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139
Exponential integrators for large systems of differential equations,
- SIAM J. Sci. Comput.
, 1998
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The scaling and squaring method for the matrix exponential revisited
- SIAM REV
, 2009
"... The calculation of the matrix exponential e A maybeoneofthebestknownmatrix problems in numerical computation. It achieved folk status in our community from the paper by Moler and Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, ” published in this journal in 1978 (and revisit ..."
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Cited by 100 (20 self)
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The calculation of the matrix exponential e A maybeoneofthebestknownmatrix problems in numerical computation. It achieved folk status in our community from the paper by Moler and Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, ” published in this journal in 1978 (and revisited in this journal in 2003). The matrix exponential is utilized in a wide variety of numerical methods for solving differential equations and many other areas. It is somewhat amazing given the long history and extensive study of the matrix exponential problem that one can improve upon the best existing methods in terms of both accuracy and efficiency, but that is what the SIGEST selection in this issue does. “The Scaling and Squaring Method for the Matrix Exponential Revisited ” by N. Higham, originally published in the SIAM Journal on Matrix Analysis and Applications in 2005, applies a new backward error analysis to the commonly used scaling and squaring method, as well as a new rounding error analysis of the Padé approximant of the scaled matrix. The analysis shows, and the accompanying experimental results verify, that a Padé approximant of a higher order than currently used actually results in a more accurate
One point isometric matching with the heat kernel
- Computer Graphics Forum
"... HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 68 (4 self)
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HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Exponential integrators
, 2010
"... In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential eq ..."
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Cited by 68 (5 self)
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in
Error estimation and evaluation of matrix functions via the Faber transform
- SIAM J. Numer. Anal
"... Abstract. The need to evaluate expressions of the form f(A) orf(A)b, wheref is a nonlinear function, A is a large sparse n × n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved ..."
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Cited by 39 (13 self)
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Abstract. The need to evaluate expressions of the form f(A) orf(A)b, wheref is a nonlinear function, A is a large sparse n × n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed.
A NEW SCALING AND SQUARING ALGORITHM FOR THE MATRIX EXPONENTIAL
, 2009
"... The scaling and squaring method for the matrix exponential is based on the approximation eA ≈ (rm(2−sA)) 2s, where rm(x) is the [m/m] Padé approximant to ex and the integers m and s are to be chosen. Several authors have identified a weakness of existing scaling and squaring algorithms termed overs ..."
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Cited by 37 (23 self)
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The scaling and squaring method for the matrix exponential is based on the approximation eA ≈ (rm(2−sA)) 2s, where rm(x) is the [m/m] Padé approximant to ex and the integers m and s are to be chosen. Several authors have identified a weakness of existing scaling and squaring algorithms termed overscaling, in which a value of s much larger than necessary is chosen, causing a loss of accuracy in floating point arithmetic. Building on the scaling and squaring algorithm of Higham [SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179–1193], which is used by the MATLAB function expm, we derive a new algorithm that alleviates the overscaling problem. Two key ideas are employed. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of from powers of rm. The second idea is to base the backward error analysis that underlies the algorithm on members of the sequence {‖Ak‖1/k} instead of ‖A‖, since for nonnormal matrices it is possible that ‖Ak‖1/k is much smaller than ‖A‖, andindeed this is likely when overscaling occurs in existing algorithms. The terms ‖Ak‖1/k are estimated without computing powers of A by using a matrix 1-norm estimator in conjunction with a bound of the form ‖Ak‖1/k ≤ max ( ‖Ap‖1/p, ‖Aq‖1/q) that holds for certain fixed p and q less than k. The improvements to the truncation error bounds have to be balanced by the potential for a large ‖A‖
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 35 (6 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative
COMPUTING THE ACTION OF THE MATRIX EXPONENTIAL, WITH AN APPLICATION TO EXPONENTIAL INTEGRATORS
, 2010
"... A new algorithm is developed for computing etAB, where A is an n × n matrix and B is n×n0 with n0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n0 matrices, and the only input parameter is a backward error tolerance. The algorithm ..."
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Cited by 31 (9 self)
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A new algorithm is developed for computing etAB, where A is an n × n matrix and B is n×n0 with n0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the recent analysis of Al-Mohy and Higham [SIAM J. Matrix Anal. Appl. 31 (2009), pp. 970-989], which provides sharp truncation error bounds expressed in terms of the quantities ‖Ak‖1/k for a few values of k, where the norms are estimated using a matrix norm estimator. Shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm. Numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems, and analysis of rounding errors and of the conditioning of the problem provides theoretical support. Experimental comparisons with two Krylov-based MATLAB codes show the new algorithm to be sometimes much superior in terms of computational cost and accuracy. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form ∑p k=0 ϕk(A)uk that arise in exponential integrators, where the ϕk are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension n + p built by augmenting A with additional rows and columns, and the algorithm of this paper can therefore be employed.
Real-time Control of Physically Based Simulations using Gentle Forces
"... Figure 1: Real-time control ensures fixed simulation outcome regardless of runtime user forces: First: the rest configuration of the “T”-shape structure and the two target balls. Second: reference motion from an external simulator; the two ends of the “T ” impact the two balls. Third: user-perturbed ..."
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Cited by 29 (4 self)
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Figure 1: Real-time control ensures fixed simulation outcome regardless of runtime user forces: First: the rest configuration of the “T”-shape structure and the two target balls. Second: reference motion from an external simulator; the two ends of the “T ” impact the two balls. Third: user-perturbed real-time simulation, without control. The two ends miss the target. Fourth: controlled user-perturbed real-time simulation, with gentle control forces, tracks the reference motion and successfully impacts the target. The perturbation force load (green arrow; applied 1/5 through the simulation, only in the third and fourth motion) pushes the “T ” in the opposite direction of motion. Recent advances have brought real-time physically based simulation within reach, but simulations are still difficult to control in real time. We present interactive simulations of passive systems such as deformable solids or fluids that are not only fast, but also directable: they follow given input trajectories while simultaneously reacting to user input and other unexpected disturbances. We achieve such directability using a real-time controller that runs in tandem with a real-time physically based simulation. To avoid stiff and overcontrolled systems where the natural dynamics are overpowered, the injection of control forces has to be minimized. This search for gentle forces can be made tractable in real-time by linearizing the system dynamics around the input trajectory, and then using a time-varying linear quadratic regulator to build the controller. We show examples of controlled complex deformable solids and fluids, demonstrating that our approach generates a requested fixed outcome for reasonable user inputs, while simultaneously providing runtime motion variety.
