## Organisation

# NUMA, Numerical Analysis and Applied Mathematics Section

### Division

NUMERICAL APPROXIMATION AND NUMERICAL LINEAR ALGEBRA GROUP (A. Bultheel): * Fast and stable algorithms to solve many problems in mathematics connected to continued fractions, orthogonal functions, rational approximation & interpolation, etc. with applications in signal processing, linear systems, etc.. * Numerical solution of recurrence relations with continued fractions. * Fast and stable algorithms to solve structured linear systems of equations. * Understanding and developing robust numerical methods for the calculation of the (rightmost) eigenvalue(s) of large sparse matrices with applications in the study of stability of equilibrium solutions of dynamical systems. Wavelet based techniques in image processing: compression, denoising, etc.. * Analysis and applications of spline functions: smoothing of curves and surfaces with and without constraints. Modelling of surface with splines in CAGD. Development of a software package smoothing using splines. * Development of educational software for mathematics. NUMERICAL INTEGRATION, NONLINEAR EQUATIONS AND SOFTWARE (R.Cools) Construction of cubature formulas for approximating multivariate integrals using group theory, ideal theory & invariant theory. * Derivation of error expansions and lower bounds for the cost of a cubature formula. * Development of software packages for automatic numerical integration based on heuristic error estimators, adaptive subdivision strategies, etc.. * Root counting of systems of polynominal equations and systems of nonlinear analytic equations. * Understanding and developing homotopy continuation methods for computing all common zeros of systems of polynominal equations. * Development of a software package for solving systems of polynomial equations using homotopy continuation methods and for solving sytems of analytic equations using numerical integration and structural linear systems. SCIENTIFIC COMPUTING (D. Roose) The research group focuses on the development of numerical methods for solving large scale simulation problems in science & engineering. Efficiency, robustness and amenability to implementation on high performance (parallel) computers, are important aspects in the design of the algorithms. * Fast solvers for stationary and time-dependent partial differential equations: iterative methods for elliptic and hyperbolic problems, predonditionig for Krylov subspace methods, multigrid accelaration, domain decomposition; acceleration of waveform relaxation methods for tume-dependent parabolic problems; application and integration of these solvers in software for fluid dynamics. * Fast and robust methods for solving large-scale eigenvalue problems, and application in linear stability analysis * Numerical methods for nonlinear dynamical systems and for bifurcation analysis of nonlinear parameter-dependent problems. Emphasis on partial differential equations and delay differential equations. * Parallel computing aspects of numerical simulation: parallelisation of numerical software for fluid dynamics, hydro- dynamics, etc., algorithms and tools for load balancing and grid partioning for geometrically parallel applications on irregular adaptively refined grids. * Application of wavelet-based methods in image processing: image compression, noise reduction, edge detection; combination of wavelet-based techniques with Bayesian statistics; application to large images in Geographical Image Systems, and to video sequences.