Preconditioners for Krylov subspace methods: An overview

When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind

Null-space preconditioners for saddle point systems

The null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive-definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization, and compare their performance with the equivalent Schur-complement based preconditioners. We also describe how to apply the non-symmetric preconditioners proposed using the conjugate gradient method (CG) with a non-standard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications

A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part I

We propose a class of preconditioners, which are also tailored for symmetric linear systems from linear algebra and nonconvex optimization. Our preconditioners are specifically suited for large linear systems and may be obtained as by-product of Krylov subspace solvers. Each preconditioner in our class is identified by setting the values of a pair of parameters and a scaling matrix, which are user-dependent, and may be chosen according with the structure of the problem in hand. We provide theoretical properties for our preconditioners. In particular, we show that our preconditioners both shift some eigenvalues of the system matrix to controlled values, and they tend to reduce the modulus of most of the other eigenvalues. In a companion paper we study some structural properties of our class of preconditioners, and report the results on a significant numerical experience.preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods

Application of domain decomposition methods to problems in topology optimisation

Determination of the optimal layout of structures can be seen in everyday life, from nature to industry, with research dating back to the eighteenth century. The focus of this thesis involves investigation into the relatively modern field of topology optimisation, where the aim is to determine both the optimal shape and topology of structures. However, the inherent large-scale nature means that even problems defined using a relatively coarse finite element discretisation can be computationally demanding. This thesis aims to describe alternative approaches allowing for the practical use of topology optimisation on a large scale. Commonly used solution methods will be compared and scrutinised, with observations used in the application of a novel substructuring domain decomposition method for the subsequent large-scale linear systems. Numerical and analytical investigations involving the governing equations of linear elasticity will lead to the development of three different algorithms for compliance minimisation problems in topology optimisation. Each algorithm will involve an appropriate preconditioning strategy incorporating a matrix representation of a discrete interpolation norm, with numerical results indicating mesh independent performance

Numerical Optimisation Problems in Finance

This thesis consists of four projects regarding numerical optimisation and financial derivative pricing. The first project deals with the calibration of the Heston stochastic volatility model. A method using the Levenberg-Marquardt algorithm with the analytical gradient is developed. It is so far the fastest Heston model calibrator and meets the speed requirement of practical trading. In the second project, a triply-nested iterative method for the implementation of interior-point methods for linear programs is proposed. It is the first time that an interior-point method entirely based on iterative solvers succeeds in solving a fairly large number of linear programming instances from benchmark libraries under the standard stopping criteria. The third project extends the Black-Scholes valuation to a complex volatility parameter and presents its singularities at zero and infinity. Fractals that describe the chaotic nature of the Newton-Raphson calculation of the implied volatility are shown for different moneyness values. Among other things, these fractals visualise dramatically the effect of an existing modification for improving the stability and convergence of the search. The project studies scientifically an interesting problem widespread in the financial industry, while revealing artistic values stemming from mathematics. The fourth project investigates the consistency of a class of stochastic volatility models under spot rate inversion, and hence their suitability in the foreign exchange market. The general formula of the model parameters for the inversion rate is given, which provides basis for further investigation. The result is further extended to the affine stochastic volatility model. The Heston model, among the other members in the stochastic volatility family, is the only one that we found to be consistent under the spot inversion. The conclusion on the Heston model verifies the arbitrage opportunity in the variance swap

Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting $\rm L^1$ term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method we introduce fast and efficient preconditioners which enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically