This paper is concerned with the investigation of the nonlinear eigenvalue problem describing the natural oscillations of a plate with a load that elastically attached to it. We study properties of eigenvalues and eigenfunctions of this eigenvalue problem and prove the existence theorem for eigensolutions. Theoretical results are illustrated by numerical experiments.

When the eigenvalues of a given eigenvalue problem are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure.

This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

We are interested in a nonlinear filtering problem motivated by an information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. We solve this problem by `change of measure method\\\'' and show the existence of the density of the unnormalized conditional distribution which is a solution to the Zakai equation. Zakai equation is a linear SPDE which, in general, cannot be solved analytically. We apply Galerkin method to solve it numerically and show the convergence of Galerkin approximation in mean square. Lastly, we design an adaptive Galerkin filter with a basis of Hermite polynomials and we present numerical examples to illustrate the effectiveness of the proposed method. The work is closely related to the paper Frey and Schmidt (2010).

NJ, Univ., Dep. of Economics, Diss.--Princeton, 2004. / Kopie, ersch. im Verl. UMI, Ann Arbor, Mich. - Enth. 3 Beitr.

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter. To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors and inner products of vectors. We investigate the convergence and derive grid-independent error estimates of these methods for computing eigenvalues. Numerical experiments demonstrate practical effectiveness of the proposed methods for a class of mechanical problems.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

In this document we review the status of existing techniques for nonlinear model order reduction by investigating how well these techniques perform for typical industrial needs. In particular the TPWL-method (Trajectory Piecewise Linear-method) and the POD-approach (Proper Orthogonal Decomposion) is taken under consideration. We address several questions that are (closely) related to both the theory and application of nonlinear model order reduction techniques. The goal of this document is to provide an overview of available methods together with a classification of nonlinear problems that in principle could be handled by these methods.

In this thesis we present some results on cross-diffusion and nonlocal interaction. In the first part we study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero. Finally we briefly discuss coarsening dynamics in the system, which can be observed in numerical results and is motivated by rewriting the PDEs as a system of nonlocal Cahn-Hilliard equations. Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and very few results exist in this direction. In the second part of this thesis, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in [60]. Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of nonnegative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak-strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions. In the third part we focus on a class of integral functionals known as nonlocal perimeters. Intuitively, these functionals express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K which might be singular. We show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau’s problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. Furthermore, a Γ-convergence result is discussed. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel which might be singular in the origin but that have faster-than-L 1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we give explicitly.

We consider the solution of nonlinear optimal control problems subject to stochastic perturbations with incomplete observations. In particular, we generalize results obtained by Ito and Kunisch in [8] where they consider a receding horizon control (RHC) technique based on linearizing the problem on small intervals. The linear-quadratic optimal control problem for the resulting time-invariant (LTI) problem is then solved using the linear quadratic Gaussian (LQG) design. Here, we allow linearization about an instationary reference trajectory and thus obtain a linear time-varying (LTV) problem on each time horizon. Additionally, we apply a model predictive control (MPC) scheme which can be seen as a generalization of RHC and we allow covariance matrices of the noise processes not equal to the identity. We illustrate the MPC/LQG approach for a three dimensional reaction-diffusion system. In particular, we discuss the benefits of time-varying linearizations over time-invariant ones.