Magdeburg, Universiẗat, Diss., 2005.

Berlin, Freie Universiẗat, Diss., 2002. / Dateiformat: zip, Dateien im PDF-Format.

Zugl.: Bochum, Universiẗat, Diss., 1992.

Techn. University, Diss., 2005--Dresden.

The topic of this thesis is the theoretical and numerical research of optimal control problems for uncertain nonlinear systems, described by semilinear parabolic differential equations with additive noise, where the state is not completely available. Based on a paper by Kazufumi Ito and Karl Kunisch, which was published in 2006 with the title "Receding Horizon Control with Incomplete Observations", we analyze a Model Predictive Control (MPC) approach where the resulting linear problems on small intervals are solved with a Linear Quadratic Gaussian (LQG) design. Further we define a performance index for the MPC/LQG approach, find estimates for it and present bounds for the solutions of the underlying Riccati equations. Another large part of the thesis is devoted to extensive numerical studies for an 1+1- and 3+1-dimensional problem to show the robustness of the MPC/LQG strategy. The last part is a generalization of the MPC/LQG approach to infinite-dimensional problems.

Techn. University, Diss., 2003--Berlin.

Techn. University, Diss., 2003--Berlin.

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.

Complex spatial structures in biology arise from random interactions of molecules. These molecular interactions can be studied using spatial stochastic models, such as Reaction Diffusion Master Equation (RDME), a mesoscopic model that subdivides the spatial domain into smaller, well mixed grid cells, in which the macroscopic diffusion-controlled reactions take place. While RDME has been widely used to study how fluctuations in number of molecules affect spatial patterns, simulations are computationally expensive and it requires a lower bound for grid cell size to avoid an apparent unphysical loss of bimolecular reactions. In this thesis, we propose Gaussian Reaction Diffusion Master Equation (GRDME), a novel model in the RDME framework, based on the discretization of the Laplace operator with Particle Strength Exchange (PSE) method with a Gaussian kernel. We show that GRDME is a computationally efficient model compared to RDME. We further resolve the controversy regarding the loss of bimolecular reactions and argue that GRDME can flexibly bridge the diffusion-controlled and ballistic regimes in mesoscopic simulations involving multiple species. To efficiently simulate GRDME, we develop Gaussian Next Subvolume Method (GNSM). GRDME simulated with GNSM up to six-times lower computational cost for a three-dimensional simulation, providing a significant computational advantage for modeling three-dimensional systems. The computational cost can be further lowered by increasing the so-called smoothing length of the Gassian jumps. We develop a guideline to estimate the grid resolution below which RDME and GRDME exhibit loss of bimolecular reactions. This loss of reactions has been considered unphysical by others. Here we show that this loss of bimolecular reactions is consistent with the well-established theory on diffusion-controlled reaction rates by Collins and Kimball, provided that the rate of bimolecular propensity is interpreted as the rate of the ballistic step, rather than the macroscopic reaction rate. We show that the reaction radius is set by the grid resolution. Unlike RDME, GRDME enables us to explicitly model various sizes of the molecules. Using this insight, we explore the diffusion-limited regime of reaction dynamics and discover that diffusion-controlled systems resemble small, discrete systems. Others have shown that a reaction system can have discreteness-induced state inversion, a phenomenon where the order of the concentrations differs when the system size is small. We show that the same reaction system also has diffusion-controlled state inversion, where the order of concentrations changes, when the diffusion is slow. In summary, we show that GRDME is a computationally efficient model, which enables us to include the information of the molecular sizes into the model.:1 Modeling Mesoscopic Biology 1.1 RDME Models Mesoscopic Stochastic Spatial Phenomena 1.2 A New Diffusion Model Presents an Opportunity For A More Efficient RDME 1.3 Can A New Diffusion Model Provide Insights Into The Loss Of Reactions? 1.4 Overview 2 Preliminaries 2.1 Reaction Diffusion Master Equation 2.1.1 Chemical Master Equation 2.1.2 Diffusion-controlled Bimolecular Reaction Rate 2.1.3 RDME is an Extention of CME to Spatial Problems 2.2 Next Subvolume Method 2.2.1 First Reaction Method 2.2.2 NSM is an Efficient Spatial Stochastic Algorithm for RDME 2.3 Discretization of the Laplace Operator Using Particle Strength Exchange 2.4 Summary 3 Gaussian Reaction Diffusion Master Equation 3.1 Design Constraints for the Diffusion Model in the RDME Framework 3.2 Gaussian-jump-based Model for RDME 3.3 Summary 4 Gaussian Next Subvolume Method 4.1 Constructing the neighborhood N 4.2 Finding the Diffusion Event 4.3 Comparing GNSM to NSM 4.4 Summary 5 Limits of Validity for (G)RDME with Macroscopic Bimolecular Propensity Rate 5.1 Previous Works 5.2 hmin Based on the Kuramoto length of a Grid Cell 5.3 hmin of the Two Limiting Regimes 5.4 hmin of Bimolecular Reactions for the Three Cases of Dimensionality 5.5 hmin of GRDME in Comparison to hmin of RDME 5.6 Summary 6 Numerical Experiments To Verify Accuracy, Efficiency and Validity of GRDME 6.1 Accuracy of the Diffusion Model 6.2 Computational Cost 6.3 hmin and Reaction Loss for (G)RDME With Macroscopic Bimolecular Propensity Rate kCK 6.3.1 Homobiomlecular Reaction With kCK at the Ballistic Limit 6.3.2 Homobiomlecular Reaction With kCK at the Diffusional Limit 6.3.3 Heterobiomlecular Reaction With kCK at the Ballistic Limit 6.4 Summary 7 (G)RDME as a Spatial Model of Collins-Kimball Diffusion-controlled Reaction Dynamics 7.1 Loss of Reactions in Diffusion-controlled Reaction Systems 7.2 The Loss of Reactions in (G)RDME Can Be Explained by Collins Kimball Theory 7.3 Cell Width h Sets the Reaction Radius σ∗ 7.4 Smoothing Length ε′ Sets the Size of the Molecules in the System 7.5 Heterobimolecular Reactions Can Only Be Modeled With GRDME 7.6 Zeroth Order Reactions Impose a Lower Limit on Diffusivity Dmin 7.6.1 Consistency of (G)RDME Could Be Improved by Redesigning Zeroth Order Reactions 7.7 Summary 8 Difussion-Controlled State Inversion 8.1 Diffusion-controlled Systems Resemble Small Systems 8.2 Slow Diffusion Leads to an Inversion of Steady States 8.3 Summary 9 Conclusion and Outlook 9.1 Two Physical Interpretations of (G)RDME 9.2 Advantages of GRDME 9.3 Towards Numerically Consistent (G)RDME 9.4 Exploring Mesoscopic Biology With GRDME Bibliography

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.