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Contributions to Credit Portfolio Modeling and Optimization
The devastating impacts of the recent global financial crisis underscore the need for both financial institutions and banking supervision to develop more appropriate credit risk models to ensure the stability of the financial system. This work contributes to quantitative credit portfolio risk modeling in three ways. First, it introduces a general credit portfolio modeling concept that comprises specific credit risk management models as special cases. Second, analytical techniques are presented for specifying asset correlations in a credit portfolio through systematic factors. Finally, a new approach for clustering of obligors in a credit portfolio is proposed using threshold accepting, a stochastic optimization technique. In particular, a computationally tractable technique to validate ex-post the precision of the clustering system is suggested and applied to a real world retail credit portfolio. The contributions of this book should provide benefit to practitioners, academics and graduate students in the field of financial risk management.
Low Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients
Abstract: We study the solution of linear systems resulting from the discreitization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure and, thus, solving them can be both time- and computer memory-consuming. Under certain assumptions, we show that the solution of such linear systems can be approximated with a vector of low tensor rank. We then solve the linear systems using low rank preconditioned iterative solvers. Numerical experiments demonstrate that these low rank preconditioned solvers are effective.
Block-diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs
Abstract: This paper is aimed at the efficient numerical simulation of optimization problems governed by either steady-state or unsteady partial differential equations involving random coefficients. This class of problems often leads to prohibitively high dimensional saddle point systems with tensor product structure, especially when discretized with the stochastic Galerkin finite element method. Here, we derive and analyze robust Schur complement-based block-diagonal preconditioners for solving the resulting stochastic optimality systems with all-at-once low-rank solvers. Moreover, we illustrate the effectiveness of our solvers with numerical experiments.
Low-rank Solvers for Unsteady Stokes-Brinkman Optimal Control Problem with Random Data
Abstract: We consider the numerical simulation of an optimal control problem constrained by the unsteady Stokes-Brinkman equation involving random data. More precisely, we treat the state, the control, the target (or the desired state), as well as the the viscosity, as analytic functions depending on uncertain parameters. This allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic Galerkin finite element method discretization of the model. The discrete problem yields a prohibitively high dimensional saddle point system with Kronecker product structure. We develop a new alternating iterative tensor method for an efficient reduction of this system by the low-rank Tensor Train representation. Besides, we propose and analyze a robust Schur complement-based preconditioner for the solution of the saddle-point system. The performance of our approach is illustrated with extensive numerical experiments based on two- and three-dimensional examples. The developed Tensor Train scheme reduces the solution storage by two orders of magnitude.
Numerical Control: Part A
Numerical Control: Part A, Volume 23 in the Handbook of Numerical Analysis series, highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this volume include Numerics for finite-dimensional control systems, Moments and convex optimization for analysis and control of nonlinear PDEs, The turnpike property in optimal control, Structure-Preserving Numerical Schemes for Hamiltonian Dynamics, Optimal Control of PDEs and FE-Approximation, Filtration techniques for the uniform controllability of semi-discrete hyperbolic equations, Numerical controllability properties of fractional partial differential equations, Optimal Control, Numerics, and Applications of Fractional PDEs, and much more. Provides the authority and expertise of leading contributors from an international board of authors Presents the latest release in the Handbook of Numerical Analysis series Updated release includes the latest information on Numerical Control
Matrix Methods
Operators preserving primitivity for matrix pairs / L.B. Beasley, A.E. Guterman -- Decompositions of quaternions and their matrix equivalents / D. Janovská, G. Opfer -- Sensitivity analysis of Hamiltonian and reversible systems prone to dissipation-induced instabilities / O.N. Kirillov -- Block triangular miniversal deformations of matrices and matrix pencils / L. Klimenko, V.V. Sergeichuk -- Determining the Schein rank of boolean matrices / E.E. Marenich -- Lattices of matrix rows and matrix columns. Lattices of invariant column eigenvectors / V. Marenich -- Matrix algebras and their length / O.V. Markova -- On a new class of singular nonsymmetric matrices with nonnegative integer spectra ...
Introduction to Derivative-Free Optimization
The first contemporary comprehensive treatment of optimization without derivatives. This text explains how sampling and model techniques are used in derivative-free methods and how they are designed to solve optimization problems. It is designed to be readily accessible to both researchers and those with a modest background in computational mathematics.
