Probabilistic ODE solvers with Runge-Kutta means

Advances in Neural Information Processing Systems

Volumn 1, Issue January, 2014, Pages 739-747

  Schober, Michael ^a   Duvenaud, David ^b   Hennig, Philipp ^a  

^a MAX PLANCK INSTITUTE FOR INTELLIGENT SYSTEMS   (Germany)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); DIFFERENTIAL EQUATIONS; INFORMATION SCIENCE; MARKOV PROCESSES; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PROBABILITY DISTRIBUTIONS;

COMPUTATIONAL COSTS; GAUSS-MARKOV PROCESS; POINT ESTIMATE; POSTERIOR MEANS; POSTERIOR PROBABILITY; PROBABILISTIC OUTPUT; RESEARCH QUESTIONS; STATE OF THE ART;

RUNGE KUTTA METHODS;

EID: 84937116034     PISSN: 10495258     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (21)

1. T. Graepel. "Solving noisy linear operator equations by Gaussian processes: Application to ordinary and partial differential equations". In: International Conference on Machine Learning (ICML). 2003.
2. B. Calderhead, M. Girolami, and N. Lawrence. "Accelerating Bayesian inference over nonlinear differential equations with Gaussian processes." In: Advances in Neural Information Processing Systems (NIPS). 2008.
3. F. Dondelinger et al. "ODE parameter inference using adaptive gradient matching with Gaussian processes". In: Artificial Intelligence and Statistics (AISTATS). 2013, pp. 216-228.
4. Y. Wang and D. Barber. "Gaussian Processes for Bayesian Estimation in Ordinary Differential Equations". In: International Conference on Machine Learning (ICML). 2014.
5. J. Skilling. "Bayesian solution of ordinary differential equations". In: Maximum Entropy and Bayesian Methods, Seattle (1991).
6. P. Hennig and S. Hauberg. "Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics". In: Proc. of the 17th int. Conf. on Artificial Intelligence and Statistics (AISTATS). Vol. 33. JMLR, W&CP, 2014.
7. M. Schober et al. "Probabilistic shortest path tractography in DTI using Gaussian Process ODE solvers". In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2014. Springer, 2014.
8. O. Chkrebtii et al. "Bayesian Uncertainty Quantification for Differential Equations". In: arXiv prePrint 1306.2365 (2013).
9. C. Runge. "Über die numerische Auflösung von Differentialgleichungen". In: Mathematische Annalen 46 (1895), pp. 167-178.
10. W. Kutta. "Beitrag zur näherungsweisen Integration totaler Differentialgleichungen". In: Zeitschrift für Mathematik und Physik 46 (1901), pp. 435-453.
11. E. Hairer, S. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I - Nonstiff Problems. Springer, 1987.
12. J. R. Dormand and P. J. Prince. "A family of embedded Runge-Kutta formulae". In: Journal of computational and applied mathematics 6.1 (1980), pp. 19-26.
13. J. Butcher. "Coefficients for the study of Runge-Kutta integration processes". In: Journal of the Australian Mathematical Society 3.02 (1963), pp. 185-201.
14. F. Ceschino and J. Kuntzmann. Problèmes différentiels de conditions initiales (méthodes numériques). Dunod Paris, 1963.
15. E. B. Shanks. "Solutions of Differential Equations by Evaluations of Functions". In: Mathematics of Computation 20.93 (1966), pp. 21-38.
16. E. Hairer and C. Lubich. "Numerical solution of ordinary differential equations". In: The Princeton Companion to Applied Mathematics, ed. by N. Higham. PUP, 2012.
17. N. Wiener. "Extrapolation, interpolation, and smoothing of stationary time series with engineering applications". In: Bull. Amer. Math. Soc. 56 (1950), pp. 378-381.
18. S. Särkkä. Bayesian filtering and smoothing. Cambridge University Press, 2013.
19. C. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. MIT, 2006.
20. R. Shumway and D. Stoffer. "An approach to time series smoothing and forecasting using the EM algorithm". In: Journal of time series analysis 3.4 (1982), pp. 253-264.
21. Z. Ghahramani and G. Hinton. Parameter estimation for linear dynamical systems. Tech. rep. Technical Report CRG-TR-96-2, University of Totronto, Dept. of Computer Science, 1996.