| 1. |
- Achieng, Pauline, 1990-
(author)
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Reconstruction of solutions of Cauchy problems for elliptic equations in bounded and unbounded domains using iterative regularization methods
- 2023
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Doctoral thesis (other academic/artistic)abstract
- Cauchy problems for elliptic equations arise in applications in science and engineering. These problems often involve finding important information about an elliptical system from indirect or incomplete measurements. Cauchy problems for elliptic equations are known to be disadvantaged in the sense that a small pertubation in the input can result in a large error in the output. Regularization methods are usually required in order to be able to find stable solutions. In this thesis we study the Cauchy problem for elliptic equations in both bounded and unbounded domains using iterative regularization methods. In Paper I and II, we focus on an iterative regularization technique which involves solving a sequence of mixed boundary value well-posed problems for the same elliptic equation. The original version of the alternating iterative technique is based on iterations alternating between Dirichlet-Neumann and Neumann-Dirichlet boundary value problems. This iterative method is known to possibly work for Helmholtz equation. Instead we study a modified version based on alternating between Dirichlet-Robin and Robin-Dirichlet boundary value problems. First, we study the Cauchy problem for general elliptic equations of second order with variable coefficients in a limited domain. Then we extend to the case of unbounded domains for the Cauchy problem for Helmholtz equation. For the Cauchy problem, in the case of general elliptic equations, we show that the iterative method, based on Dirichlet-Robin, is convergent provided that parameters in the Robin condition are chosen appropriately. In the case of an unbounded domain, we derive necessary, and sufficient, conditions for convergence of the Robin-Dirichlet iterations based on an analysis of the spectrum of the Laplacian operator, with boundary conditions of Dirichlet and Robin types.In the numerical tests, we investigate the precise behaviour of the Dirichlet-Robin iterations, for different values of the wave number in the Helmholtz equation, and the results show that the convergence rate depends on the choice of the Robin parameter in the Robin condition. In the case of unbounded domain, the numerical experiments show that an appropriate truncation of the domain and an appropriate choice of Robin parameter in the Robin condition lead to convergence of the Robin-Dirichlet iterations.In the presence of noise, additional regularization techniques have to implemented for the alternating iterative procedure to converge. Therefore, in Paper III and IV we focus on iterative regularization methods for solving the Cauchy problem for the Helmholtz equation in a semi-infinite strip, assuming that the data contains measurement noise. In addition, we also reconstruct a radiation condition at infinity from the given Cauchy data. For the reconstruction of the radiation condition, we solve a well-posed problem for the Helmholtz equation in a semi-infinite strip. The remaining solution is obtained by solving an ill-posed problem. In Paper III, we consider the ordinary Helmholtz equation and use seperation of variables to analyze the problem. We show that the radiation condition is described by a non-linear well-posed problem that provides a stable oscillatory solution to the Cauchy problem. Furthermore, we show that the ill–posed problem can be regularized using the Landweber’s iterative method and the discrepancy principle. Numerical tests shows that the approach works well.Paper IV is an extension of the theory from Paper III to the case of variable coefficients. Theoretical analysis of this Cauchy problem shows that, with suitable bounds on the coefficients, can iterative regularization methods be used to stabilize the ill-posed Cauchy problem.
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| 2. |
- Argatov, Ivan, et al.
(author)
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A simple mathematical model for the resonance frequency analysis of dental implant stability : Implant clamping quotient.
- 2019
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In: Mechanics research communications. - : Elsevier. - 0093-6413 .- 1873-3972. ; 95, s. 67-70
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Journal article (peer-reviewed)abstract
- A simple mathematical model for free vibrations of an elastically clamped beam is suggested to interpret the results of the resonance frequency analysis developed for implant stability measurements in terms of the Implant Stability Quotient (ISQ) units. It is shown that the resonance frequency substantially depends on the lateral compliance of the implant/bone system. Based on the notion of the lateral stiffness of the implant/bone system, a new measure of the implant stability is introduced in the form similar to the ISQ scale and is called the Implant Clamping Quotient (ICQ), because it characterizes the jawbone’s clamp of the implant. By definition, the ICQ unit is equal to a percentage of the original scale for the lateral stiffness of the implant/bone system.
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| 3. |
- Argatov, Ivan, et al.
(author)
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How to define the storage and loss moduli for a rheologically nonlinear material?
- 2017
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In: Continuum Mechanics and Thermodynamics. - : Springer. - 0935-1175 .- 1432-0959. ; 29:6, s. 1375-1387
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Journal article (peer-reviewed)abstract
- A large amplitude oscillatory shear (LAOS) is considered in the strain-controlled regime, and the interrelation between the Fourier transform and the stress decomposition approaches is established. Several definitions of the generalized storage and loss moduli are examined in a unified conceptual scheme based on the Lissajous–Bowditch plots. An illustrative example of evaluating the generalized moduli from a LAOS flow is given.
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| 4. |
- Argatov, Ivan, et al.
(author)
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Rayleigh surface waves in functionally graded materials : long-wave limit
- 2019
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In: Quarterly Journal of Mechanics and Applied Mathematics. - : Oxford Univeristy Press. - 0033-5614 .- 1464-3855. ; 72:2, s. 197-211
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Journal article (peer-reviewed)abstract
- A first-order asymptotic model for describing waves propagating along the surface of a functionally graded isotropic elastic half-space is constructed in the long-wave limit under the assumption of a finitely supported perturbation of the half-space properties. Explicit approximations for the Rayleigh waves are derived under the assumption that the semi-infinite elastic medium is slightly inhomogeneous
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| 5. |
- Argatov, Ivan, et al.
(author)
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Resonance spectrum for a continuously stratified layer : application to ultrasonic testing
- 2013
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In: Waves in Random and Complex Media. - : Taylor & Francis. - 1745-5030 .- 1745-5049. ; 23:1, s. 24-42
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Journal article (peer-reviewed)abstract
- Ultrasound wave propagation in a nonhomogeneous linearly elastic layer of constant thickness immersed between homogeneous fluid and solid media is considered. The resonances (scattering poles) for the corresponding acoustic propagator are studied. It is shown that the distribution of the resonances depends on the smoothness of the coefficients that characterize physical properties of the layer and the ambient media. Namely, if the coefficients have jump discontinuities at the boundaries, then the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. On the contrary, if the coefficients are continuous, then it is shown that the resonances are asymptotically distributed along a logarithmic curve. The developed mathematical model is applied to the ultrasonic testing of the articular cartilage (AC) layer attached to the subchondral bone from one side and being in contact with a solution on the other side. It is conjectured that the spacing between two successive resonances may be sensitive to AC degeneration. The application of the obtained results to the development of ultrasonic testing for quantitative evaluation of AC is discussed.
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| 6. |
- Chepkorir, Jennifer, 1989-
(author)
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Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations
- 2024
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Doctoral thesis (other academic/artistic)abstract
- In this thesis, we study Cauchy problems for the elliptic and degenerate elliptic equations. These problems are ill-posed. We split the boundary of the domain into two parts. On one of them, say Γ0, we have available Cauchy data and on remaining part Γ1 we introduce unknown Robin data. To construct the operator equation which replaces our Cauchy problem we use two boundary value problems (BVP). The first one is the mixed BVP with Robin condition on Γ1 and with Dirichlet condition on Γ0 and the second BVP with Dirichlet Data on Γ1 and with Robin data on Γ0. The well–posedness of these problems is achieved by an appropriate choice of parameters in Robin boundary conditions. The first Dirichlet–Robin BVP is used to construct the operator equation replacing the Cauchy problem and the second Robin–Dirichlet problem for adjoint operator. Using these problems we can apply various regularization methods for stable reconstruction of the solution. In Paper I, the Cauchy problem for the elliptic equation with variable coefficients, which includes Helmholtz type equations, is analyzed. A proof showing that the Dirichlet–Robin alternating algorithm is convergent is given, provided that the parameters in the Robin conditions are chosen appropriately. Numerical experiments that shows the behaviour of the algorithm are given. In particular, we show how the speed of convergence depends on the choice of Robin parameters. In Paper II, the Cauchy problem for the Helmholtz equation, for moderate wave numbers k2, is considered. The Cauchy problem is reformulated as an operator equation and iterative method based on Krylov subspaces are implemented. The aim is to achieve faster convergence in comparison to the Alternating algorithm from the previous paper. Methods such as the Landweber iteration, the Conjugate gradient method and the generalized minimal residual method are considered. We also discuss how the algorithms can be adapted to also cover the case of non–symmetric differential operators. In Paper III, we look at a steady state heat conduction problem in a thin plate. The plate connects two cylindrical containers and fix their relative positions. A two dimensional mathematical model of heat conduction in the plate is derived. Since the plate has sharp edges on the sides we obtained a degenerate elliptic equation. We seek to find the temperature on the interior cylinder by using data on the exterior cylinder. We reformulate the Cauchy problem as an operator equation, with a compact operator. The operator equation is solved using the Landweber method and the convergence is investigated. In Paper IV, the Cauchy problem for a more general degenerate elliptic equation is considered. We stabilize the computations using Tikhonov regularization. The normal equation, in the Tikhonov algorithm, is solved using the Conjugate gradient method. The regularization parameter is picked using either the L–curve or the Discrepancy principle. In all papers, numerical examples are given where we solve the various boundary value problems using a finite difference scheme. The results show that the suggested methods work quite well.
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| 7. |
- de Hoop, Maarten, et al.
(author)
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Semiclassical analysis of elastic surface waves
- 2017
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Other publication (other academic/artistic)abstract
- In this paper, we present a semiclassical description of surface waves or modes in an elastic medium near a boundary, in spatial dimension three. The medium is assumed to be essentially stratified near the boundary at some scale comparable to the wave length. Such a medium can also be thought of as a surficial layer (which can be thick) overlying a half space. The analysis is based on the work of Colin de Verdi\`ere on acoustic surface waves. The description is geometric in the boundary and locally spectral "beneath" it. Effective Hamiltonians of surface waves correspond with eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. Using these Hamiltonians, we obtain pseudodifferential surface wave equations. We then construct a parametrix. Finally, we discuss Weyl's formulas for counting surface modes, and the decoupling into two classes of surface waves, that is, Rayleigh and Love waves, under appropriate symmetry conditions.
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| 8. |
- de Hoop, Maarten V., et al.
(author)
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Analysis of wavenumber resonances for the Rayleigh system in a half space
- 2023
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In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : Royal Society. - 1364-5021 .- 1471-2946. ; 479:2277
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Journal article (peer-reviewed)abstract
- We present a comprehensive analysis of wavenumber resonances or leaking modes associated with the Rayleigh operator in a half space containing a heterogeneous slab, being motivated by seismology. To this end, we introduce Jost solutions on an appropriate Riemann surface, a boundary matrix and a reflection matrix in analogy to the studies of scattering resonances associated with the Schrödinger operator. We analyse their analytic properties and characterize the distribution of these wavenumber resonances. Furthermore, we show that the resonances appear as poles of the meromorphic continuation of the resolvent to the nonphysical sheets of the Riemann surface as expected.
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| 9. |
- De Hoop, Maarten V, et al.
(author)
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Inverse problem for Love waves in a layered, elastic half-space
- 2024
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In: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; :045013, s. 1-44
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Journal article (peer-reviewed)abstract
- In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves.
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| 10. |
- de Hoop, Maarten V., et al.
(author)
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Inverse problem for the Rayleigh system with spectral data
- 2022
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In: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 63:3, s. 1-33
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Journal article (peer-reviewed)abstract
- We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lamé parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and we point out parallels with a corresponding problem for the Schrödinger equation. The Jost function can be identified with spectral data. We derive a Gel’fand-Levitan type equation and obtain uniqueness with two distinct frequencies.
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