### Browsing by Author "Mustonen, Lauri"

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Item Approximations and Surrogates for Computational Inverse Boundary Value Problems(Aalto University, 2017) Mustonen, Lauri; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandInverse boundary value problems are closely related to imaging techniques where measurements on the surface are used to estimate, or reconstruct, inner properties of the imaged object. In this thesis, improved reconstruction methods and new computational approaches are presented for elliptic and parabolic inverse boundary value problems. Two imaging applications that are addressed are electrical impedance tomography (EIT) and thermal tomography. The inverse problems that are considered are nonlinear and the reconstructions are sought by least squares minimization. Algorithms for such minimization often rely on iterative evaluation of the target function and its partial derivatives. In this thesis, we approximate these target functions, which themselves are solutions to partial differential equations, with polynomial surrogates that are simple to evaluate and differentiate. In the context of EIT, this method is used to estimate the shape of the object in addition to its electrical properties. The method is also shown to be feasible for thermal tomography, including the case of uncertain object shape. We also present a novel logarithmic linearization method for EIT. Transforming the voltage measurements in a certain logarithmic way reduces the nonlinearity in the relationship between the electrical properties and the measurements, allowing a reconstruction with fewer or only one minimization step. Furthermore, we propose a modification to the complete electrode model for EIT. The new model is shown to be compatible with experimental measurement data, while the increased regularity of the predicted electromagnetic potential improves convergence properties of numerical methods.Item Detecting stochastic inclusions in electrical impedance tomography(IOP PUBLISHING LTD, 2017-10) Barth, Andrea; Harrach, Bastian; Hyvönen, Nuutti; Mustonen, Lauri; Department of Mathematics and Systems Analysis; Numerical Analysis; University of Stuttgart; Goethe University FrankfurtThis work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.Item Generalized linearization techniques in electrical impedance tomography(SPRINGER HEIDELBERG, 2018-09) Hyvönen, Nuutti; Mustonen, Lauri; Department of Mathematics and Systems AnalysisElectrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current–voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. In this paper, we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumann-to-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Fréchet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography.Item An inverse boundary value problem for the p-Laplacian(IOP PUBLISHING LTD, 2019-01-28) Hannukainen, Antti; Hyvönen, Nuutti; Mustonen, Lauri; Department of Mathematics and Systems Analysis; Stanford UniversityThis work tackles an inverse boundary value problem for a p-Laplace type partial differential equation parametrized by a smoothening parameter T ≥ 0. The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. The numerical studies are based on a straightforward linearization of the forward map, and they demonstrate that the accuracy of such an approach depends nontrivially on 1 < p < ∞ and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a Hölder continuous conductivity coefficient to the solution of the Neumann problem, is Fréchet differentiable, excluding the degenerate case T = 0 that corresponds to the classical (weighted) -Laplace equation.Item Käänteisen inkluusio-ongelman tilastollinen ratkaisu käyttäen Möbius-kuvauksia(2012-09-05) Mustonen, Lauri; Hyvönen, Nuutti; Perustieteiden korkeakoulu; Kinnunen, JuhaItem Optimizing Heating Patterns in Thermal Tomography(2018-10-02) Puska, Juha-Pekka; Mustonen, Lauri; Perustieteiden korkeakoulu; Hyvönen, NuuttiThermal tomography is a promising method for non-destructive testing of materials based on measuring the boundary temperature of an object that is exposed to a known heat flux. The estimation of the internal structure, i.e. the spatially varying heat capacity and thermal conductivity, is an inverse problem, to which the statistical inversion approach is applied. The question of optimal experiment design is how to conduct an experiment so that the maximum amount of information is gained. In this thesis, the goal is to optimize the time-dependent heating patterns in thermal tomography. The actual measurements were numerically simulated using finite element modeling. The resulting parameter reconstructions with the optimized heating patterns were compared to a reference pattern to see if, on average, the optimized patterns resulted in smaller reconstruction errors. The results indicate that on average the reconstruction error was mostly dependent on the rate of increase in the heating and the total amount of heat transferred. The optimization procedure also consistently resulted in patterns with maximum heating within the given constraints.Item Parametric differential equations and inverse diffusivity problem(2014) Mustonen, Lauri; Leinonen, Matti; Perustieteiden korkeakoulu; Perustieteiden korkeakoulu; Hyvönen, NuuttiParametric differential equations have received increased attention during the past decade, mostly due to their applications to quantifying stochastic systems. In this thesis, we formulate the parametric time-dependent diffusion equation and solve it numerically by using the finite element method in the spatial domain and the spectral Galerkin method in the parameter domain. The obtained solution is used as a tool for an inverse boundary value problem, where the unknown is the diffusion coefficient. The absence of random variables in the forward problem allows choosing compactly supported functions such as splines to represent the diffusivity, whereas the stochastic equations are usually parametrized by using orthogonal Karhunen-Loève eigenfunctions. We analyze how the locality of the diffusivity functions affects the sparsity of the resulting large linear system. In the context of direct equation solvers, the reduction of fill-in is addressed with numerical examples. Although discretization errors are clearly visible, diffusivity reconstructions indicate that the solution to the parametric equation may provide a feasible algorithm for the inverse diffusivity problem, which further can be considered as a basis for thermal tomography.Item Smoothened complete electrode model(Society for Industrial and Applied Mathematics Publications, 2017-12) Hyvönen, Nuutti; Mustonen, Lauri; Department of Mathematics and Systems Analysis; Emory UniversityThis work reformulates the complete electrode model of electrical impedance tomography in order to enable more efficient numerical solution. The model traditionally assumes constant contact conductances on all electrodes, which leads to a discontinuous Robin boundary condition since the gaps between the electrodes can be described by vanishing conductance. As a consequence, the regularity of the electromagnetic potential is limited to less than two square-integrable weak derivatives, which negatively affects the convergence of, e.g., the finite element method. In this paper, a smoothened model for the boundary conductance is proposed, and the unique solvability and improved regularity of the ensuing boundary value problem are proven. Numerical experiments demonstrate that the proposed model is both computationally feasible and compatible with real-world measurements. In particular, the new model allows faster convergence of the finite element method.Item Thermal tomography with unknown boundary(SIAM PUBLICATIONS, 2018-01-01) Hyvönen, Nuutti; Mustonen, Lauri; Department of Mathematics and Systems Analysis; Emory UniversityThermal tomography is an imaging technique for deducing information about the internal structure of a physical body from temperature measurements on its boundary. This work considers time-dependent thermal tomography modeled by a parabolic initial/boundary value problem without accurate information on the exterior shape of the examined object. The adaptive sparse pseudospectral approximation method is used to form a polynomial surrogate for the dependence of the temperature measurements on the thermal conductivity, the heat capacity, the boundary heat transfer coefficient, and the body shape. These quantities can then be efficiently reconstructed via nonlinear, regularized least squares minimization employing the surrogate and its derivatives. The functionality of the resulting reconstruction algorithm is demonstrated by numerical experiments based on simulated data in two spatial dimensions.