Gustaf Söderlind

Index reduction in differential-algebraic equations using dummy derivatives

A new index reduction algorithm for DAEs is developed. In the usual manner, parts of the DAE are differentiated analytically and appended to the original system. For each additional equation, a derivative is selected to be replaced by a new algebraic variable called a dummy derivative. The resulting augmented system is at most index 1, but no longer overdetermined. The dummy derivatives are not subject to discretization; their purpose is to annihilate part of the dynamics in the DAE, leaving only what corresponds to the dynamics of a state-space form. No constraint stabilization is necessary in the subsequent numerical treatment. Numerical tests indicate that the method yields results with an accuracy comparable to that obtained for the corresponding state-space ODE.

Automatic Control and Adaptive Time-Stepping

Adaptive time-stepping is central to the efficient solution of initial value problems in ODEs and DAEs. The error committed in the discretization method primarily depends on the time-step size h, which is varied along the solution in order to minimize the computational effort subject to a prescribed accuracy requirement. This paper reviews the recent advances in developing local adaptivity algorithms based on well established techniques from linear feedback control theory, which is introduced in a numerical context. Replacing earlier heuristics, this systematic approach results in a more consistent and robust performance. The dynamic behaviour of the discretization method together with the controller is analyzed. We also review some basic techniques for the coordination of nonlinear equation solvers with the primary stepsize controller in implicit time-stepping methods.

On nonlinear difference and differential equations

A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODEs. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODEs, estimates based on the Dahlquist constant always give sharper results.The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems.

Bounds on nonlinear operators in finite-dimensional banach spaces

SummaryWe consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operators Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.

Stage Value Predictors and Efficient Newton Iterations in Implicit Runge--Kutta Methods

The prediction of stage values in implicit Runge--Kutta methods is important both for overall efficiency as well as for the design of suitable control strategies for the method. The purpose of this paper is to construct good stage value predictors for implicit methods and to verify their behavior in practical computations. We show that for stiffly accurate methods of low stage order it is necessary to use several predictors. In other words, a continuous extension for the method will not yield the best results. We also investigate how to gain additional efficiency in the Newton iterations used to correct the prediction error. This leads to new control strategies with respect to refactorization of Jacobians that seek to globally minimize total work per unit time of integration.

Volume Tracking: A new method for quantitative assessment and visualization of intracardiac blood flow from three-dimensional, time-resolved, three-component magnetic resonance velocity mapping

BackgroundFunctional and morphological changes of the heart influence blood flow patterns. Therefore, flow patterns may carry diagnostic and prognostic information. Three-dimensional, time-resolved, three-directional phase contrast cardiovascular magnetic resonance (4D PC-CMR) can image flow patterns with unique detail, and using new flow visualization methods may lead to new insights. The aim of this study is to present and validate a novel visualization method with a quantitative potential for blood flow from 4D PC-CMR, called Volume Tracking, and investigate if Volume Tracking complements particle tracing, the most common visualization method used today.MethodsEight healthy volunteers and one patient with a large apical left ventricular aneurysm underwent 4D PC-CMR flow imaging of the whole heart. Volume Tracking and particle tracing visualizations were compared visually side-by-side in a visualization software package. To validate Volume Tracking, the number of particle traces that agreed with the Volume Tracking visualizations was counted and expressed as a percentage of total released particles in mid-diastole and end-diastole respectively. Two independent observers described blood flow patterns in the left ventricle using Volume Tracking visualizations.ResultsVolume Tracking was feasible in all eight healthy volunteers and in the patient. Visually, Volume Tracking and particle tracing are complementary methods, showing different aspects of the flow. When validated against particle tracing, on average 90.5% and 87.8% of the particles agreed with the Volume Tracking surface in mid-diastole and end-diastole respectively. Inflow patterns in the left ventricle varied between the subjects, with excellent agreement between observers. The left ventricular inflow pattern in the patient differed from the healthy subjects.ConclusionVolume Tracking is a new visualization method for blood flow measured by 4D PC-CMR. Volume Tracking complements and provides incremental information compared to particle tracing that may lead to a better understanding of blood flow and may improve diagnosis and prognosis of cardiovascular diseases.

Adaptivity and computational complexity in the numerical solution of ODEs

In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable τ and finding a grid deformation map, t=Θ(τ), that maps an equidistant grid {τj} to a non-equidistant grid in the original independent variable, {tj}. An optimal deformation map Θ is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method.

Logarithmic Norms and Nonlinear DAE Stability

Logarithmic norms are often used to estimate stability and perturbation bounds in linear ODEs. Extensions to other classes of problems such as nonlinear dynamics, DAEs and PDEs require careful modifications of the logarithmic norm. With a conceptual focus, we combine the extension to nonlinear ODEs [15] with that of matrix pencils [10] in order to treat nonlinear DAEs with a view to cover certain unbounded operators, i.e. partial differential algebraic equations. Perturbation bounds are obtained from differential inequalities for any given norm by using the relation between Dini derivatives and semi-inner products. Simple discretizations are also considered.

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by β-blocking, for Euler-Lagrange DAEs of index 2. Thus we may use “nonstiff” multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed “half-explicit” Runge-Kutta methods.

Global Bounds on Numerical Error for Ordinary Differential Equations

Classical upper bounds on global numerical errors are much too large for most ordinary differential systems of practical interest. The explanation is that perturbation bounds are generally poorly represented in estimates based on the Lipschitz constant. Consequently, complexity studies that are based on classical Lipschitz estimates are either exceedingly pessimistic or need be restricted to a small subset of problems of marginal relevance. In this paper we review the shortcomings of the classical bounds, explaining the reasons for their inadequacy. We present two modern alternative techniques, that produce much more realistic error bounds: the Alekseev-Grobner lemma and Lipschitz algebra bounds using the Dahlquist functional. The exposition of both techniques is illustrated with examples of their implementation, and compared with the classical approach. In the concluding section we review a new and speculative approach to approximate ordinary differential systems. This method abandons all standard concepts of time-stepping, matching of terms in a Taylor expansion, etc. Instead, the solution is derived by truncating a Dirichlet expansion. It is accompanied by a useful global error bound and yields itself to easy complexity analysis.