Research blog H.Z. Munthe-Kaas
http://hans.munthe-kaas.no/work/Blog/Blog.html
This blog has entries on my scientific papers, technical reports, talks and unpublished notes. You will also find ‘hindsight notes’ on some of the older papers, and errata (if such are discovered). <br/>iWeb 3.0.4Article: On the Lie enveloping algebra of a post-Lie algebra
http://hans.munthe-kaas.no/work/Blog/Entries/2014/10/24_Article__On_the_Lie_enveloping_algebra_of_a_post-Lie_algebra.html
266c6257-d618-4fdb-a8b0-e3e51d267f5cFri, 24 Oct 2014 12:30:02 +0100Authors: Kurusch Ebrahimi-Fard, Alexander Lundervold, Hans Munthe-Kaas <br/><br/>Journal: Journal of Lie Theory, 25(4):1139-1165. <br/>Abstract: We consider pairs of Lie algebras g and ̄g*, defined over a common vector space, where the Lie brackets of g and ̄g* are related via a post- Lie algebra structure. The latter can be extended to the Lie enveloping algebra U(g). This permits us to define another associative product on U(g), which gives rise to a Hopf algebra isomorphism between U(g) and a new Hopf algebra assembled from U(g*) with the new product. <br/>For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also pro- vides a concise way to develop Butcher’s order theory for Runge–Kutta methods. <br/> <br/><br/><br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Integrators on homogeneous spaces: Isotropy choice and connections
http://hans.munthe-kaas.no/work/Blog/Entries/2014/9/9_Article__Integrators_on_homogeneous_spaces__Isotropy_choice_and_connections.html
7f19cb70-052c-4326-9c28-ca5277d518d9Tue, 9 Sep 2014 12:36:14 +0100Authors: Hans Munthe-Kaas, Olivier Verdier <br/><br/>Journal: To appear in J. Found. Comput. Math. <br/>Abstract: We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtain homogeneous space integrators by combining a Lie group integrator with an isotropy choice. We show that equivariant isotropy choices combined with equivariant Lie group integrators produces an equivariant homogeneous space integrator. Moreover, we show that the RKMK, Crouch–Grossman or commutator-free methods are equivariant. In order to show this, we give a novel description of Lie group integrators in terms of stage trees and motion maps, which unifies the known Lie group integrators. We then proceed to study the equivariant isotropy maps of order zero, which we call connections, and show that they can be identified with reductive structures and invariant principal connections. We give concrete formulas for connections in standard homogeneous spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar decomposition manifolds. Finally, we show that the space of matrices of fixed rank possesses no connection. <br/> <br/><br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: B-series methods are exactly the local, affine equivariant methods
http://hans.munthe-kaas.no/work/Blog/Entries/2014/9/3_Article__B-series_methods_are_exactly_the_local,_affine_equivariant_methods.html
ffff408d-fa14-4c37-b44a-dca9d40ffccbWed, 3 Sep 2014 12:26:27 +0100Authors: <a href="http://arxiv.org/find/math/1/au:+McLachlan_R/0/1/0/all/0/1">Robert I. McLachlan</a>, <a href="http://arxiv.org/find/math/1/au:+Modin_K/0/1/0/all/0/1">Klas Modin</a>, <a href="http://arxiv.org/find/math/1/au:+Munthe_Kaas_H/0/1/0/all/0/1">Hans Munthe-Kaas</a>, <a href="http://arxiv.org/find/math/1/au:+Verdier_O/0/1/0/all/0/1">Olivier Verdier</a> <br/>Journal: Submitted. <br/>Abstract: Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined in all dimensions, so they correspond to sequences of maps — one map for each dimension. A long-standing problem has been to characterise those sequences of maps that arise from B-series. This problem is solved here: we prove that a sequence of smooth maps between vector fields on affine linear spaces has a B-series expansion if and only if it is local and affine equivariant, meaning it respects all affine linear maps between affine spaces. <br/><br/><br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Aromatic Butcher series
http://hans.munthe-kaas.no/work/Blog/Entries/2014/5/9_Article__Aromatic_Butcher_series.html
dccd5bba-0b74-4378-9182-e7672e85232eFri, 9 May 2014 12:21:43 +0100Authors: Hans Munthe-Kaas, Olivier Verdier <br/>Journal: To appear in J. Found. Comp. Math. <br/>Abstract: We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge–Kutta methods, that extends the class of Runge–Kutta methods, and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance. <br/><br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Laureates Meet Young Researchers In Heidelberg
http://hans.munthe-kaas.no/work/Blog/Entries/2014/3/1_Article__Laureates_Meet_Young_Researchers_In_Heidelberg.html
8180519a-39ef-4668-abb4-33c27486e9caSat, 1 Mar 2014 18:45:25 +0000Authors: Helge Holden, Hans Munthe-Kaas, Dierk Schleicher<br/><br/>Journal: EMS Newsletter March 2014. <br/>Abstract: Editorial article on the first Heidelberg Laureate Forum.<br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Computing of B-series by automatic differentiation
http://hans.munthe-kaas.no/work/Blog/Entries/2013/6/1_Article__Computing_of_B-series_by_automatic_differentiation.html
5a69c7d0-a146-40f7-9493-85ea32fc7c71Sat, 1 Jun 2013 12:00:12 +0100Authors: Ferenc A. Bartha, Hans Z. Munthe-Kaas <br/>Journal: Discrete and Continuous Dynamical Systems - Ser. A, (2014) 34:903 - 914 <br/>Abstract: We present an algorithm based on Automatic Differentiation for computing general B-series of vector fields. The algorithm has a computational complexity depending linearly on n, and provides a practical way of computing B-series up to a moderately high order d. Compared to Automatic Differentiation for computing Taylor series solutions of differential equations, the proposed algorithm is more general, since it can compute any B-series. However the computational cost of the proposed algorithm grows much faster in d than a Taylor series method, thus very high order B-series are not tractable by this approach.<br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Symmetric spaces and Lie triple systems in numerical analysis of differential equations
http://hans.munthe-kaas.no/work/Blog/Entries/2013/1/31_Article__Symmetric_spaces_and_Lie_triple_systems_in_numerical_analysis_of_differential_equations.html
1844d349-59c5-4662-8249-33676dd58322Thu, 31 Jan 2013 10:45:37 +0000Authors: H.Z. Munthe-Kaas, G.R.W. Quispel, A. Zanna <br/>Journal: BIT Numerical Mathematics (2014) 54:257–282 <br/>Abstract: A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the math- ematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. The proposed techniques has the property that all the time-steps are positive. <br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: On post-Lie algebras, Lie–Butcher series and moving frames
http://hans.munthe-kaas.no/work/Blog/Entries/2012/3/21_Article__Multidimensional_pseudo-spectral_methods_on_lattice_grids_2.html
0c344571-7bf5-46cb-bc6c-76c2f72bfabcWed, 21 Mar 2012 14:44:01 +0000Authors: H. Z. Munthe-Kaas, A. Lundervold<br/>Journal: Foundations of Computational Mathematics 2013 (4), 583-613<br/>Abstract: Pre-Lie (or Vinberg) algebras arise from a flat and torsion free connection on a differential manifold. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series is an algebraic tool used to study geometric properties of flows on euclidean spaces, which is founded on pre-Lie algebras. Motivated by analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the def- inition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. The generalized Lie–Butcher series combining Butcher series with Lie series are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, is explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie–Butcher series are related to invariants of curves described by moving frames.<br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Multidimensional pseudo-spectral methods on lattice grids
http://hans.munthe-kaas.no/work/Blog/Entries/2012/3/1_Article__Multidimensional_pseudo-spectral_methods_on_lattice_grids.html
b25655d2-8a81-4953-968e-e21fd1d23b0bThu, 1 Mar 2012 12:12:26 +0000Authors: H. Z. Munthe-Kaas, T. Sørevik<br/>Journal: <a href="http://www.sciencedirect.com/science/article/pii/S0168927411002042">Applied Numerical Mathematics,Volume 62, Issue 3, March 2012, Pages 155–165.</a><br/>Abstract: When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L2-integrable, the contributions from many of the Ns terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Algebraic structure of stochastic expansions and universally accurate simulation
http://hans.munthe-kaas.no/work/Blog/Entries/2012/3/1_Article__Algebraic_structure_of_stochastic_expansions_and_universally_accurate_simulation.html
8b1f6463-dccc-4396-8efd-ef2ee13d72dfThu, 1 Mar 2012 11:57:35 +0000Authors: Kurusch Ebrahimi-Fard, Alexander Lundervold, Simon J. A. Malham, Hans Munthe-Kaas, Anke Wiese<br/>Journal: <a href="http://rspa.royalsocietypublishing.org/content/early/2012/04/10/rspa.2012.0024.short">Proc. Royal Soc. A, 2012.</a><br/>Abstract: We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators. These are approximations that generate strong numerical integration schemes that are more accurate than the corresponding stochastic Taylor approximation, independent of the governing vector fields and to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is one example. Herein we: show that the natural context to study stochastic integrators and their properties is the convolution shuffle algebra of endomorphisms; establish a new whole class of efficient integrators; and then prove that, within this class, the sinhlog integrator generates the optimal efficient stochastic integrator at all orders.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Through the Kaleidoscope. Symmetries, Groups and Chebyshev Approximations from a Computational Point of View
http://hans.munthe-kaas.no/work/Blog/Entries/2012/1/5_Article__Through_the_Kaleidoscope._Symmetries,_Groups_and_Chebyshev_Approximations_from_a_Computational_Point_of_View.html
1ef1ae9d-d387-4b07-8715-74090fd89d00Wed, 4 Jan 2012 23:00:00 +0000Authors: H. Z. Munthe-Kaas, M. Nome, B. N. Ryland<br/>Journal: Foundations of Computational Mathematics, Budapest 2011 403, 188. London Mathematical Society Lecture Note Series.<br/>Abstract: In this paper we survey parts of group theory, with emphasis on structures that are important in design and analysis of numerical algorithms and in software design. In particular, we provide an extensive introduction to Fourier analysis on locally compact abelian groups, and point to- wards applications of this theory in computational mathematics. Fourier analysis on non-commutative groups, with applications, is discussed more briefly. In the final part of the paper we provide an introduction to multivariate Chebyshev polynomials. These are constructed by a kaleidoscope of mirrors acting upon an abelian group, and have recently been applied in numerical Clenshaw–Curtis type numerical quadrature and in spec- tral element solution of partial differential equations, based on triangular and simplicial subdivisions of the domain.<br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: On algebraic structures of numerical integration on vector spaces and manifolds
http://hans.munthe-kaas.no/work/Blog/Entries/2011/12/19_Article__On_algebraic_structures_of_numerical_integration_on_vector_spaces_and_manifolds.html
7629d3e5-f1b3-44ef-b4ca-f6e64bbd431dMon, 19 Dec 2011 12:49:56 +0000Author: Alexander Lundervold, Hans Z. Munthe-Kaas<br/>Journal: Submitted.<br/>Abstract: Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Hopf algebras of formal diffeomorphisms and numerical integration on manifolds
http://hans.munthe-kaas.no/work/Blog/Entries/2011/11/1_Article__Hopf_algebras_of_formal_diffeomorphisms_and_numerical_integration_on_manifolds.html
c88dce31-ef00-4756-8450-03b6fa80c337Tue, 1 Nov 2011 12:33:05 +0000Authors: H. Z. Munthe-Kaas, A. Lundervold<br/>Journal: <a href="http://books.google.co.uk/books?hl=en&lr=&id=HUYkE-ac1oIC&oi=fnd&pg=PA295&dq=munthe-kaas+lundervold+hopf+algebras&ots=WMsbAK53Z3&sig=x3NSEzj8uh1c6P15Szb_fQ5W6-M&redir_esc=y#v=onepage&q=munthe-kaas%20lundervold%20hopf%20algebras&f=false">Contemporary mathematics, vol 539 AMS 2011.</a><br/>Abstract: B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge-Kutta methods. Connections to renormalization theory in perturbative quantum field theory have been established in recent years. The algebraic structure of classical Runge-Kutta methods is described by the Connes-Kreimer Hopf algebra. Lie-Butcher theory is a generalization of B-series aimed at studying Lie-group integrators for differential equations evolving on manifolds. Lie-group integrators are based on general Lie group actions on a manifold, and classical Runge-Kutta integrators appear in this setting as the special case of R^n acting upon itself by translations. Lie--Butcher theory combines classical B-series on R^n with Lie-series on manifolds. The underlying Hopf algebra combines the Connes-Kreimer Hopf algebra with the shuffle Hopf algebra of free Lie algebras. We give an introduction to Hopf algebraic structures and their relationship to structures appearing in numerical analysis, aimed at a general mathematical audience. In particular we explore the close connection between Lie series, time-dependent Lie series and Lie--Butcher series for diffeomorphisms on manifolds. The role of the Euler and Dynkin idempotents in numerical analysis is discussed. A non-commutative version of a Faa di Bruno bialgebra is introduced, and the relation to non-commutative Bell polynomials is explored.<br/><br/><a href="http://scholar.google.com/scholar?cites=9031822368777836568&as_sdt=2005&sciodt=0,5&hl=en">Search for citing papers.</a><br/><br/>Hindsight notes: None yet. <br/><br/>Talk: On Multivariate Chebyshev Polynomials; from group theory to PDE solvers
http://hans.munthe-kaas.no/work/Blog/Entries/2011/7/5_Talk__On_Multivariate_Chebyshev_Polynomials%3B_from_group_theory_to_PDE_solvers.html
989fa8ce-f28c-48ea-b0ba-84a659b36dfbMon, 4 Jul 2011 23:00:00 +0100Speaker: H. Z. Munthe-Kaas<br/>Place: <a href="http://www.damtp.cam.ac.uk/user/na/FoCM11/plenary_speaker.html">FoCM 2011 Plenary lecture</a><br/>Abstract: Classical univariate Chebyshev polynomials are fundamental objects in computational mathematics. Their ubiquity in applications is summarized by the quote: "Chebyshev polynomials are everywhere dense in numerical analysis" (perhaps due to George Forsythe). Most of the useful properties of Chebyshev polynomials arise from their tight connections to group theory. From this perspective, multivariate Chebyshev polynomials appear as natural generalizations, constructed by a kaleidoscope of mirrors acting upon R^n (i.e. affine Weyl groups). Multivariate Chebyshev polynomials were first introduced by Koornwinder already in 1974, but they have only very recently been applied in computational mathematics. The fact that the multivariate polynomials are orthogonal on domains related to triangles, tetrahedra and higher dimensional simplexes, rises the important question of their applicability in triangle-based spectral element methods for PDEs. We have developed general software tools with such applications in mind. Finally, we remark upon connections to the representation theory of compact Lie groups, which may lead to applications in very different areas of computational mathematics.<br/><br/>References: <br/> • <a href="Entries/2006/4/24_Article__On_group_Fourier_analysis_and_symmetry_preserving_discretizations_of_PDEs.html">H. Z. Munthe-Kaas: On group Fourier analysis and symmetry preserving discretizations of PDEs, J. Phys. A: Math. Gen. 39 (2006) 5563–5584.</a> <br/> • <a href="Entries/1989/5/12_Tech.rep.__Symmetric_FFTs,_a_general_approach.html">H. Munthe-Kaas: Symmetric FFTs; a general approach, Tech. rep. Department of Math. Sciences, NTNU Trondheim, May 1989</a>.<br/> • <a href="perma://BLPageReference/62678E80-78CA-4D9A-8457-B624E177F287">H. Z. Munthe-Kaas, M. Nome, B. N. Ryland: Through the Kaleidoscope; Symmetries, Groups and Chebyshev Approximations from a Computational Point of View, to appear in FoCM Budapest proceedings</a>.<br/> • <a href="Entries/2011/5/1_Article__Topics_in_Structure_Preserving_Integration.html">S. Christiansen, H. Z. Munthe-Kaas, B. Owren: Topics in structure preservation. Acta Numerica 2011</a>. <br/> • <a href="Entries/2011/3/1_Article__On_multivariate_Chebyshev_polynomials_and_spectral_approximations_on_triangles.html">B.N. Ryland, H.Z. Munthe-Kaas: On multivariate Chebyshev polynomials and spectral approximation on triangles, Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in computational science and engineering, Springer 2011.</a>Article: Backward error analysis and the substitution law for Lie group integrators
http://hans.munthe-kaas.no/work/Blog/Entries/2011/6/6_Article__Backward_error_analysis_and_the_substitution_law_for_Lie_group_integrators.html
2bcfc7da-4a80-42f8-b4bf-b82b98fad5e6Mon, 6 Jun 2011 13:00:02 +0100Author: Alexander Lundervold, Hans Z. Munthe-Kaas<br/>Journal: Foundations of Computational Mathematics 2013 (2), 161-186.<br/>Abstract: Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. They are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the algebraic theory of backward error analysis for methods based on Lie--Butcher series.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Topics in Structure Preserving Integration
http://hans.munthe-kaas.no/work/Blog/Entries/2011/5/1_Article__Topics_in_Structure_Preserving_Integration.html
cbfbbcbb-bd01-4449-aa2b-aa3ebe154f20Sun, 1 May 2011 10:57:51 +0100Authors: S. H. Christiansen, H. Z. Munthe-Kaas, B. Owren<br/>Journal: <a href="http://www.cambridge.org/gb/knowledge/isbn/item6459638/?site_locale=en_GB">Acta Numerica 2011, pp. 1-119.</a><br/>Abstract: In the last few decades the concepts of structure-preserving discretization, ge- ometric integration and compatible discretizations have emerged as subfields in the numerical approximation of ordinary and partial differential equations. The article discusses certain selected topics within these areas; discretiza- tion techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial differential equations, followed by a discussion of how time integrators can be designed to preserve first integrals in the differential equation using discrete gradients and discrete variational derivatives.<br/>Lie group integrators depend crucially on fast and structure-preserving al- gorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algo- rithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes.<br/>The theory of mixed finite elements is developed in terms of special inverse systems of complexes of differential forms, where the inclusion of cells cor- responds to pullback of forms. The theory covers, for instance, composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose prod- uct is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to vari- ational discretizations.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: On multivariate Chebyshev polynomials and spectral approximations on triangles
http://hans.munthe-kaas.no/work/Blog/Entries/2011/3/1_Article__On_multivariate_Chebyshev_polynomials_and_spectral_approximations_on_triangles.html
1f290614-dffa-4e19-aeb1-01d0164b1901Tue, 1 Mar 2011 14:25:12 +0000Authors: H. Z. Munthe-Kaas, B. N. Ryland<br/>Journal: <a href="http://www.springerlink.com/content/uu0n68588p530477/">Lecture Notes in Computational Science and Engineering, 2011, Volume 76, 19-41.<br/></a>Abstract: In this paper we describe the use of multivariate Chebyshev polynomials in computing spectral derivations and Clenshaw–Curtis type quadratures. The multivariate Chebyshev polynomials give a spectrally accurate approximation of smooth multivariate functions. In particular we investigate polynomials derived from the A2 root system. We provide analytic formulas for the gradient and integral of A2 bivariate Chebyshev polynomials. This yields triangular based Clenshaw–Curtis quadrature and spectral derivation algorithms with O (N log N ) computational complexity. Through linear and nonlinear mappings, these methods can be applied to arbitrary triangles and non-linearly transformed triangles. A MATLAB toolbox and a C++ library have also been developed for these methods.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Generalized polar coordinates on Lie groups and numerical integrators
http://hans.munthe-kaas.no/work/Blog/Entries/2008/8/18_Article__Generalized_polar_coordinates_on_Lie_groups_and_numerical_integrators.html
7600dae2-912a-4f91-afac-7febf14da92bMon, 18 Aug 2008 13:46:17 +0100Authors: Stein Krogstad · Hans Z. Munthe-Kaas · Antonella Zanna<br/>Journal:<a href="http://www.springerlink.com/content/tt1rl665m53hr708/">NUMERISCHE MATHEMATIK Volume 114, Number 1 (2009), 161-187.</a><br/>Abstract: Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coor- dinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres.<br/><br/>Hindsight notes: For non-scientific reasons it took six years to publish this work! (My fault!). <br/><br/>Article: On the Hopf Algebraic Structure of Lie Group Integrators
http://hans.munthe-kaas.no/work/Blog/Entries/2007/4/25_Article__On_the_Hopf_Algebraic_Structure_of_Lie_Group_Integrators.html
477143dc-a415-48db-9039-f5d72584364cWed, 25 Apr 2007 08:19:00 +0100Authors: H. Z. Munthe-Kaas, W. Wright<br/>Journal: <a href="http://www.springerlink.com/content/d33718x870l07248/?p=0120219aa9394f2f8cb3e25add545bd8&pi=9">Found. Comput.Math.(2007).</a><br/>Abstract: A commutative but not cocommutative graded Hopf algebra HN, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees HC, developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that HN is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between HN and four other Hopf algebras is discussed. The dual of HN is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra HC of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of HN using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra HSh is obtained from HN by a quotient construction. The Hopf algebra HP of ordered trees by Foissy differs from HN in the definition of the product (noncommutative concatenation for HP and shuffle for HN) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Explicit Volume-Preserving Splitting Methods for Linear and Quadratic Divergence-Free Vector Fields
http://hans.munthe-kaas.no/work/Blog/Entries/2006/11/24_Article__Explicit_Volume-Preserving_Splitting_Methodsfor_Linear_and_Quadratic_Divergence-Free_Vector_Fields.html
ae8a9af7-f862-4050-bac1-512db269ed41Fri, 24 Nov 2006 14:01:03 +0000Authors: R.I. McLachlan · H.Z. Munthe-Kaas · G.R.W. Quispel · A. Zanna<br/>Journal: <a href="http://www.springerlink.com/content/a78315q388101556/">FOUNDATIONS OF COMPUTATIONAL MATHEMATICS<br/>Volume 8, Number 3 (2008), 335-355<br/></a>Abstract: We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine dif- ferent treatments of the diagonal and off-diagonal parts, giving rise to a number of possible combinations.<br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: On group Fourier analysis and symmetry preserving discretizations of PDEs
http://hans.munthe-kaas.no/work/Blog/Entries/2006/4/24_Article__On_group_Fourier_analysis_and_symmetry_preserving_discretizations_of_PDEs.html
b8d29099-3db3-4517-ac93-a90121b6ec2eMon, 24 Apr 2006 10:13:27 +0100Author: H. Z. Munthe-Kaas<br/>Journal: J. Phys. A: Math. Gen. 39 (2006) 5563–5584.<br/>Abstract: In this paper we review some group theoretic techniques applied to discretizations of PDEs. Inspired by the recent years active research in <br/>Lie group- and exponential-time integrators for differential equations, we <br/>will in the first part of the paper present algorithms for computing matrix <br/>exponentials based on Fourier transforms on finite groups. As an example, <br/>we consider spherically symmetric PDEs, where the discretization preserves <br/>the 120 symmetries of the icosahedral group. This motivates the study of <br/>spectral element discretizations based on triangular subdivisions. In the <br/>second part of the paper, we introduce novel applications of multivariate <br/>non-separable Chebyshev polynomials in the construction of spectral element <br/>bases on triangular and simplicial sub-domains. These generalized Chebyshev <br/>polynomials are intimately connected to the theory of root systems and Weyl <br/>groups (used in the classification of semi-simple Lie algebras), and these <br/>polynomials share most of the remarkable properties of the classical Chebyshev <br/>polynomials, such as near-optimal Lebesgue constants for the interpolation <br/>error, the existence of FFT-based algorithms for computing interpolants and <br/>pseudo-spectral differentiation and existence of Gaussian integration rules. The <br/>two parts of the paper can be read independently.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Globally conservative properties and error estimation of a multi-symplectic scheme ..
http://hans.munthe-kaas.no/work/Blog/Entries/2006/1/2_Article__Globally_conservative_properties_and_error_estimation_of_a_multi-symplectic_scheme_...html
fc7a7182-3702-45e4-ad01-3921622741a3Mon, 2 Jan 2006 01:14:05 +0000Authors: Jialin Hong, Ying Liu, Hans Munthe-Kaas, Antonella Zanna<br/>Journal: Applied Numerical Mathematics 56 (2006) 814–843.<br/>Abstract: Based on the multi-symplecticity of the Schrödinger equations with variable coefficients, we give a multi-symplectic numerical scheme, and investigate some conservative properties and error estimation of it. We show that the scheme satisfies discrete normal conservation law corresponding to one possessed by the original equation, and propose global energy transit formulae in temporal direction. We also discuss some discrete properties corresponding to energy conservation laws of the original equations. In numerical experiments, the comparisons with modified Goldberg scheme and Modified Crank–Nicolson scheme are given to illustrate some properties of the multi-symplectic scheme in the numerical implementation, and the global energy transit is monitored due to the scheme does not preserve energy conservation law. Our numerical experiments show the match between theoretical and corresponding numerical results. <br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Computable scalar fields: A basis for PDE software
http://hans.munthe-kaas.no/work/Blog/Entries/2004/12/15_Article__Computable_scalar_fields__A_basis_for_PDE_software.html
de1305d2-e583-45e5-aa88-64f5a68b9cf2Wed, 15 Dec 2004 11:59:34 +0000Author: M. Haveraaen, H. A. Friis, H Munthe-Kaas<br/>Journal: The Journal of Logic and Algebraic Programming 65 (2005) 36-49.<br/>Abstract: Partial differential equations (PDEs) are fundamental in the formulation of mathematical models of the physical world. Computer simulation of PDEs is an efficient and important tool in science and engineering. Implicit in this is the question of the computability of PDEs. In this context we present the notions of scalar and tensor fields, and discuss why these abstractions are useful for the practical formulation of solvers for PDEs.<br/>Given computable scalar fields, the operations on tensor fields will also be computable. As a consequence we get computable solvers for PDEs. The traditional numerical methods for achieving computability by various approximation techniques (e.g., finite difference, finite element or finite volume methods), all have artifacts in the form of numerical inaccuracies and various forms of noise in the solutions. We hope these observations will inspire the development of a theory for computable scalar fields, which either lets us understand why these artefacts are inherent, or provides us with better tools for constructing these basic building blocks.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Eigenvalues for equivariant matrices
http://hans.munthe-kaas.no/work/Blog/Entries/2004/9/15_Article__Eigenvalues_for_equivariant_matrices.html
ddd1d004-94a1-4bfe-b2fd-3b0ca8a31f7aWed, 15 Sep 2004 12:18:33 +0100Author: K. Åhlander, H. Munthe-Kaas<br/>Journal: Journal of Computational and Applied Mathematics 192 (2006) 89 – 99.<br/>Abstract: An equivariant matrix A commutes with a group of permutation matrices. Such matrices often arise in numerical applications where the computational domain exhibits geometrical symmetries, for instance triangles, cubes, or icosahedra.<br/>The theory for block diagonalizing equivariant matrices via the generalized Fourier transform (GFT) is reviewed and applied to eigenvalue computations. For dense matrices which are equivariant under large symmetry groups, we give theoretical estimates that show a substantial performance gain. In case of cubic symmetry, the gain is about 800 times, which is verified by numerical results.<br/>It is also shown how the multiplicity of the eigenvalues is determined by the symmetry, which thereby restricts the number of distinct eigenvalues. The inverse GFT is used to compute the corresponding eigenvectors. It is emphasized that the inverse transform in this case is very fast, due to the sparseness of the eigenvectors in the transformed space.<br/><br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Applications of the Generalized Fourier Transform in Numerical Linear Algebra
http://hans.munthe-kaas.no/work/Blog/Entries/2004/7/1_Article__APPLICATIONS_OF_THE_GENERALIZED_FOURIER_TRANSFORM_IN_NUMERICAL_LINEAR_ALGEBRA.html
232c8b64-f967-40af-a190-c5c5be070a2dThu, 1 Jul 2004 12:28:38 +0100Author: K. Åhlander, H. Munthe-Kaas<br/>Journal: BIT Numerical Mathematics (2005) 45: 819–850.<br/>Abstract: Equivariant matrices, commuting with a group of permutation matrices, are con- sidered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields sub- stantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential.<br/>The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices be- ing diagonalized with the Discrete Fourier Transform, we generalize the classical con- volution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices.<br/>Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: On enumeration problems in Lie–Butcher theory
http://hans.munthe-kaas.no/work/Blog/Entries/2002/11/1_Article__On_enumeration_problems_in_LieButcher_theory.html
83709024-c365-492c-9c3a-8c2944ccd44cFri, 1 Nov 2002 12:38:13 +0000Author: H. Munthe-Kaas, S. Krogstad<br/>Journal: Future Generation Computer Systems 19 (2003) 1197–1205.<br/>Abstract: The algebraic structure underlying non-commutative Lie–Butcher series is the free Lie algebra over ordered trees. In this paper we present a characterization of this algebra in terms of balanced Lyndon words over a binary alphabet. This yields a systematic manner of enumerating terms in non-commutative Lie–Butcher series.<br/><br/>Remarks in hindsight: It was a strange journal to publish this work! The journal had a special issue devoted to Geometric Integration, with many interesting papers. But, I regret that we didn’t send this to a more visible Journal. <br/>The paper establishes an explicit basis for the Free post Lie algebra in one generator, see <a href="Entries/2012/3/21_Article__Multidimensional_pseudo-spectral_methods_on_lattice_grids_2.html">“On Post-Lie algebras ...”</a> for details.<br/><br/>Article: On Object-Oriented Frameworks and Coordinate Free Formulations of PDEs
http://hans.munthe-kaas.no/work/Blog/Entries/2002/1/1_Article__On_Object-Oriented_Frameworks_and_Coordinate_Free_Formulations_of_PDEs.html
fa66c74b-033f-4065-96c7-7a73183db018Tue, 1 Jan 2002 13:56:38 +0000Author: K. Åhlander, M. Haveraaen, H. Munthe-Kaas<br/>Journal: Engineering with Computers (2002) 18: 286–294.<br/>Abstract: An object-oriented (OO) framework for Partial Differential Equations (PDEs) provides software abstractions for numerical simulation of PDEs. The design of such frame- works is not trivial, and the outcome of the design is highly dependent on which mathematical abstractions one chooses to support. In this paper, coordinate free abstractions for PDEs are advocated. The coordinate free formulation of a PDE hides the underlying coordinate system. Therefore, software based on these concepts has the prospect of being more modular, since the PDE formulation is separated from the representation of the coordinates. Use of coordinate free methods in two independent OO frameworks are presented, in order to exemplify the viability of the concepts. The described applications simulate seismic waves for various classes of rock models and the incompressible Navier-Stokes equations on curvi-linear grids, respectively. In both cases, the possibility to express the equations in a domain inde- pendent fashion is crucial. Similarities and differences between the two coordinate free frameworks are discussed. A number of places where such frameworks should be designed for modification is identified. This identification is of interest both for framework developers and for tentative framework users.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Square-Conservative Schemes for a Class of Evolution Equations Using Lie-Group Methods
http://hans.munthe-kaas.no/work/Blog/Entries/2001/1/22_Article__Square-Conservative_Schemes_for_a_Class_of_Evolution_Equations_Using_Lie-Group_Methods.html
ea550665-905f-45fe-91e5-a81f37c2b45eMon, 22 Jan 2001 14:04:08 +0000Author: J-B Chen, H Munthe-Kaas, M-Z Qin<br/>Journal: SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 2164–2178 (2002).<br/><br/>Abstract: A new method for constructing square-conservative schemes for a class of evolution equations using Lie-group methods is presented. The basic idea is as follows. First, we discretize the space variable appropriately so that the resulting semidiscrete system of equations can be cast into a system of ordinary differential equations evolving on a sphere. Second, we apply Lie-group methods to the semidiscrete system, and then square-conservative schemes can be constructed since the obtained numerical solution evolves on the same sphere. Both exponential and Cayley coordinates are used. Numerical experiments are also reported.<br/><br/>Remarks in hindsight: None yet. <br/><br/>Article: Use of coordinate-free numerics in elastic wave simulation
http://hans.munthe-kaas.no/work/Blog/Entries/2001/1/1_Article__Use_of_coordinate-free_numerics_in_elastic_wave_simulation.html
ebca2ae3-8af1-4c33-84da-a19e1b259913Mon, 1 Jan 2001 14:13:28 +0000Authors: H.A. Friis, T. A. Johansen, M. Haveraaen, H. Munthe-Kaas, Å. Drottning<br/><br/>Journal: Applied Numerical Mathematics 39 (2001) 151–171<br/>Abstract: The modeling of elastic waves in solids and fluids is widely applied in physics and geophysics and often requires large computer codes for its realization. Despite the fact that different wave propagation problems have many similarities from the point of view of the abstract mathematical formulation, such codes are usually hard to adapt to new problems and changing requirements.<br/>In this paper we discuss a so-called coordinate-free approach for the general computer implementation of tensor- field equations. We show how it applies to the modeling of seismic waves in isotropic and anisotropic structures, sonic waves in boreholes and ultrasonic waves in poro-elastic fluid-filled materials. The results clearly indicate that the coordinate-free approach is very flexible for the implementation of various wave propagation problems.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Multistep methods integrating ordinary differential equations on manifolds
http://hans.munthe-kaas.no/work/Blog/Entries/2001/1/1_Article__Multistep_methods_integrating_ordinary_differential_equations_on_manifolds.html
31feb20c-7687-4c5c-83a2-282596a912b8Mon, 1 Jan 2001 12:47:35 +0000Authors: S. Faltinsen, A. Marthinsen, H. Z. Munthe-Kaas<br/>Journal: Applied Numerical Mathematics 39 (2001) 349–365.<br/>Abstract: This paper presents a family of generalized multistep methods that evolves the numerical solution of ordinary differential equations on configuration spaces formulated as homogeneous manifolds. Any classical multistep method may be employed as an invariant method, and the order of the invariant method is as high as in the classical setting. We present numerical results that reflect some of the properties of the multistep methods.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: DiffMan: An object-oriented MATLAB toolbox for solving differential equations on manifolds
http://hans.munthe-kaas.no/work/Blog/Entries/2001/1/1_Article__DiffMan__An_object-oriented_MATLAB_toolbox_for_solving_differential_equations_on_manifolds.html
ffef87a5-6d58-4ae8-8e39-0cdb5068c0b0Mon, 1 Jan 2001 12:37:00 +0000Authors: K. Engø, A. Marthinsen, H. Z. Munthe-Kaas<br/>Journal: Applied Numerical Mathematics 39 (2001) 323–347.<br/>Abstract: We describe an object-oriented MATLAB toolbox for solving differential equations on manifolds. The software reflects recent development within the area of geometric integration. Through the use of elements from differential geometry, in particular Lie groups and homogeneous spaces, coordinate free formulations of numerical integrators are developed. The strict mathematical definitions and results are well suited for implementation in an object- oriented language, and, due to its simplicity, the authors have chosen MATLAB as the working environment. The basic ideas of DiffMan are presented, along with particular examples that illustrate the working of and the theory behind the software package.<br/><br/>Hindsight notes: The DiffMan package was not upgraded for many years, but after many requests, it has finally been upgraded to work with Matlab 2012 A release. Download <a href="http://www.diffman.no/">here</a>.<br/><br/>Article: Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms
http://hans.munthe-kaas.no/work/Blog/Entries/2000/9/12_Article__Generalized_Polar_Decompositions_on_Lie_Groups_with_Involutive_Automorphisms.html
1974aa70-2fc7-4685-b009-a42290efcf51Tue, 12 Sep 2000 14:43:38 +0100Authors: H. Z. Munthe-Kaas, G. R. W. Quispel, A. Zanna<br/>Journal: Found. Comput. Math. (2001) 1:297–324.<br/>Abstract: The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group.<br/><br/>In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion.<br/>Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G.<br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Generalized polar decompositions for the approximation of the matrix exponential
http://hans.munthe-kaas.no/work/Blog/Entries/2000/9/5_Article__Generalized_polar_decompositions_for_the_approximation_of_the_matrix_exponential.html
23e9c3fc-822e-45db-8782-5342ea913739Tue, 5 Sep 2000 14:53:52 +0100Authors: A. Zanna, H. Z. Munthe-Kaas<br/>Journal: SIAM J. Matrix Anal. Appl. (2002), Vol. 23, No. 3, pp. 840–862.<br/><br/>Abstract: Pre-Lie (or Vinberg) algebras arise from a flat and torsion free connection on a differential manifold. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series is an algebraic tool used to study geometric properties of flows on euclidean spaces, which is founded on pre-Lie algebras. Motivated by analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the def- inition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. The generalized Lie–Butcher series combining Butcher series with Lie series are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, is explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie–Butcher series are related to invariants of curves described by moving frames.<br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Adjoint and selfadjoint Lie group methods
http://hans.munthe-kaas.no/work/Blog/Entries/2000/9/5_Article__Adjoint_and_selfadjoint_Lie_group_methods.html
d32677c6-d72e-4148-a699-45dd168b101dTue, 5 Sep 2000 12:13:33 +0100Authors: A. Zanna, K. Engø, H. Z. Munthe-Kaas<br/>Journal: BIT 2001, Vol. 41, No. 2, pp. 395–421.<br/>Abstract: In the past few years, a number of Lie-group methods based on Runge–Kutta schemes have been proposed. One might extrapolate that using a selfadjoint Runge–Kutta scheme yields a Lie-group selfadjoint scheme, but this is generally not the case: Lie- group methods depend on the choice of a coordinate chart which might fail to comply to selfadjointness.<br/>In this paper we discuss Lie-group methods and their dependence on centering coor- dinate charts. The definition of the adjoint of a numerical method is thus subordinate to the method itself and the choice of the chart. We study Lie-group numerical methods and their adjoints, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint Runge–Kutta schemes and symmetric coordinates, based on a geodesic or on a flow midpoint. As a result, the proposed selfadjoint Lie-group numerical schemes obey time-symmetry both for linear and nonlinear problems.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Generalized polar decompositions for approximation of the matrix exponential
http://hans.munthe-kaas.no/work/Blog/Entries/2000/9/5_Article__Generalized_polar_decompositions_for_approximation_of_the_matrix_exponential.html
4316bae5-a56b-4145-aa2d-2c3d8bd42735Tue, 5 Sep 2000 12:03:28 +0100Authors: A. Zanna, H. Z. Munthe-Kaas<br/>Journal: SIAM J. Matrix Anal. Appl. Vol. 23, No. 3, pp. 840–862 (2002).<br/><br/>Abstract: In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297–324] to approximate a matrix exponential. The algorithms presented have the property that, if Z ∈ g, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of g. This property is very relevant when solving Lie-group ODEs and is not usually fulfilled by standard approximations to the matrix exponential.<br/>We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy.<br/>The proposed methods have a complexity of O(κn3), with constant κ small, depending on the order and the Lie algebra g. The algorithms are recommended in cases where it is of fundamental importance that the approximation for the exponential resides in G, and when the order of approximation needed is not too high.We present in detail algorithms up to fourth order.<br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: On the Role of Mathematical Abstractions for Scientific Computing
http://hans.munthe-kaas.no/work/Blog/Entries/2000/7/12_Article__On_the_Role_of_Mathematical_Abstractions_for_Scientific_Computing.html
265629ec-6923-433d-876e-13ae57acbb82Wed, 12 Jul 2000 14:34:21 +0100Authors: Krister Åhlander , Magne Haveraaen , Hans Munthe-Kaas<br/>Journal: The architecture of scientific software (Boisvert and Tang eds.), Kluwer acad. publ. (2001), pp. 145-159. <br/>Abstract: A distinguished feature of scientific computing is the necessity to design software abstractions for approximations. The approximations are themselves abstractions of mathematical models, which also are abstractions. In this paper, the relation between different mathematical abstraction levels and scientific computing software is discussed, in particular with respect to the simulation of partial differential equations (PDEs). It is found that software based on continuous abstractions have more chances of being modular, than software based on discrete approximations of the continuous abstractions. Moreover, it is stated that coordinate-free abstractions are a solid foundation for the simulation of PDEs.<br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Lie-group methods
http://hans.munthe-kaas.no/work/Blog/Entries/2000/1/1_Article__Lie-group_methods.html
904ebac4-b1d9-4def-ba47-0d8ae825bdcbSat, 1 Jan 2000 12:20:44 +0000Authors: A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, A. Zanna<br/>Journal: Acta Numerica (2000), pp. 215–365, CUP.<br/>Abstract: Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geo- metry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.<br/><br/><br/>Hindsight notes: This paper surveys the state of knowledge of Lie group methods up to year 2000. Later came other developments such as generalized polar decompositions and commutator free methods, which are not discussed in this paper.<br/><br/>Article: High order Runge-Kutta methods on manifolds
http://hans.munthe-kaas.no/work/Blog/Entries/1999/1/1_Article__High_orderRunge-Kutta_methods_on_manifolds.html
7b40352f-14e5-4394-9272-919fab52a2e3Fri, 1 Jan 1999 13:53:46 +0000Authors: H. Munthe-Kaas<br/>Journal: Applied Numerical Mathematics29 (1999) 115-127<br/>Abstract: We present a family of Runge-Kutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical Runge-Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement. These are defined in a general abstract framework, based on a Lie algebra acting on the manifold. The general framework gives rise to a wide range of different concrete applications; we present some examples.<br/><br/><br/><br/>Hindsight notes: None yet. <br/><br/>Article: Computations in a Free Lie-algebra
http://hans.munthe-kaas.no/work/Blog/Entries/1999/1/1_Article__Computations_in_a_Free_Lie-algebra.html
4e4e8fde-4566-4d23-8da7-fc235801e337Fri, 1 Jan 1999 13:44:35 +0000Authors: H. Z. Munthe-Kaas, B. Owren<br/>Journal: Phil. Trans. R. Soc. Lond. A (1999) 357, 957–981.<br/>Abstract: Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker–Campbell–Hausdorff formula and the recently developed Lie group methods for integration of differential equa- tions on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computa- tions. Witt’s formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides good bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exempli- fied by the new Matlab toolbox ‘DiffMan’.<br/><br/>Hindsight notes: None yet. <br/><br/>Article: Runge-Kutta methods on Lie groups
http://hans.munthe-kaas.no/work/Blog/Entries/1998/1/1_Article__Runge-Kutta_methods_on_Lie_groups.html
52beb803-a26f-44ab-b815-2785250d9727Thu, 1 Jan 1998 21:52:48 +0000Author: Hans Munthe-Kaas<br/>Journal: BIT 38:1 (1998), 92-111.<br/>Abstract: We construct generalized Runge-Kutta methods for integration of differential equations evolving on a Lie group. The methods are using intrinsic operations on the <br/>group, and we are hence guaranteed that the numerical solution will evolve on the <br/>correct manifold. Our methods must satisfy two different criteria to achieve a given order: <br/>• Coefficients A_i,j and b_j must satisfy the classical order conditions. This is done by picking the coefficients of any classical RK scheme of the given order. <br/>• We must construct functions to correct for certain non-commutative effects to the given order.<br/>These tasks are completely independent, so once correction functions are found to the <br/>given order, we can turn any classical RK scheme into an RK method of the same order <br/>on any Lie group. <br/>The theory in this paper shows the tight connections between the algebraic structure <br/>of the order conditions of RK methods and the algebraic structure of the so called <br/>'universal enveloping algebra' of Lie algebras. This may give important insight also <br/>into the classical RK theory. <br/><br/>Remarks in hindsight: We did not at the time know how to go beyond order 4. Arbitrary order was found by the (much simpler) approach of THIS paper. <br/><br/>