Computer simulation is now pervasive throughout science and engineering. Discretision of the underlying model results in discrepancy between the true solution and the computer's predictions. The big question is how much can you trust this data? We have developed rigorous bounds to quantify and control the accuracy of simulations of a diverse range of applications from electromagnetics working with BAe Systems, to the certification of pipelines in the North Sea Oil industry.

Broadly speaking, my research interests are the development of efficient and practical computational algorithms for the adaptive solution of systems of partial differential equations (PDEs), their rigorous mathematical analysis and their application to relevant physical problems. My interest in this area is rooted in the firm belief that computational solution of partial differential equations will continue to play an increasingly important role in science and engineering, particularly in inter-disciplinary projects, where the relevant problems and applications are generally highly complex and rule out analytical treatment. Ultimately, the value of numerical simulations lies in the accuracy and reliability of the computed output data, and the practicality of the algorithm, which can only be established through rigorous mathematical analysis.

My research is motivated by the desire to address the issues of reliable and efficient practical computational resolution of relevant problems. As such, I seek to deal with all aspects of the computational approach ranging from: development of new methods and mathematical analysis of their properties; efficient numerical solution of algebraic equations and preconditioning procedures; quantitative error assessment and control; adaptive solution procedures and the development of hierarchic and multi-scale models. More recently, I have been interested in issues related to development of algorithms on high performance computing architectures through the NAIS Centre, of which I was Director, supported by the award of £4.5M for Numerical Algorithms and Intelligent Software for the Evolving HPC Platform.

In collaboration with Professor K. Morgan (University of Wales, Swansea) and British Aerospace plc, under the contract Computational Simulation of Electromagnetic Scattering (PI Ainsworth, supported by EPSRC under contract GR/M59426) effective algorithms for the numerical approximation of Maxwell's equations were developed. This project resulted in a number of novel results concerning finite element approximation using the function space H(curl) which is the natural setting for Maxwell's equations. A novel, systematic approach to the construction of finite element spaces in three dimensions was presented in joint work [15] with a post-doctoral research assistant, Joe Coyle (now Associate Professor at Monmouth University, NJ), while a full theoretical analysis of the conditioning was studied in [14]. Rigorous approximation theoretic properties of the spaces in two dimensions are derived in [25] where the possibility of obtaining exponential rates of convergence is proved, while practical examples of this and the important issue of dispersive properties of the finite elements are discussed in [13]. This work has been the subject of a plenary lectures presented at the IMACS conference on Adaptive Methods for PDEs held at the Fields Institute for Research in Mathematical Sciences, Toronto (2002)and Second International Conference on Scientific Computing and PDEs and First East Asia SIAM Symposium, held in Hong Kong (2005).

This contract stimulated my interest in the study of dispersive behavior of numerical methods for computational wave propagation, where a number of ground-breaking results were obtained in [4, 5] and developed further in a series of papers[33, 34, 6] with a graduate student, Hafiz Wajid (now an Associate Professor at COMSATS Institute of Technology, Pakistan). I was delighted to be invited to present a plenary lecture on this topic at the main conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2003) held in Jyväskylä, Finland and at International Conference on High Order and Spectral Methods (ICOSAHOM) held at Brown in Providence (2004). Numerical wave propagation was the subject of a two week Durham LMS Symposium in July 2002, co-organized by Ainsworth, supported by London Mathematical Society and EPSRC.

The development of robust algorithms for the approximation of problems with small parameters was the subject of Adaptive Methods for Singularly Perturbed Elliptic Systems (PI Ainsworth, supported by EPSRC under contract GR/L90507). In collaboration with a graduate student, Patrick Coggins (PhD, 2000), the stability of high order finite element methods for Stokes' and Navier Stokes equations was studied [11, 12], with particular emphasis on the use of high aspect ratio elements needed to capture the boundary layers typically found in singularly perturbed problems. This work was the subject of invited lectures presented at the Dundee Biennial Conference on Numerical Analysis in 1999 and Third International Workshop on Analysis and Numerical Approximation of Singular Problems IWANASP2008 held in Lisbon (2008). Subsequently, in collaboration with a post-doctoral research assistant, Katia Pinchedez [25, 26], robust high order methods for numerical approximation of the Reissner-Mindlin plate model were developed and analysed. The issue of the accuracy of dimensionally reduced models was studied in collaboration with a graduate student, Mark Arnold (PhD, 1999), where quantitative bounds on the accuracy of the actual PDE model were derived in [3, 10], while sequences of optimal, hierarchic models were derived and analysed in [9]. This formed the subject of invited lectures at the conference on Reliability of Modelling and Computation in Applied Mechanics held in Hannover (1996) and on Adaptive Finite Element Methods in Computational Mechanics held in Stuttgart (1997). Research in this direction was later supported by a Leverhulme Research Fellowship to study Robust Algorithms for Computational PDEs.

In collaboration with Professor D.W. Kelly (Manufacturing Engineering at UNSW, Sydney), I was associate investigator on the contract Post-processing and a posteriori error bounds for finite element analysis (supported by award of A$311,914 from the Australian Research Council), whereby techniques were developed for computing reliable error bounds for quantities of interest arising the solution of problems in structural analysis modelled using the Lamé-Navier equations of linear elasticity [18, 19].

Problems of reliability of computational simulation and the derivation of quantitative bounds on the accuracy of discrete approximations of PDEs form the subject of a posteriori error estimation. A long standing collaboration with Professor J.T. Oden (University of Texas at Austin) culminated in the publication of a monograph, A Posteriori Error Estimation in Finite Element Analysis published by John Wiley & Sons (2000), which has been cited quite widely. My research in the area of reliability and a posteriori error estimation has been recognised by regular invitations to present plenary lectures at several international meetings over the past decade. A posteriori error estimation was the subject of a special programme held at the Mathematical Sciences Research Institute at Berkeley from 6th March-28th April 2000, organised by Ainsworth, along with Professors Babuˇska (UT Austin), Bank (UCSD) and Verf¨urth (Bochum), on the topic of A Posteriori Error Estimation and Adaptive Approaches in the Finite Element Method supported by MSRI and NSF.

The development of efficient procedures for the solution of algebraic systems is a vital component of any competitive numerical simulation. My research in this area has mainly been concerned with the development and analysis of preconditioning procedures. The systems arising from an adaptive high order FEM and BEM can present problems since the matrices are not generally sparse and are highly ill-conditioned. Ainsworth [1, 2] produced the first domain decomposition technique for adaptive hp-FEM,along with a rigorous mathematical analysis proving that the algorithms reduce the condition number to logarithmic growth in the mesh parameters. Later work demonstrated that the method is easily implemented and remains effective even on the very highly non-uniform h-p meshes that arise in practical computations [32]. This work was the subject of an invited talk presented at the prestigious international conference Domain Decomposition XI (1998). I was honoured to be invited to speak again in this conference series at Domain Decomposition XVII held in St. Wolfgang, Austria (2006).

In collaboration with Professor Benqi Guo (University of Manitoba), supported by EPSRC Visiting Fellowship GR/M09155, we developed a rigorous analysis of a practical domain decomposition techniques for p [16] and h-version [17] boundary element approximation in 3D. The problem of conditioning and preconditioning of boundary element equations on adaptively refined meshes was studied in collaboration with W. McLean (UNSW, Sydney, Australia) supported by Australian Research Council [21, 22], where a complete and very general analysis of the conditioning on non-quasi-uniform meshes was presented, along with a rigorous proof of the fact that diagonal scaling removes the effect of the mesh non-uniformity. This collaboration was later supported by a EPSRC Visiting Fellowship GR/N21970, where we developed fully robust multilevel preconditioners for boundary element
approximation on non-quasi-uniform meshes [20] and presented a rigorous proof based on the new concept of a cumulative smoothing property.

Professors Elliott (Warwick), S¨uli (Oxford) and I organised a programme on Computational Challenges in Partial Differential Equations held in Cambridge from January-July 2003, at the Isaac Newton Institute for Mathematical Sciences. One of the aims of this programme was to identify new and relevant areas in science and engineering where computational mathematics has the potential to have a significant impact. The programme was the subject of an article which appeared in SIAM News, May 2003. We organised a follow-up WIMCS/INI Workshop on Computational Challenges in PDEs held at Swansea University, UK from 4-8th April 2011.

A new research front was opened by the research contract Multi-scale Modelling and Numerical Analysis of Masonry (EPSRC GR/ S10153, PI: Ainsworth) in collaboration with Professor Bi´cani´c (Regius Professor of Civil Engineering at the University of Glasgow) whereby techniques were developed for the reliable simulation of multi-body contact of bricks and stone. This information is needed by engineers seeking to assess the structural integrity of old and historic buildings where cracks and large deformations are present in the masonry. This problem is of inherent interest in its own right, but I also view this as a relevant and practical example of the challenging problems arising in multi-scale modelling in general, where there is currently a significant move towards the analysis and development of computational models at the atomistic level. The full model will never be tractable using a direct numerical approach. However, in a sequence of joint papers [24, 23, 35, 36] with my post-doctoral research assistant, Angela Mihai (now on the faculty of Cardiff University, UK), an adaptive multi-scale modeling approach was developed which allowed us to simulate the dynamic response of real-life masonry structures, such as Clare College Bridge [35], using very modest computational resources. This work was the topic of an invited talk presented at INdAM Workshop on Multiscale Problems held in Cortona, Italy (2006). My work in this area was partially supported by a Scottish Executive Research Fellowship obtained competitively from the Royal Society of Edinburgh.

A collaborative project with Hewlett-Packard Laboratories in Bristol, Adaptive Numerical Methods for Optoelectronic Devices (EPSRC EP/E040993/1, PI Ainsworth), along with additional financial support from Hewlett-Packard, provided the opportunity to further develop and synthesize much of my earlier work into a C++ code in the context of tackling the problem of simulating liquid crystal displays. My post-doctoral research assistant Xinhui Ma (now on the faculty of Abertay University, UK) and I used template meta-programming techniques to develop a flexible, but highly efficient object oriented C++ code that forms the basis of a code now being used by Hewlett-Packard.

In a similar vein, the collaborative project Assessing and Quantifying the Accuracy of FEA of Thin-Walled Structures (PI Ainsworth) with the industrial partner, Ideas Ltd, led to the development of fully computable error bounds [30, 31] for linear elastic problems in both 2D and 3D, that are now being incorporated by Ideas Ltd in their commercial code which is widely used in the North Sea Oil industry. The basis for this work was begun in [8, 7] and developed further in a sequence of papers on fully computable error bounds [29, 28, 27] in collaboration with a post-doctoral research assistant, and former research student, Richard Rankin (now has a position at Rice University).

References

[1] M. Ainsworth, Hierarchical domain decomposition preconditioner for h-p finite element approximation on locally refined meshes, SIAM J. Sci. Comp., 17 (1996), pp. 1395–1413.

[2] A preconditioner based on domain decomposition for h-p finite element approximation on quasi-uniform meshes, SIAM J. Numer. Anal., 33 (1996), pp. 1358–1376.

[3] A posteriori error estimators for fully discrete hierarchical modelling on thin domains., Numer. Math., 80 (1998), pp. 325–362.

[4] Discrete dispersion relation for hp-version finite element approximation at high wave number, SIAM J. Numer. Anal., 42 (2004), pp. 553–575.

[5] Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, J. Comput. Phys., 198 (2004), pp. 106–130.

[6] Dispersive properties of high order Nédélec/edge element approximation of the time-harmonic Maxwell equations, Phil. Trans. Roy. Soc. Series A, 362 (2004), pp. 471–491.

[7] Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., 42 (2005), pp. 2320–2341 (electronic).

[8] A posteriori error estimation for a discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 45 (2007), pp. 1777–1798.

[9] M. Ainsworth and M. Arnold, Construction and analysis of optimal hierarchic models of boundary value problems on thin circular and spherical geometries, SIAM J. Sci. Comp., 22 (2000), pp. 673–703.

[10] Computable error bounds for some simple dimensionally reduced models on thin domains, IMA J. Numer. Anal., 21 (2001), pp. 81–105.

[11] M. Ainsworth and P. Coggins, The stability of mixed hp-finite element methods for Stokes flow on high aspect ratio elements, SIAM J. Numer. Anal., 38 (2000), pp. 1721–1761.

[12] A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow, IMA J. Numer. Anal., 22 (2002), pp. 307–327.

[13] M. Ainsworth and J. Coyle, Hierarchic hp-edge element families for Maxwell's equations on hybrid quadrilateral/triangular meshes, Comput. Methods Appl. Mech.Engrg., 190 (2001), pp. 6709–6733.

[14] Conditioning of hierarchic p-version Nedelec element on meshes of curvilinear quadrilaterals and hexahedra, SIAM J. Numer. Anal., 41 (2003), pp. 731–750.

[15] Hierarchic finite element bases on unstructured tetrahedral meshes, Internat. J. Numer. Methods Engrg., 58 (2003), pp. 2103–2130.

[16] M. Ainsworth and B. Guo, An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions, Numer. Math., 85 (2000), pp. 343–366.

[17] , Analysis of iterative sub-structuring techniques for boundary element approximation of the hypersingular operator in three dimensions, Applicable Anal., 81 (2002), pp. 241–280.

[18] M. Ainsworth and D. Kelly, A posteriori error estimators and adaptivity for finite element approximation of the non-homogeneous dirichlet problem, Adv. Comp. Math., 15 (2001), pp. 3–23.

[19] M. Ainsworth, D. Kelly, I. Sloan, and S.Wang, Post processing with computable error bounds for finite element approximation of a non-linear heat conduction problem, IMA J. Numer. Anal., 17 (1997), pp. 547–561.

[20] M. Ainsworth and W. McLean, Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes, Numer. Math., 93 (2003), pp. 387–413.

[21] M. Ainsworth, W. McLean, and T. Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling, SIAM J. Numer. Anal., 36 (1999), pp. 1901–1932.

[22] Diagonal scaling of stiffness matrices in the galerkin boundary element method, ANZIAM J., 42 (2000), pp. 141–150.

[23] M. Ainsworth and L. A. Mihai, Modelling and numerical analysis of masonry structures, Numer. Meth. PDE., 23 (2007), pp. 798–816.

[24] An adaptive multi-scale approach to the modelling of masonry structures, Internat. J. Numer. Methods Engrg., 78 (2009), pp. 1135–1163.

[25] M. Ainsworth and K. Pinchedez, hp-approximation theory for BDFM and RT finite elements on quadrilaterals, SIAM J. Numer. Anal., 40 (2002), pp. 2047–2068 (electronic) (2003).

[26] The hp-MITC finite element method for the Reissner-Mindlin plate problem, J. Comput. Appl. Math., 148 (2002), pp. 429–462.

[27] M. Ainsworth and R. Rankin, Fully computable bounds for the error in non-conforming finite element approximations of arbitrary order on triangular elements, SIAM J. Numer. Anal., 46 (2008), pp. 3207–3232.

[28] Robust a posteriori error estimation for the nonconforming Fortin-Soulie finite element approximation, Math. Comp., 77 (2008), pp. 1917–1939.

[29] Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes, SIAM J. Numer. Anal., 47 (2010), pp. 4112–4141.

[30] Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity, Internat. J. Numer. Methods Engrg., 82 (2010), pp. 1114–1157.

[31] M. Ainsworth and R. Rankin, Realistic computable error bounds for three dimensional finite element analyses in linear elasticity, Comput. Methods Appl. Mech. Engrg., (Accepted for publication).

[32] M. Ainsworth and B. Senior, Aspects of an adaptive hp-finite element method: Adaptive strategy, conforming approximation and efficient solvers, Comput. Methods Appl. Mech. Engrg., 150 (1997), pp. 65–87.

[33] M. Ainsworth and H. A. Wajid, Dispersive and dissipative behavior of the spectral element method, SIAM J. Numer. Anal., 47 (2009), pp. 3910–3937.

[34] Optimally blended spectral-finite element scheme for wave propagation and nonstandard reduced integration, SIAM J. Numer. Anal., 48 (2010), pp. 346–371.

[35] L. A. Mihai and M. Ainsworth, An adaptive multi-scale computational modeling of Clare College Bridge, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1839–1847.

[36] A finite element procedure for rigorous numerical enclosures on the limit load in the analysis of multibody structures, Comput. Methods Appl. Mech. Engrg., 199 (2009), pp. 48–60.

2015 ARO MURI: Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications
(Co-PI). October 2015-October 2020.

2015 ASCR SIRIUS: Science Driven Data Management for Multi-tiered Storage (Co-PI).

2013- UT Batelle Joint Faculty Appointment ORNL (PI). 

2012 AFOSR Methods for High-order Multi-Scale and Stochastic Problems: Analysis, Algorithms, and Applications (Co-Investigator).  7/15/12-7/14/16.

2011 EPSRC EP/J007242/1 Optimal Design of Drug Eluting Stents (Co-PI).

2009 EPSRC EP/G036136/1 Numerical Algorithms and Intelligent Software for the Evolving HPC Platform
(Principal Investigator at Strathclyde). £4,550,814 plus additional contribution from Scottish
Funding Council. This is a collaborative award with Universities of Edinburgh, Heriot-Watt, and
Edinburgh Parallel Computer Centre (UK’s national supercomputer HeCTOR).

2008 EPSRC/Strathclyde RAIS Award Assessing and Quantifying the Accuracy of FEA of Thin-Walled Structures (Principal Investigator). Plus contribution from industrial partner, Ideas Ltd, Glasgow.

2007 EPSRC EP/E040993/1 Adaptive Numerical Methods for Optoelectronic Devices. (Principal Investigator). 

2007 Hewlett-Packard Laboratories, UK. Adaptive Numerical Methods for Optoelectronic Devices. (Principal
Investigator).

2005 Royal Society of Edinburgh/Scottish Executive Research Fellowship. (Principal Investigator). Provides
funds for appointment of lecturer to release recipient from all academic duties for one year.

2003 EPSRC GR/S10153/01 Multiscale Modelling and Numerical Analysis of Masonry. (Principal Investigator).

2003 Leverhulme Trust Fellowship Robust Algorithms in Computational PDEs. (Principal Investigator).
 

2003 EPSRC GR/S01276/01 Computational Challenges in Partial Differential Equations. (Principal Investigator). 

2002 EPSRC GR/R40173/01 Computational Methods for Wave Propagation in Direct Scattering: An LMS
Durham Research Symposium. (Co-investigator). 

2000 EPSRC Visiting Fellowship GR/N21970 - Dr W. McLean Multilevel preconditioners for boundary element methods on adaptively refined meshes. (Principal Investigator)

1999 EPSRC Visiting Fellowship GR/M55961 - Dr W. Dörfler Numerical analysis of convection diffusion
problems. (Principal Investigator). 

1999 EPSRC GR/M59426 Computational Simulation of Electromagnetic Scattering. (Principal Investigator).
£78,000 plus additional funds to Prof. K. Morgan (Swansea University).

1998 EPSRC Visiting Fellowship GR/M09155 - Professor Benqi Guo Domain decomposition methods for
boundary element methods in three dimensions. (Principal Investigator). 

1997 Australian Research Council A Post-processing and a posteriori error bounds for finite element analysis. (Associate Investigator. PIs were Prof. Kelly and Sloan, UNSW).

1997 EPSRC GR/L90507 Adaptive Methods for Singularly Perturbed Elliptic Systems. (Principal Investigator). 

Numerical Analysis Summer Schools. ( 1994 Supported by EPSRC grant; 1996 Supported by
EPSRC grant and 1998 Supported by EPSRC under grant GR/M05034 ).

Numerous travel grants etc. from Royal Society, Glasgow Mathematical Journal, Edinburgh Mathematical
Society, London Mathematica Society, etc.

 

Member of Edinburgh Mathematical Society travel and research awards committee. 2003-2012.

Member of London Mathematical Society research meetings committee. 2003-2009.

Member of London Mathematical Society computer science committee. 2009-2012.

Member of Royal Society of Edinburgh sectional committee in mathematics. 2003-2005.

External examiner for MSc in Computational Mathematics at University of Manchester. 2004-2008.

External examiner for undergraduate final honours degree at Heriot-Watt University, Edinburgh. 2001-
2003.

External PhD Thesis Examiner

University of New South Wales, Norwegian Institute of Science and Technology, University of Bath, University of Cape Town, ...

Member of EPSRC Mathematics Peer Review College 1997–present.

Member of SIAM; London Mathematical Society; Edinburgh Mathematical Society.

Journal Reviews:  Regularly review papers for many leading international journals in mathematics, scientific computing and engineering.

Book Reviews: Published book reviews in SIAM Review, Mathematics of Computation and Mathematical Reviews.

2014 Elected to Fellowship of the Society for Industrial and Applied Mathematics (SIAM Fellow)
2010 Elected to Fellowship of the Institute of Mathematics and Its Applications (FIMA)
2004 Awarded Whitehead Prize for mathematics from London Mathematical Society.
2004 Awarded J.L. Lions Prize for Computational Mathematics from ECCOMAS
2003 Elected Fellow of Royal Society of Edinburgh (Scottish National Academy of Science and Letters,
FRSE)
2002 Leverhulme Research Fellowship.
1989 Leslie Fox Prize for Numerical Analysis. Runner up.

Member of London Mathematical Society research meetings committee. 2003-2009.

Member of London Mathematical Society computer science committee. 2009-2012.

Member of Royal Society of Edinburgh sectional committee in mathematics. 2003-2005.

External examiner for MSc in Computational Mathematics at University of Manchester. 2004-2008.

External examiner for undergraduate final honours degree at Heriot-Watt University, Edinburgh. 2001-
2003.

External PhD Thesis Examiner

University of New South Wales, Norwegian Institute of Science and Technology, University of Bath, University of
Cape Town

Member of EPSRC Mathematics Peer Review College 1997–present.

Member of SIAM; London Mathematical Society; Edinburgh Mathematical Society.

Member of editorial board of SIAM Journal on Numerical Analysis (2013-present).

Member of editial board of IMA Journal on Numerical Analysis (2004-present).

Member of editorial board of SIAM Journal on Scientific Computing 2001-2016 (Served five terms).

Senior Editor of Applied Numerical Mathematics (2003-2013). Associate editor (1998-2003).

Member of editorial board of Journal on Scientific Computing 2012-2014.

Member of editorial board of International Journal for Numerical Methods in Engineering (2003-present).

Member of editorial board of International Journal of Numerical Analysis and Modelling (2004-2012).

Member of editorial board of Numerical Methods for Partial Differential Equations (2004-present).

Member of editorial board of International Journal for Numerical Methods in Fluids (2005-present).

Member of editorial board of Proceedings of Royal Society of Edinburgh. (2003-2007). Section A: Mathematics (2003-2007). Invited to become Editor-in-Chief 2008 (declined).

London Mathematical Society Prize Committee 2014–2016

Director of Centre for Numerical Algorithms and Intelligent Software, 2011-2012. £5M multi-disciplinary centre formed by Universities of Strathclyde, Edinburgh, Heriot-Watt and Edinburgh Parallel Computing Centre. Ainsworth was one of four founding members of the centre.

Mathematical Sciences Research Institute (Berkeley): Programme on Numerical and Applied Mathematics
(6th March-28th April 2000). Co-organiser along with Professors Babuška, Bank and Verfürth, of MSRI workshop on A Posteriori Error Estimation and Adaptive Approaches in the Finite Element Method supported by MSRI and NSF.

Isaac Newton Institute for Mathematical Sciences (Cambridge): Programme on Computational Challenges
in Partial Differential Equations held in Cambridge, January-July 2003. Co-organiser with Professors
C.M. Elliot and E. Süli.

EMS Prize Committee: Member of prize committee of Edinburgh Mathematical Society 2004-2012.

Leslie Fox Prize Committee: Member of prize committee for Leslie Fox Prize in Numerical Analysis
2008-2013. Chair of adjudicating panel 2013.