A Variational Finite Element Method for Source Inversion for Convective-Diffusive Transport

We consider the inverse problem of determining an arbitrary source in a time-dependent convective-diffusive transport equation, given a velocity field and pointwise measurements of the concentration. Applications that give rise to such problems include determination of groundwater or airborne pollutant sources from measurements of concentrations, and identification of sources of chemical or biological attacks. To address ill-posedness of the problem, we employ Tikhonov and total variation regularization. We present a variational formulation of the first order optimality system, which includes the initial-boundary value state problem, the final-boundary value adjoint problem, and the space-time boundary value source problem. We discretize in the space-time volume using Galerkin finite elements. Several examples demonstrate the influence of the density of the sensor array, the effectiveness of total variation regularization for discontinuous sources, the invertibility of the source as the transport becomes increasingly convection-dominated, the ability of the space-time inversion formulation to track moving sources, and the optimal convergence rate of the finite element approximation

Estimation of the eddy thermal conductivity for lake Botonega

This paper presents a part of a computer model that is suitable for limited temperature prediction and its application for Lake Botonega, which is located in Istria, Croatia. The main assumption of this study is that the heat transfer can be described by the eddy diffusivity model to formulate the model of the heating and cooling of a lake using a series of water and air temperature measurements. The coefficient of thermal diffusion, which is a function of the lake depth, is determined using the inverse model of eddy thermal diffusivity. The inverse model is linearized using the finite element approach. The model of lake thermal diffusivity consists of a conductive part and a radiative part, with the latter part being replaced with the heat flux on the boundary. The model parameters are calculated in two steps—a predictor step and a corrector step—and the coefficient of conduction is calculated instead of the diffusion. The estimated parameters are intended for inclusion in a simple three-dimensional thermal model, which provides the lake temperature prediction that is based on previous temperature measurements, as demonstrated in the examples

Calibration and Rescaling Principles for Nonlinear Inverse Heat Conduction and Parameter Estimation Problems

This dissertation provides a systematic method for resolving nonlinear inverse heat conduction problems based on a calibration formulation and its accompanying principles. It is well-known that inverse heat conduction problems are ill-posed and hence subject to stability and uniqueness issues. Regularization methods are required to extract the best prediction based on a family of solutions. To date, most studies require sophisticated and combined numerical methods and regularization schemes for producing predictions. All thermophysical and geometrical properties must be provided in the simulations. The successful application of the numerical methods relies on the accuracy of the related system parameters as previously described. Due to the existence of uncertainties in the system parameters, these numerical methods possess bias of varying magnitudes. The calibration based approaches are proposed to minimize the systematic errors since system parameters are implicitly included in the mathematical formulation based on several calibration tests. To date, most calibration inverse studies have been based on the assumption of constant thermophysical properties. In contrast, this dissertation focuses on accounting for temperature-dependent thermophysical properties that produces a nonlinear heat equation. A novel rescaling principle is introduced for linearzing the system. This concept generates a mathematical framework similar to that of the linear formulation. Unlike the linear formulation, the present approach does require knowledge of thermophysical properties. However, all geometrical properties and sensor characterization are completely removed from the system. In this dissertation, a linear one-probe calibration method is first introduced as background. After that, the calibration method is generalized to the one-probe and two-probe, one-dimensional thermal system based on the assumption of temperature-dependent thermophysical properties. All previously proposed calibration equations are expressed in terms of a Volterra integral equation of the first kind for the unknown surface (net) heat flux and hence requires regularization owning to the ill-posed nature of first kind equations. A new strategy is proposed for determining the optimal regularization parameter that is independent of the applied regularization approach. As a final application, the described calibration principle is used for estimating unknown thermophysical properties above room temperature

Data Driven Estimation of Wall Shear Stress from Magnetic Resonance Imaging

Flows occur in various applications in engineering and medicine. Dynamic quantities, in particular the forces, that the flowing viscous fluid exerts on the neighbouring material, are of special interest. Following the widely accepted flow model, these dynamic quantities are representable by derivatives of the velocity field. A modification of magnetic resonance imaging is capable to measure besides morphological data, also velocity fields in the interior of an object. As a non-invasive method, it is in particular suited for in vivo investigations of the cardiovascular system. This thesis deals with the problem of reconstructing the wall shear stress, the distribution of the shear forces, that the blood flow exerts on the aortic vessel wall. This involves the reconstruction of both the flow geometry and velocity from magnetic resonance data and the evaluation of the normal velocity derivative, the shear rate. At first glance, this problem might seem trivial. However, there are several issues: - Magnetic resonance imaging acquires local means of the flow velocity with comparatively low spatial and temporal resolution. Additionally, the measurements are contaminated by noise. - The blood flow exhibits boundary layers, where the flow field dramatically changes over small distances. This complicates an accurate approximation of the velocity field in the region near to the boundary. - The flow geometry and the flow velocity are structurally connected: Just the shear-rate, that has to be evaluated, exhibits a discontinuity at the boundary. In the first part of this work we present a framework for the purely data driven wall shear stress reconstruction. For this purpose, we approximate the flow geometry first, and then the flow velocity using parametric representations. The reconstruction method allows for a continuous analysis as regularization procedure for two coupled inverse problems. Since the corresponding forward operators satisfy a conditional stability estimate, convergence of the reconstruction method can be established under reasonable smoothness assumptions on the geometry and the flow velocity. These results widely carry over to the discrete setting, where we introduce discrete versions of the forward operators to minimize the data error. In the second part of this work we use methods of data assimilation, to enhance the purely data driven reconstruction using a fluid dynamical model. In a first study, we utilize a variational approach for the enhancement of the velocity reconstruction under known geometry, that minimizes a functional consisting of a data error and a model error and was formerly developed and analyzed in our research group. The variational approach is formally equivalent to an optimal control problem. Secondly, we demonstrate the basic possibility to enhance the geometry identification based on a fixed fluid dynamical model. For both methods, we utilize the widely accurate purely data driven reconstructions for linearization and localization of the applied fluid dynamic model. For the assessment of the developed methods we have conducted a comprehensive validation in collaboration with experts of fluid dynamics and radiology from the Institute for Fluid Mechanics and Aerodynamics, Technical University Darmstadt, and the Department of Radiology – Medical Physics, University Hospital Freiburg, respectively. A big issue is the lack of ground truth. The wall shear stress is highly sensitve to perturbations of the environmental conditions. Hence, the experiments have to be performed with meticulous diligence, to ensure reproducibility and hence validity of the reference values, that are obtained from high-resolution laser Doppler anemometry or computational fluid dynamics. Additionally, we have developed highly robust and accurate but to specific flows limited reconstruction methods, to estimate the wall shear stress directly from the magnetic resonance data. Furthermore we fall back on virtual in silico data in some cases. Already the purely data driven reconstruction method provides largely convincing results. However, the reconstruction is highly sensitve to perturbations of the geometry identification and reveals sometimes systematic errors due to the coarse resolution. Our analysis identifies the error sources and their contribution to the overall error. This offers a guideline for a suitable choice of several parameters in the measurement setup. Furthermore, the described shortcomings of the purely data driven reconstruction are essentially corrected by the provided data assimilation techniques. The specialization to the application in the aorta leads to a fully integrated reconstruction method with low computational effort, typical running times for all provided methods are in the range of several minutes using common hardware. Therefore, a valid estimation of wall shear stress in the aorta is feasible, even under the limitations of clinical routine. This thesis was funded by the DFG via grant EG-331/1-1. The collaboration with the project partners resulted in the following papers: - H. Egger and G. Teschner. On the Stable Estimation of Flow Geometry and Wall Shear Stress from Magnetic Resonance Images. Inverse Problems, 35:095001, 2019. - A. Bauer, S. Wegt, M. Bopp, S. Jakirlic, C. Tropea, A. J. Krafft, N. Shokina, J. Hennig, G. Teschner and H. Egger. Comparison of Wall Shear Stress Estimates Obtained by Laser Doppler Velocimetry, Magnetic Resonance Imaging and Numerical Simulations. Experiments in Fluids, 60:1–16, 2019. - N. Shokina, A. Bauer, G. Teschner, W. B. Buchenberg, C. Tropea, H. Egger, J. Hennig and A. J. Krafft. MR-based Wall Shear Stress Measurements in Fully Developed Turbulent Flow using the Clauser Plot Method. Journal of Magnetic Resonance, 305:16–21, 2019. - N. Shokina, G. Teschner, A. Bauer, C. Tropea, H. Egger, J. Hennig and A. J. Krafft. Quantification of Wall Shear Stress in Large Blood Vessels using Magnetic Resonance Imaging. Computational Technologies, 24:4–27, 2019. - N. Shokina, G. Teschner, A. Bauer, C. Tropea, H. Egger, J. Hennig and A. J. Krafft. Parametric Sequential Method for MRI-based Wall Shear Stress Quantification. IEEE Transactions on Medical Imaging, 40:1105–1112, 2020. In this thesis, we will summarize the findings of the papers mentioned above and appropriatly extend them to an entire analysis of the wall shear stress reconstruction

Numerical and optimal control methods for partial differential equations arising in computational finance

The chosen title for my PhD thesis is "Numerical and optimal control methods for partial differential equations arising in computational finance". The body of my research is divided into two parts. The first part is devoted to the application of an alternating direction implicit numerical method for solving stochastic volatility option pricing models. The second part focuses on a partial-integro differential equation constrained optimal control approach to parameter estimation for the forward jump-diffusion option pricing model. The body of the thesis is preceded by an extensive introduction, which seeks to contextualize my work with respect to the field of computational finance, this is followed by a brief conclusion. Finally, the thesis is completed by a list of refer ences. The first project proposes a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. The approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence. The second project proposes to solve a parameter calibration problem for the forward jump-diffusion option pricing model proposed by Andersen and Andreasen. A distributed optimal control approach is employed, with a partial-integro differential equation as our state equation. By approaching the problem from a functional analysis perspective, I investigate the necessary regularity conditions for our parameters of interest. Following this, the existence of optimal solutions is proven under certain analytical conditions. Furthermore, the first-order necessary conditions for optimality are also established. Finally, a projected-gradient optimization method is applied numerically to empirical market data and results are given

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

Mathematical models describing the behavior of viscoelastic materials are often based on evolution equations that measure the change in stress depending on its material parameters such as stiffness, viscosity or relaxation time. In this article, we introduce a Maxwell-based rheological model, define the associated forward operator and the inverse problem in order to determine the number of Maxwell elements and the material parameters of the underlying viscoelastic material. We perform a relaxation experiment by applying a strain to the material and measure the generated stress. Since the measured data varies with the number of Maxwell elements, the forward operator of the underlying inverse problem depends on parts of the solution. By introducing assumptions on the relaxation times, we propose a clustering algorithm to resolve this problem. We provide the calculations that are necessary for the minimization process and conclude with numerical results by investigating unperturbed as well as noisy data. We present different reconstruction approaches based on minimizing a least squares functional. Furthermore, we look at individual stress components to analyze different displacement rates. Finally, we study reconstructions with shortened data sets to obtain assertions on how long experiments have to be performed to identify conclusive material parameters.Comment: 23 pages, 11 figures, 6 table