# NA Group Research

Research Areas:

- Complexity in Optimization
- Computational Non-Newtonian Fluid Dynamics
- Finance
- Fluid Dynamics
- Medicine and Biology
- Numerical Linear Algebra
- Numerical Optimization
- Numerical Solution of Partial Differential Equations
- SVD of Distributed Data
- Symmetric Cone Programming

#### Complexity in Optimization

The complexity theory of numerical optimization problems deals with the question of determining the computational cost for solving an optimization problem as a function of its input size. Different complexity models can be formulated, depending on the type of conceptual computer one assumes available (what type of numbers can it operate on, and what are the operations?) and on whether one is interested in the worst-case or average-case behaviour (in the latter case, an appropriate family of distributions has to be chosen for the random input data). Our research focuses on two aspects of this theory: Investigation of the most general class of convex optimization problems that are tractable, and investigation of the most general family of input distributions for which an average-case or smoothed analysis is possible.

#### Finance

This application area involves collaborations with other Oxford academics in the Mathematical Finance Group, and financial institutions such as Royal Bank of Scotland.

#### Fluid Dynamics

In fluid dynamics we cover a range of applications from incompressible flow (IJS, LNT, AJW) to compressible flow (MBG), acoustics (MBG) and multiphase flow (IJS, AJW), working with companies such as Rolls-Royce and Schlumberger.

#### Medicine and Biology

Medical and biological modelling is another growth area, involving collabrations with the Computational Biology Group and the Centre for Mathematical Biology.

#### Numerical Linear Algebra

Fundamental aspects of linear algebra, and aspects relating to applications, particularly in partial differential equations are both key concerns. Eigenvalues, pseudospectra and stability are fundamental topics with relevance to problems as apparently diverse as card shuffling and the stability/instability of fluid flows. Preconditioning is an important practical research subject which is required in most large-scale scientific computing.

#### Numerical Optimization

Numerical optimization (or mathematical programming) deals with the problem of minimizing or maximizing functions of finitely or infinitely many variables over domains that are implicitly defined via constraints (systems of equations and inequalities). We are interested in all aspects of this area: Optimality conditions and duality theory, the design of and analysis of algorithms, complexity, robustness, and the discovery of new applications via mathematical modelling.

#### Numerical Solution of Partial Differential Equations

This is a common theme running through much of the research within the Numerical Analysis Group. We use a wide variety of methods, including finite element methods, finite volume methods and spectral methods. Some of the work concerns the development of new numerical methods (for example for strongly anisotropic applications) and some the analysis of existing methods leading to a posteriori error analysis and grid adaptation.

#### Singular Value Decomposition of Distributed Data

The computation of the leading part of the singular value decomposition (SVD) of a matrix is an important problem in numerical linear algebra. In applications SVDs appear in dimension-reduction reduction techniques (principal component analysis (PCA), etc.), and the matrices are often very large, and in some applications also dense and distributed over a network of machines. We are interested in computational approaches that deal specifically with the distributed nature of the available computational resources, which may run at very different speeds and may be prone to failure, making it necessary to develop methods that don't rely on synchronous communication between the computational nodes.

#### Symmetric Cone Programming

Conic programming deals with an important class of tractable convex optimization problems that include linear programming, convex quadratic programming, and semidefinite programming. All these problems are treated in a unified theoretical framework for the design of algorithms, the analysis of their complexity, duality theory, perturbation analysis etc.