Papers by Gustaf Soderlind
Linear Algebra and its Applications, Apr 1, 1987
In studying stability of solutions of linear differential equations one naturally encounters Liap... more In studying stability of solutions of linear differential equations one naturally encounters Liapunov equations. In a suitable setting they can be interpreted as equations for the normalized "directions" of these solutions. When applying discretizations to the Liapunov equations one is led to a problem which in its most elementary form can be stated as: Given a matrix A and a vector b, determine a vector x with xrx = 1 and a scalar p such that Axb = px. Here p is called the inhomogeneous eigenvalue. We consider the question of how many solution pairs (CL, X) of this problem exist. We also give some numerical methods to compute such a pair; they are based on (generalizations of) shifted power iterations. Finally we consider the case that jlbll is small so that the inhomogeneous eigenvalues can be viewed as perturbations of the homogeneous ones (i.e. b = 0).
To increase their computational efficiency, time-stepping methods for the solution of ODEs and DA... more To increase their computational efficiency, time-stepping methods for the solution of ODEs and DAEs are usually adaptive. A control system for the adaptivity includes the automatic control of step size as well as method order, and can be constructed using proven design principles from control theory. It is a ''superstructure,'' surrounding the basic method, that manages the computational process. As the control system is a dynamical system in its own right, its properties need to be properly analyzed and tested, motivating a separate study of control system dynamics. Here we propose a complete specification of a control system for adaptive time-stepping, based on controlling local error estimates. The control system is specified in Matlab to facilitate ease of implementation and is executable on its own. The entire control system is independent of the basic discretization method, and can be tested stand--alone, without any specific ODE or DAE method. With respect to step size control, a collection consisting of PI controllers and digital filters is included. For controlling the order, error estimates for two alternative methods of different orders are monitored. The estimates are fed to the same step size controller, which will produce different step size suggestions to estimate relative efficiency. The order change is based on simple time integration of the efficiency measure. Anti-windup is used to counteract excessive step size increases for method orders currently not in use. We further discuss the choice and handling of the norm used for measuring local errors. (Less)
Technical Reports; TFRT, 1987
ACM Transactions on Mathematical Software, Mar 23, 2018
Siam Journal on Mathematical Analysis, 1985
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Computing, Sep 1, 1992
... Gustaf S6derlind Department of Computer Science Lund University PO Box 118 S-221 00 Lund Swed... more ... Gustaf S6derlind Department of Computer Science Lund University PO Box 118 S-221 00 Lund Sweden Verleger: Springer-Veriag KG, Sachsenplatz 4-6, A-120t Wien. - Herausgeber: Prof. Dr. Hans J. Stener, Institut for Angewandte und Numerische Mathematik der ...
Encyclopedia of Applied and Computational Mathematics, 2015
Annals of Biomedical Engineering, Jul 18, 2012
Recent studies suggest that vortex ring formation during left ventricular (LV) rapid filling is a... more Recent studies suggest that vortex ring formation during left ventricular (LV) rapid filling is an optimized mechanism for blood transport, and that the volume of the vortex ring is an important measure. However, due to lack of quantitative methods, the volume of the vortex ring has not previously been studied. Lagrangian Coherent Structures (LCS) is a new flow analysis method, which enables in vivo quantification of vortex ring volume. Therefore, we aimed to investigate if vortex ring volume in the human LV can be reliably quantified using LCS and magnetic resonance velocity mapping (4D PC-MR). Flow velocities were measured using 4D PC-MR in nine healthy volunteers and four patients with dilated ischemic cardiomyopathy. LV LCS were computed from flow velocities and manually delineated in all subjects. Vortex volume in the healthy volunteers was 51 ± 6% of the left ventricular volume, and 21 ± 5% in the patients. Interobserver variability was-1 ± 13% and interstudy variability was-2 ± 12%. Compared to idealized flow experiments, the vortex rings showed additional complexity and asymmetry, related to endocardial trabeculation and papillary muscles. In conclusion, LCS and 4D PC-MR enables measurement of vortex ring volume during rapid filling of the LV.
Abstract A technique for solving high-index problems by combining symbolic and numerical methods ... more Abstract A technique for solving high-index problems by combining symbolic and numerical methods is presented. The technique is a variant of index reduction. In the usual manner, parts of the differential-algebraic equation (DAE) are differentiated analytically and ...
CWI quarterly, Mar 1, 1998
This paper reviews the recent advances in developing automatic control algorithms for the numeric... more This paper reviews the recent advances in developing automatic control algorithms for the numerical integration of ordinary dierential equations (ODEs) and dierential-algebraic equations (DAEs). By varying the stepsize, the error committed in a single step of the discretization method can be aected. Modern time-stepping methods provide an estimate of this error, and by comparing the estimate to a specied accuracy requirement a control algorithm selects the next stepsize. To construct ecient controllers it is necessary to analyze the dynamic behaviour of the discretization method together with the controller. Based on feedback control theory, this systematic approach has replaced earlier heuristics and resulted in a more consistent and robust performance. Other quantities affected by the stepsize are convergence rates of xed-point and Newton iterations, and we therefore also review new techniques for the coordination of nonlinear equation solvers with the primary stepsize controller. Taken together, the recent development provides principles and guidelines for constructing ODE/DAE software where heuristics and tuning parameters have largely been eliminated.
Journal of Computational and Applied Mathematics, Dec 1, 1997
For implicit Runge{Kutta methods intended for sti ODEs or DAEs, it is often dicult to embed a loc... more For implicit Runge{Kutta methods intended for sti ODEs or DAEs, it is often dicult to embed a local error estimating method which gives realistic error estimates for sti/algebraic components. If the embedded method's stability function is unbounded at z = 1, sti error components are grossly overestimated. In practice some codes \improve" such inadequate error estimates by p remultiplying the estimate by a \lter" matrix which damps or removes the large, sti error components. Although improving computational performance, this technique is somewhat arbitrary and lacks a sound theoretical backing. In this scientic note we resolve this problem by introducing an implicit error estimator. It has the desired properties for sti/algebraic components without invoking articial improvements. The error estimator contains a free parameter which determines the magnitude of the error, and we show h o w this parameter is to be selected on the basis of method properties. The construction principles for the error estimator can be adapted to all implicit Runge{Kutta methods, and a better agreement between actual and estimated errors is achieved, resulting in better performance.
ACM Transactions on Mathematical Software, Mar 20, 2020
We present a software package, M odes , offering h -adaptive and p -adaptive linear multistep met... more We present a software package, M odes , offering h -adaptive and p -adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods ( p = k for explicit and p = k +1 for implicit methods) are supported. For stiff computation, implicit methods of order p = k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h - and p -adaptivity. Implemented as a M atlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. M odes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.
Applied Mathematics and Computation, 2012
This paper develops a semi-analytic technique for generating smooth nonuniform grids for the nume... more This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter (1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h i = x i+1 À x i , with 0 = x 0 < x 1 < Á Á Á < x N = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small , neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L 2 , for large as well as small values of N, and over a wide range of values of. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness.
Journal of Cardiovascular Magnetic Resonance, 2013
Bit Numerical Mathematics, Sep 1, 1984
We derive an error bound for fixed-point iterations x.+~ = f(x~) by using monotonicity in the sen... more We derive an error bound for fixed-point iterations x.+~ = f(x~) by using monotonicity in the sense of [2]. The new bound is preferable to the classical one which bounds the error in terms of the Lipschitz constant of f.
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Papers by Gustaf Soderlind