@phdthesis{Gathungu2018,
  author    = {Gathungu, Duncan Kioi},
  title     = {On Multigrid and H-Matrix Methods for Partial Integro-Differential Equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-156430},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2018},
  abstract  = {The main theme of this thesis is the development of multigrid and hierarchical matrix solution procedures with almost linear computational complexity for classes of partial integro-differential problems. An elliptic partial integro-differential equation, a convection-diffusion partial integro-differential equation and a convection-diffusion partial integro-differential optimality system are investigated. In the first part of this work, an efficient multigrid finite-differences scheme for solving an elliptic Fredholm partial integro-differential equation (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization and a Simpson's quadrature rule to approximate the PIDE problem and a multigrid scheme and a fast multilevel integration method of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework that includes numerical experiments for elliptic PIDE problems with singular kernels. The experience gained in this part of the work is used for the investigation of convection diffusion partial-integro differential equations in the second part of this thesis. Convection-diffusion PIDE problems are discretized using a finite volume scheme referred to as the Chang and Cooper (CC) scheme and a quadrature rule. Also for this class of PIDE problems and this numerical setting, a stability and accuracy analysis of the CC scheme combined with a Simpson's quadrature rule is presented proving second-order accuracy of the numerical solution. To extend and investigate the proposed approximation and solution strategy to the case of systems of convection-diffusion PIDE, an optimal control problem governed by this model is considered. In this case the research focus is the CC-Simpson's discretization of the optimality system and its solution by the proposed multigrid strategy. Second-order accuracy of the optimization solution is proved and results of local Fourier analysis are presented that provide sharp convergence estimates of the optimal computational complexity of the multigrid-fast integration technique. While (geometric) multigrid techniques require ad-hoc implementation depending on the structure of the PIDE problem and on the dimensionality of the domain where the problem is considered, the hierarchical matrix framework allows a more general treatment that exploits the algebraic structure of the problem at hand. In this thesis, this framework is extended to the case of combined differential and integral problems considering the case of a convection-diffusion PIDE. In this case, the starting point is the CC discretization of the convection-diffusion operator combined with the trapezoidal quadrature rule. The hierarchical matrix approach exploits the algebraic nature of the hierarchical matrices for blockwise approximations by low-rank matrices of the sparse convection-diffusion approximation and enables data sparse representation of the fully populated matrix where all essential matrix operations are performed with at most logarithmic optimal complexity. The factorization of part of or the whole coefficient matrix is used as a preconditioner to the solution of the PIDE problem using a generalized minimum residual (GMRes) procedure as a solver. Numerical analysis estimates of the accuracy of the finite-volume and trapezoidal rule approximation are presented and combined with estimates of the hierarchical matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical estimates and the optimal computational complexity of the proposed hierarchical matrix solution procedure. These results include an extension to higher dimensions and an application to the time evolution of the probability density function of a jump diffusion process.},
  subject      = {Mehrgitterverfahren},
  language  = {en}
}
@phdthesis{Gaviraghi2017,
  author    = {Gaviraghi, Beatrice},
  title     = {Theoretical and numerical analysis of Fokker-Planck optimal control problems for jump-diffusion processes},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145645},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2017},
  abstract  = {The topic of this thesis is the theoretical and numerical analysis of optimal control problems, whose differential constraints are given by Fokker-Planck models related to jump-diffusion processes. We tackle the issue of controlling a stochastic process by formulating a deterministic optimization problem. The key idea of our approach is to focus on the probability density function of the process, whose time evolution is modeled by the Fokker-Planck equation. Our control framework is advantageous since it allows to model the action of the control over the entire range of the process, whose statistics are characterized by the shape of its probability density function. We first investigate jump-diffusion processes, illustrating their main properties. We define stochastic initial-value problems and present results on the existence and uniqueness of their solutions. We then discuss how numerical solutions of stochastic problems are computed, focusing on the Euler-Maruyama method. We put our attention to jump-diffusion models with time- and space-dependent coefficients and jumps given by a compound Poisson process. We derive the related Fokker-Planck equations, which take the form of partial integro-differential equations. Their differential term is governed by a parabolic operator, while the nonlocal integral operator is due to the presence of the jumps. The derivation is carried out in two cases. On the one hand, we consider a process with unbounded range. On the other hand, we confine the dynamic of the sample paths to a bounded domain, and thus the behavior of the process in proximity of the boundaries has to be specified. Throughout this thesis, we set the barriers of the domain to be reflecting. The Fokker-Planck equation, endowed with initial and boundary conditions, gives rise to Fokker-Planck problems. Their solvability is discussed in suitable functional spaces. The properties of their solutions are examined, namely their regularity, positivity and probability mass conservation. Since closed-form solutions to Fokker-Planck problems are usually not available, one has to resort to numerical methods. The first main achievement of this thesis is the definition and analysis of conservative and positive-preserving numerical methods for Fokker-Planck problems. Our SIMEX1 and SIMEX2 (Splitting-Implicit-Explicit) schemes are defined within the framework given by the method of lines. The differential operator is discretized by a finite volume scheme given by the Chang-Cooper method, while the integral operator is approximated by a mid-point rule. This leads to a large system of ordinary differential equations, that we approximate with the Strang-Marchuk splitting method. This technique decomposes the original problem in a sequence of different subproblems with simpler structure, which are separately solved and linked to each other through initial conditions and final solutions. After performing the splitting step, we carry out the time integration with first- and second-order time-differencing methods. These steps give rise to the SIMEX1 and SIMEX2 methods, respectively. A full convergence and stability analysis of our schemes is included. Moreover, we are able to prove that the positivity and the mass conservation of the solution to Fokker-Planck problems are satisfied at the discrete level by the numerical solutions computed with the SIMEX schemes. The second main achievement of this thesis is the theoretical analysis and the numerical solution of optimal control problems governed by Fokker-Planck models. The field of optimal control deals with finding control functions in such a way that given cost functionals are minimized. Our framework aims at the minimization of the difference between a known sequence of values and the first moment of a jump-diffusion process; therefore, this formulation can also be considered as a parameter estimation problem for stochastic processes. Two cases are discussed, in which the form of the cost functional is continuous-in-time and discrete-in-time, respectively. The control variable enters the state equation as a coefficient of the Fokker-Planck partial integro-differential operator. We also include in the cost functional a \$L^1\$-penalization term, which enhances the sparsity of the solution. Therefore, the resulting optimization problem is nonconvex and nonsmooth. We derive the first-order optimality systems satisfied by the optimal solution. The computation of the optimal solution is carried out by means of proximal iterative schemes in an infinite-dimensional framework.},
  subject      = {Fokker-Planck-Gleichung},
  language  = {en}
}