The thin plate spline, as introduced by Duchon, interpolates a smooth surface through scattered data. It is computationally expensive when there are many data points. The finite element thin plate spline (TPSFEM) possesses similar smoothing properties and is efficient for large data sets. Its efficiency is further improved by adaptive refinement that adapts the precision of the finite element grid. Adaptive refinement processes and error indicators developed for partial differential equations may not apply to the TPSFEM as it incorporates information about the scattered data. This additional information results in features not evident in partial differential equations. An iterative adaptive refinement process and five error indicators were adapted for the TPSFEM. We give comprehensive depictions of the process in this article and evaluate the error indicators through a numerical experiment with a model problem and two bathymetric surveys in square and L-shaped domains.

As the generation of exascale high-performance clusters begins, it has become evident that numerical algorithms will greatly benefit from built-in resilience features that can handle system faults. Prior studies of fault-tolerant multigrid methods have focused on structured grids. In this work, however, we study the resilience of multigrid solvers on unstructured grids with adaptive refinement. The challenge lies in the fact that unstructured grids distributed across multiple processors may manifest as local hierarchical grids with unaligned boundaries. Our numerical experiments highlight that this disparity can result in divergence when employing standard local multigrid for fault recovery. We analyze this phenomenon by using an energy control condition. To tackle the divergence issue, we propose a simple variation of the multigrid V-cycle that scales the coarse problem. We present a convergence proof for the new algorithm. By implementing this new method for local recovery, our numerical experiments confirm that convergence can be recovered on unstructured grids while the algorithm agrees with the standard multigrid V-cycle on grids with aligned boundaries. More importantly, the impact of a fault can be mitigated and delays in the global multigrid iterations can be reduced. Finally, we investigate how local regions within the adaptive mesh, associated with different faulty processors, affect the effectiveness of fault recovery.

We present a computational method for reconstructing the refractive index of an unknown complex-shaped two-dimensional medium with embedded metal inclusions from its transverse magnetic electromagnetic scattering properties. We present a novel hybrid surface-volume integral equation that generalises the Lippmann–Schwinger equation to media containing embedded scatterers. Using this hybrid equation, we show that the Frechet derivative of the forward-mapping satisfies a particular inhomogeneous wave scattering problem. We solve the inhomogeneous wave scattering problem using a novel coupled FEM-BEM formulation. Our numerical scheme is based on a thin plate spline ansatz for the refractive index, which can be constructed using a relatively small number of control points, and can be efficiently constructed and evaluated even for the complex-shaped multiply-connected domains of interest. Numerical experiments demonstrate the effectiveness of our method by reconstructing several challenging media.

The thin-plate spline is a data fitting technique that possesses many favourable properties like insensitivity to noise [6]. One obstacle of its usage is the high computational cost and memory requirement for large data sets

This work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 10 23 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.

The discrete thin plate spline smoother fits smooth surfaces to large data sets efficiently. It combines the favourable properties of the finite element surface fitting and thin plate splines. The efficiency of its finite element grid is improved by adaptive refinement, which adapts the precision of the solution. It reduces computational costs by refining only in sensitive regions, which are identified using error indicators. While many error indicators have been developed for the finite element method, they may not work for the discrete smoother. In this article we show three error indicators adapted from the finite element method for the discrete smoother. A numerical experiment is provided to evaluate their performance in producing efficient finite element grids. References F. L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Pat. Anal. Mach. Int. 11.6 (1989), pp. 567–585. doi: 10.1109/34.24792. C. Chen and Y. Li. A robust method of thin plate spline and its application to DEM construction. Comput. Geosci. 48 (2012), pp. 9–16. doi: 10.1016/j.cageo.2012.05.018. L. Fang. Error estimation and adaptive refinement of finite element thin plate spline. PhD thesis. The Australian National University. http://hdl.handle.net/1885/237742. L. Fang. Error indicators and adaptive refinement of the discrete thin plate spline smoother. ANZIAM J. 60 (2018), pp. 33–51. doi: 10.21914/anziamj.v60i0.14061. M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Commun. Stat. Simul. Comput. 19.2 (1990), pp. 433–450. doi: 10.1080/0361091900881286. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Soft. 15.4 (1989), pp. 326–347. doi: 10.1145/76909.76912. R. F. Reiniger and C. K. Ross. A method of interpolation with application to oceanographic data. Deep Sea Res. Oceanographic Abs. 15.2 (1968), pp. 185–193. doi: 10.1016/0011-7471(68)90040-5. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal. 41.1 (2003), pp. 208–234. doi: 10.1137/S0036142901383296. D. Ruprecht and H. Muller. Image warping with scattered data interpolation. IEEE Comput. Graphics Appl. 15.2 (1995), pp. 37–43. doi: 10.1109/38.365004. E. G. Sewell. Analysis of a finite element method. Springer, 2012. doi: 10.1007/978-1-4684-6331-6. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput. 63.2 (2015), pp. 374–409. doi: 10.1007/s10915-014-9898-x. O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24.2 (1987), pp. 337–357. doi: 10.1002/nme.1620240206.

On future extreme scale computers, it is expected that faults will become an increasingly serious problem as the number of individual components grows and failures become more frequent. This is driving the interest in designing algorithms with built-in fault tolerance that can continue to operate and that can replace data even if part of the computation is lost in a failure. For fault-free computations, the use of adaptive refinement techniques in combination with finite element methods is well established. Furthermore, iterative solution techniques that incorporate information about the grid structure, such as the parallel geometric multigrid method, have been shown to be an efficient approach to solving various types of partial different equations. In this article, we present an advanced parallel adaptive multigrid method that uses dynamic data structures to store a nested sequence of meshes and the iteratively evolving solution. After a fail-stop fault, the data residing on the faulty processor will be lost. However, with suitably designed data structures, the neighbouring processors contain enough information so that a consistent mesh can be reconstructed in the faulty domain with the goal of resuming the computation without having to restart from scratch. This recovery is based on a set of carefully designed distributed algorithms that build on the existing parallel adaptive refinement routines, but which must be carefully augmented and extended.

The thin plate spline method is a widely used data fitting technique which has the ability to smooth noisy data. We present some example applications of a new mixed finite element discretisation of the thin plate spline method. The new approach works with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system, thus ensuring that the scheme is numerically efficient and the formulation is stable. We overview of the theoretical foundations of the new approach and give numerical examples in both two and three dimensions. References D. N. Arnold and F. Brezzi. Some new elements for the Reissner–Mindlin plate model. In Boundary Value Problems for Partial Differential Equations and Applications, pages 287–292. Masson, Paris, 1993. D. Boffi and C. Lovadina. Analysis of new augmented Lagrangian formulations for mixed finite element schemes. Numer. Math., 75:405–419, 1997. doi:10.1007/s002110050246 X.-L. Cheng, W. Han, and H.-C. Huang. Some mixed finite element methods for biharmonic equation. J. Comput. Appl. Math., 126:91–109, 2000. doi:10.1016/S0377-0427(99)00342-8 J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, 571:85–100. Springer-Verlag, Berlin, 1977. doi:10.1007/BFb0086566 V. Girault and P.-A. Raviart. Finite Element Methods for Navier–Stokes Equations. Springer-Verlag, Berlin, 1986. doi:10.1007/978-3-642-61623-5 M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simulat. Comput., 19:433–450, 1990. doi:10.1080/03610919008812866 C. Johnson and J. Pitkaranta. Analysis of some mixed finite element methods related to reduced integration. Math. Comput., 38(158):375–400, 1982. doi:10.2307/2007276 T. Karper, K.-A. Mardal, and R. Winther. Unified finite element discretizations of coupled Darcy–Stokes flow. Numer. Meth. Part. D. E., 25:311–326, 2009. doi:10.1002/num.20349 B. P. Lamichhane. A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. J. Comput. Appl. Math., 235:5188–5197, 2011. doi:10.1016/j.cam.2011.05.005 B. P. Lamichhane, S. G. Roberts, and L. Stals. A mixed finite element discretisation of thin-plate splines. In W. McLean and A. J. Roberts (Eds), Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, ANZIAM J., 52:C518–C534, 2010. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3934 B. P. Lamichhane, S. G. Roberts, and L. Stals. A mixed finite element discretisation of thin plate splines based on biorthogonal systems. J. Sci. Comput., 1–23, July 2015. doi:10.1007/s10915-015-0068-6 G. Wahba. Spline Models for Observational Data, volume 59 of Series in Applied Mathematic. SIAM, Philadelphia, 1990. doi:10.1137/1.9781611970128

The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose size only depends linearly on the number of finite element nodes. The mixed formulation is obtained by introducing the gradient of the corresponding function as an additional unknown. The novel approach taken in this paper is to work with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system thus ensuring that the scheme is numerically efficient, and the formulation is stable. Some numerical results are presented to demonstrate the performance of our approach. A preconditioned conjugate gradient method is an efficient solver for the arising linear system of equations.

We present a new technique for solving the saddle point problem arising from a finite element based thin plate spline formulation. The solver uses the Sherman–Morrison–Woodbury formula to divide the domain into different regions depending on the properties of the data projection matrix. We analyse the conditioning of the resulting system on certain data distributions and use the results to develop effective preconditioners. We show our approach is efficient for a wide range of parameters by testing it on a number of different examples. Numerical results are given in one, two and three dimensions.

BACKGROUND Effective teaching and learning require appropriate assessment tasks and prompt feedback. Feedback is critical to inform students if they are on track to achieve the course learning goals. According to Sadler (1989) useful feedback should provide evidence of learning that fills a gap between what is understood and what is aimed to be understood. In mathematics, the most effective form of assessment is weekly assignments to learn how to write mathematics. This is a difficult skill that requires a lot of practice and guidance. It is not sufficient to write down the correct answer or to give a list of calculations without adequate explanation. Students must employ precise use of words, formulae, symbols and punctuation. It is essential for students to receive feedback not only on the correctness of their answers, but also on their writing abilities in mathematics. AIMS The main aim of this project is to improve student learning of mathematics by promoting changes in mathematics assessment and teaching methods through the adoption of electronic pens and tablets. This strategy allows (i) the online submission and marking of assignments which gives students constructive feedback on their assessment. (ii) lecturers to work out proofs and examples in class on a step-by-step fashion that is also very effective for lecture recording. This is aimed at facilitating students’ flexible learning making it easier to study mathematics in their own time wherever they are: at home, in a library or on public transport. APPROACH We have endeavored to leverage changes in mathematics assessment and teaching methods by adopting electronic pens and tablets to allow online submission, electronic assignments’ marking and delivery of lectures. We have purchased a number of Samsung tablets equipped with a Wacom e-pen for the teaching staff of our first year mathematics service and honours courses (total enrolment of 450 students). CONCLUSIONS Lecturers have found that the use of e-pens on tablet is a very powerful method to deliver lectures that also allows for easy recording. However, lecturers and tutors have found that whilst providing written feedback to students once the assignments are marked results in an excellent learning outcome, marking assignments on tablets is still not as easy as marking them on paper, probably due to limitations in the technology. This has resulted in delays in returning marked assignments, and thus feedback on learning, to students. REFERENCES Sadler, D.R. (1989). Formative assessment and the design of instructional systems. Instructional Science,18,144.

Parallel implementation of the sparse grid combination technique in high dimensions presents many complexity challenges. We enumerate these challenges, classifying them respectively as computational, algorithmic, and software complexity. We discuss strategies for overcoming the individual complexity barriers. We describe our architecture for a software framework that will allow its users to build complex multiple grid solver applications.

A key issue confronting petascale and exascale computing is the growth in probability of soft and hard faults with increasing system size. A promising approach to this problem is the use of algorithms that are inherently fault tolerant. We introduce such an algorithm for the solution of partial differential equations, based on the sparse grid approach. Here, the solution of multiple component grids are efficiently combined to achieve a solution on a full grid. The technique also lends itself to a (modified) MapReduce framework on a cluster of processors, with the map stage corresponding to allocating each component grid for solution over a subset of the processors, and the reduce stage corresponding to their combination. We describe how the sparse grid combination method can be modified to robustly solve partial differential equations in the presence of faults. This is based on a modified combination formula that can accommodate the loss of one or two component grids. We also discuss accuracy issues associated with this formula. We give details of a prototype implementation within a MapReduce framework using the dynamic process features and asynchronous message passing facilities of MPI. Results on a two-dimensional advection problem show that the errors after the loss of one or two sub-grids are within a factor of 3 of the sparse grid solution in the presence of no faults. They also indicate that the sparse grid technique with four times the resolution has approximately the same error as a full grid, while requiring (for a sufficiently high resolution) much lower computation and memory requirements. We finally outline a MapReduce variant capable of responding to faults in ways other than re-scheduling of failed tasks. We discuss the likely software requirements for such a flexible MapReduce framework, the requirements it will impose on users’ legacy codes, and the system's runtime behavior.

Nonstationary iterated Tikhonov regularization is an efficient method for solving ill-posed problems in Hilbert spaces. However, this method may not produce good results in some situations since it tends to oversmooth solutions and hence destroy special features such as sparsity and discontinuity. By making use of duality mappings and Bregman distance, we propose an extension of this method to the Banach space setting and establish its convergence. We also present numerical simulations which indicate that the method in Banach space setting can produce better results.

The Hasegawa-Wakatani models are used in the study of confinement of hot plasmas with externally imposed magnetic fields. The nonlinear terms in the Hasegawa-Wakatani models complicate the analysis of the system as they propagate local changes across the entire system. Centre manifold analysis allows us to project down onto much smaller systems that are more easily analysed. Qualitative information about the behaviour of the reduced system, such as whether it is stable or unstable, can be used to predict the behaviour of the original full system. We show how the simple structure of the linear part of the Hasegawa-Wakatani equations can be used to define these projection operators. The centre manifold analysis will be used on a few examples to highlight certain properties of the Hasegawa-Wakatani models.

Thin-plate splines are a well established technique for the interpolation and smoothing of scattered data. However, the traditional formulation of the method leads to large, dense and often ill-conditioned matrices, which reduces its applicability in practice. We present a new mixed finite element formulation based on the ideas behind the mortar finite element methods. The resulting system of equations is sparse and positive definite, and its size depends only on the number of finite elements not the number of data points. References http://geopubs.wr.usgs.gov/open-file/of00-043/bathymetry/appendices.html. C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezis and J.-L. Lions, editor, {{N}onlinear Partial Differential Equations and Their Applications}, pages 13--51. Paris, 1994. X. Cheng, W. Han, and H. Huang. Some mixed finite element methods for biharmonic equation. {Journal of Computational and Applied Mathematics}, 126(1-2):91--109, 2000. http://dx.doi.org/10.1016/S0377-0427(99)00342-8. P. G. Ciarlet. {The Finite Element Method for Elliptic Problems}. North Holland, Amsterdam, 1978. P. G. Ciarlet and P. A. Raviart. A mixed finite element method for the biharmonic equation. In C. De Boor, editor, {Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations}, pages 125--143, New York, 1974. Academic Press. J. Duchon. Splines minimizing rotation-invariant semi-norms in {S}obolev spaces. In {Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics}, volume 571, pages 85--100. Springer-Verlag, Berlin, 1977. http://www.springerlink.com/content/g27671q701166031/. R. S. Falk. Approximation of the biharmonic equation by a mixed finite element method. {SIAM Journal on Numerical Analysis}, 15(3):556--567, 1978. http://epubs.siam.org/sinum/resource/1/sjnaam/v15/i3. M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for {L}aplacian smoothing splines. {Communications in Statistics --- Simulation and Computation}, 18(3):1059--1076, 1989. http://www.tandfonline.com/toc/lssp20/18/3. C. Johnson and J. Pitkaranta. Analysis of some mixed finite element methods related to reduced integration. {Mathematics of Computation}, 38(158):375--400, 1982. http://www.ams.org/journals/mcom/1982-38-158/home.html. B. P. Lamichhane, S. Roberts, and L. Stals. A mixed finite element discretization of thin plate splines based on quasi-biorthogonal systems. To be submitted. P. Monk. A mixed finite element method for the biharmonic equation. {SIAM Journal on Numerical Analysis}, 24(4):737--749, 1987. http://epubs.siam.org/sinum/resource/1/sjnaam/v24/i4. T. Ramsay. Spline smoothing over difficult regions. {Journal of Royal Statistical Society. Series B (Statistical Methodology)}, 64(2):307--319, 2002. http://onlinelibrary.wiley.com/doi/10.1111/rssb.2002.64.issue-2/issuetoc. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. {SIAM Journal on Numerical Analysis}, 41(1):208--234, 2003. http://epubs.siam.org/sinum/resource/1/sjnaam/v41/i1. L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. {Mathematics of Computation}, 54(190):483--493, 1990. http://www.ams.org/journals/mcom/1990-54-190/home.html. L. Stals and S. Roberts. Smoothing large data sets using discrete thin plate splines. {Computing and Visualization in Science}, 9(3):185--195, 2006. http://www.springerlink.com/content/1432-9360/9/3/. G. Wahba. {Spline Models for Observational Data}, volume 59 of {Series in Applied Mathematic}. SIAM, Philadelphia, first edition, 1990.

A commonly used method for the fitting of smooth functions to noisy data sets is the thin-plate spline method. Traditional thin-plate splines use radial basis functions and consequently requires the solution of a dense linear system of equations that grows with the number of data points. We present a method based instead on low order polynomial basis functions with local support defined on finite element grids. An advantage of such an approach is that the resulting system of equations is sparse and its size depends on the number of nodes in the finite element grid.