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School of Mathematics Whilst direct formulations consist of determining the effect of a given cause, in inverse formulations the situation is completely, or partially reversed.
The interest is into the research of inverse problems for partial differential equations governing phenomena in fluid flow, elasticity, acoustics, heat transfer, mechanics of aerosols, etc. Typical practical applications relate to flows in porous media, heat conduction in materials, thermal barrier coatings, heat exchangers, corrosion, etc. The objectives are to investigate the existence, uniqueness and stability of the solution to the problem that mathematically models a physical phenomenon under investigation, and to develop new convergent, stable and robust algorithms for obtaining the desired solution. The analyses concern inverse boundary value problems, inverse initial value problems, parameter identification, inverse geometry and source determination problems.The Boundary Element Method (BEM) is attractive mainly due to the possibility of reducing the dimensionality of a boundary value problem described by linear partial differential equations. To be successful in the reduction of dimensionality it is needed to have the fundamental solution of the original partial differential equations available in an analytical or simple form. First, we show that the problem is severely ill posed in the sense of Hadamard. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill posed problem. A stability estimate of logarithmic type is established. 2017 Elsevier Ltd We employ the method of fundamental solutions (MFS) for detecting a sound soft scatterer surrounding a host acoustic homogeneous medium due to a given point source inside it. The measurements are taken inside the medium and, in addition, are contaminated louis vuitton briefcase knockoff with noise. The MFS discretization yields a nonlinear constrained regularized minimization problem which is solved using standard software. The results of several numerical experiments are presented and discussed. Gas flow in shale is a very complex phenomenon, currently investigated using a variety of techniques including the analysis of transient experiments conducted on full core and crushed shale using a range of gases. A range of gas flow mechanisms may operate in shale including continuum flow, slippage, transitional flow and Knudsen diffusion. These processes, as well as gas sorption, need to be taken into account when interpreting experimental data and extrapolating the results to the subsurface. Several models have been published that attempt to account for these differentprocesses. Unfortunately, these have a large number of unknown parameters and few studies have assessed the extent to which transient experiments may be used to invert for the key unknowns or the errors that are associated. Here we present a methodology in which various inversion techniques are applied louis vuitton neverfull damier vs monogram to assess the viability of deriving key unknowns which control gas flow in shale from transient experiments with a range of noise. A finite volume method is developed for solving the model of Civan (2010, 2011a,b) of transient gas flow in shale. The model is applicable to non linear diffusion problems, in which the permeability and fluid density both depend on the scalar variable, pressure. The governing equation incorporates the Knudsen number, allowing different flow mechanisms to be addressed, as well as the gas adsorption isotherm. The method is verified for unsteady state problems for which analytical or numerical solutions are available. The method is then applied to a pressure pulse decay test. An inverse numerical formulation is generated, using a minimization iterative algorithm, to estimate some unknown physical parameters. Both numerically simulated noisy and experimentaldata are input into the formulation of the inverse problem. Error analysis is undertaken to investigate the accuracy of results. A good agreement between inverted and exact parameter values is obtained for several parameters. However, it was found that the strong correlation louis vuitton alma handbag uk between intrinsic permeability and tortuosity meant that it was not possible to accurately invert simultaneously for these two parameters from the current pressure pulse decay model. 2017 We consider the inverse problem of determining the time dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples. This paper investigates the inverse problems of simultaneous reconstruction of time dependent thermal conductivity, convection or absorption coefficients in the parabolic heat equation governing transient heat and bio heat thermal processes. Using initial and boundary conditions, as well as heat moments as over determination conditions ensure that these inverse problems have a unique solution. However, the problems are still ill posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization. A discussion of the choice of multiple regularization parameters is provided. The finite difference method with the Crank Nicolson scheme is employed as a direct solver. The resulting inverse problems are recast as nonlinear minimization problems and are solved using the lsqnonlin routinefrom the MATLAB toolbox. Numerical results are presented and discussed. 2016 Informa UK Limited, trading as Taylor Group. An inverse problem in static thermo elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique. potential, temperature or pressure, may louis vuitton bags cost satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz partial differential equations (PDEs). For constant coefficients, the solutions of these elliptic PDEs are sought as linear combinations of explicitly available fundamental solutions (free space Greens functions), as in the method of fundamental solutions (MFS). Prior to this application of the MFS, the free term inhomogeneity represented by the intensity of the source is removed by the method of particular solutions. The resulting transmission problem then recasts as that of determining the interface between composite materials. In order to ensure a unique solution, the unknownsource domain is assumed to be star shaped. This in turn enables its boundary to be parametrized by the radial coordinate, as a function of the polar or, spherical angles. The problem is nonlinear and the numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin which is designed to minimize a sum of squares starting from an initial guess and with no gradient required to be supplied by the user. Simple bounds on the variables can also be prescribed.
Since the inverse problem is still ill posed with respect to small errors in the data and possibly additional ill conditioning introduced by the spectral feature of the MFS approximation, the least squares functional which is minimized needs to be augmented with regularizing penalty terms on the MFS coefficients and on the radial function for a stable estimation of these couple of unknowns. Thorough numerical investigations are undertaken for retrieving regular and irregular shapes of the source support from both exact and noisy input data.Hussein MS, Lesnic D, Ivanchov MI Identification of a heterogeneous orthotropic conductivity in a rectangular domain International Journal of Novel Ideas:Mathematics, 1, 2017.
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