the kernel of P reduces to {0}, we have…”

Section: The Randomised Stability Constant For Abstract Inverse Problemsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

On the randomised stability constant for inverse problems

Alberti¹,

Capdeboscq²,

Privat³

2020

Mathematics in Engineering

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“…If s = 0, thenl t = l t ,l l = l l in (16), (40) andṼ t = V t ,K t = K t for sets (17) and (41) according to (38). Since a solution to Perturbed SP Problem (26) exists, applying Identity (38), we get the solutioñ…”

Section: Shape Differentiability Of Objectives For Co Problemsmentioning

confidence: 99%

“…The constraint operator (see (3)) may become: the trace operator under contact conditions [1][2][3], the jump operator for cracks and anticracks [4][5][6], the gradient operator in plasticity [7], the divergence operator under incompressibility conditions [8][9][10], a state-constraint in mathematical programs with equilibrium constraints [11,12], and the like. The constraint problems are related to parameter identification problems (see the theory in References [13][14][15] and application to biological systems in Reference [16]), to inverse problems by the mean of observation data used in mathematical physics [17,18] and in acoustics [19][20][21], to overdetermined and free-boundary problems [22,23]. As an application, in the current paper we focus on the incompressible Brinkman flow problem under a divergence-free condition (see the related modeling of porous medium in References [24,25], well-posedness analysis in Reference [26], and fluid-porous coupling with numerics in References [27,28]).…”

Section: Introductionmentioning

confidence: 99%

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

Granada

Gwinner

Kovtunenko

2018

Axioms

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This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.

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“…In this context, inverse modeling emerges, which is nothing more than obtaining certain variable through the solution of an inverse mathematical problem. In other words, it is possible to mathematically obtaining unmeasurable parameters of a system from mensurable ones since they have a physical relationship (Hasanoğlu & Romanov, 2017 (Klute, 1986) lay fractions, t ), resp Vol. 10,No.…”

Section: Introductionmentioning

confidence: 99%

Estimation of van Genuchten Equation Parameters in Laboratory and through Inverse Modeling with Hydrus-1D

Nascimento¹,

Júnior²,

Araújo³

et al. 2018

JAS

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Soil water retention curve (SWRC) becomes important because it guides when and how much to irrigate, optimizing the use of water; can be obtained in the field or laboratory, being commonly determined in the laboratory with porous plate apparatus, and the determination is compromised by issues such as time and labor. In this context, inverse modeling emerges, which allows to obtain a variable going from the effect to the cause, using Hydrus-1D. Hence, this study aims to obtain van Genuchten equation parameters through inverse modeling with Hydrus-1D and make the respective comparisons and inferences. Matric potential data were obtained over time in an instantaneous profile-type experiment. Six sets of three tensiometers each were installed surrounding the center of the experimental plot, at depths of 0.20, 0.35 and 0.50 m. Target depth was 0.35 m, where the roots of most crops are concentrated, and the other tensiometers were used to obtain the potential gradient. Matric potential data were used to feed Hydrus-1D and obtain the van Genuchten equation parameters. Laboratory curves were obtained using porous plate apparatus, with four replicates. It was concluded that, in general, the Hydrus-1D model estimates van Genuchten equation parameters and, consequently, the SWCC of an Argissolo more consistently with field conditions than those obtained in the laboratory; and, provided it is fed with field data, the Hydrus-1D simulates well the behavior of matric potential and moisture over time, reducing the time and labor in the procedures to obtain van Genuchten equation parameters in the laboratory.

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Introduction to Inverse Problems for Differential Equations

Cited by 76 publications

References 0 publications

On the randomised stability constant for inverse problems

On the randomised stability constant for inverse problems

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

Estimation of van Genuchten Equation Parameters in Laboratory and through Inverse Modeling with Hydrus-1D

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