Symmetrically processed splitting integrators for enhanced hamiltonian monte carlo sampling

We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard leapfrog/Verlet without impairing in any way the quality of the samples. They are based on a suitable modification of the processing technique first introduced by Butcher. The idea of modified processing may also be useful for other purposes, like the construction of high-order splitting integrators with positive coefficients

DIRK Schemes with High Weak Stage Order

Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples.Comment: 10 pages, 5 figure

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

High order structure preserving explicit methods for solving linear-quadratic optimal control problems

[EN] We consider the numerical integration of linear-quadratic optimal control problems. This problem requires the solution of a boundary value problem: a non-autonomous matrix Riccati differential equation (RDE) with final conditions coupled with the state vector equation with initial conditions. The RDE has positive definite matrix solution and to numerically preserve this qualitative property we propose first to integrate this equation backward in time with a sufficiently accurate scheme. Then, this problem turns into an initial value problem, and we analyse splitting and Magnus integrators for the forward time integration which preserve the positive definite matrix solutions for the RDE. Duplicating the time as two new coordinates and using appropriate splitting methods, high order methods preserving the desired property can be obtained. The schemes make sequential computations and do not require the storrage of intermediate results, so the storage requirements are minimal. The proposed methods are also adapted for solving linear-quadratic N-player differential games. The performance of the splitting methods can be considerably improved if the system is a perturbation of an exactly solvable problem and the system is properly split. Some numerical examples illustrate the performance of the proposed methods.The author wishes to thank the University of California San Diego for its hospitality where part of this work was done. He also acknowledges the support of the Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03. The author also acknowledges the suggestions by the referees to improve the presentation of this work.Blanes Zamora, S. (2015). High order structure preserving explicit methods for solving linear-quadratic optimal control problems. Numerical Algorithms. 69:271-290. https://doi.org/10.1007/s11075-014-9894-0S27129069Abou-Kandil, H., Freiling, G., Ionescy, V., Jank, G.: Matrix Riccati equations in control and systems theory. Basel, Burkhäuser Verlag (2003)Al-Mohy, A.H., Higham, N.J.: Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators. SIAM. J. Sci. Comp. 33, 488–511 (2011)Anderson, B.D.O., Moore, J.B.: Optimal control: linear quadratic methods. Dover, New York (1990)Ascher, U.M., Mattheij, R.M., Russell, R.D.: Numerical solutions of boundary value problems for ordinary differential equations. Prentice-Hall, Englewood Cliffs (1988)Bader, P., Blanes, S., Ponsoda, E.: Structure preserving integrators for solving linear quadratic optimal control problems with applications to describe the flight of a quadrotor. J. Comput. Appl. Math. 262, 223–233 (2014)Basar, T., Olsder, G.J.: Dynamic non cooperative game theory, 2nd Ed, SIAM, Philadelphhia (1999)Blanes, S., Casas, F.: On the necessity of negative coefficients for operator splitting schemes of order higher than two. Appl. Num. Math. 54, 23–37 (2005)Blanes, S., Casas, F., Farrés, A., Laskar, J., Makazaga, J., Murua, A.: New families of symplectic splitting methods for numerical integration in dynamical astronomy. Appl. Numer. Math. 68, 58–72 (2013)Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)Blanes, S., Casas, F., Ros, J.: High order optimized geometric integrators for linear differential equations. BIT 42, 262–284 (2002)Blanes, S., Diele, F., Marangi, C., Ragni, S.: Splitting and composition methods for explicit time dependence in separable dynamical systems. J. Comput. Appl. Math. 235, 646–659 (2010)Blanes, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nystrm methods. J. Comput. Appl. Math. 142, 313–330 (2002)Blanes, S., Ponsoda, E.: Magnus integrators for solving linear-quadratic differential games. J. Comput. Appl. Math. 236, 3394–3408 (2012)Brif, C., Chakrabarti, R., Rabitz, H.: Control of quantum phenomena: past, present and future. New J. Phys. 12, 075008(68pp) (2010)Cruz, J.B., Chen, C.I.: Series Nash solution of two person non zero sum linear quadratic games. J. Optim. Theory Appl. 7, 240–257 (1971)Dieci, L., Eirola, T.: Positive definitness in the numerical solution of Riccati differential quations. Numer. Math. 67, 303–313 (1994)Engwerda, J.: LQ dynamic optimization and differential games. Wiley (2005)Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations (2nd edition). Springer Series in Computational Mathematics, 31. Springer-Verlag (2006)Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286 (2010)Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (1985)Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie group methods. Acta Numerica 9, 215–365 (2000)Iserles, A., Nørsett, S.P.: On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc. Lond. A 357, 983–1019 (1999)Jódar, L., Ponsoda, E.: Non-autonomous Riccati-type matrix differential equations: existence interval, construction of continuous numerical solutions and error bounds. IMA. J. Num. Anal. 15, 61–74 (1995)Jódar, L., Ponsoda, E., Company, R.: Solutions of coupled Riccati equations arising in differential games. Control. Cybern. 24, 117–128 (1995)Kaitala, V, Pohjola, M. In: Carraro, Filar (eds.) : Sustainable international agreement on greenhouse warming. A game theory study. Control and Game Theoretic Models of the Environment, pp 67–87. Birkhauser, Boston (1995)Keller, H.B.: Numerical solution of two point boundary value problems. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 24. SIAM, Philadelphia (1976)McLachlan, R.I.: Composition methods in the presence of small parameters. BIT 35, 258–268 (1995)McLachlan, R.I., Quispel, R.: Splitting Methods. Acta Numer. 11, 341–434 (2002)Moler, C.B., Van Loan, C.F.: Nineteen Dubious Ways to Compute the Exponential of a Matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)Na, T.Y.: Computational methods in engineering boundary value problems. In: Mathematics in Science and Engineering, Vol. 145. Accademic Press, New York (1979)Palao, J.P., Kosloff, R.: Quantum computing by an optimal control algorithm for unitry transformations. Phys. Rev. Lett. 28 (2002)Peirce, A.P., Dahleh, M.A., Rabitz, H.: Optimal control of quantum-mechanical systems: existence, numerical approximation, and applications. Phys. Rev. A 37, 4950–4967 (1988)Reid, W.T.: Riccati Differential Equations. Academic, New York (1972)Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Software 24, 130–156 (1998)Speyer, J.L., Jacobson, D.H.: Primer on optimal control theory. SIAM, Philadelphia (2010)Starr, A.W., Ho, Y.C.: Non-zero sum differential games. J. Optim. Theory and Appl 3, 179–197 (1969)Zhu, W., Rabitz, H.: A rapid monotonically convergent iteration algorithm for quantum optimal control ever the expectation value of a positive definite operator. J. Chem. Phys. 109, 385–391 (1998

Emergency hospital services utilization in Lleida (Spain): A cross-sectional study of immigrant and Spanish-born populations

<p>Abstract</p> <p>Background</p> <p>The use of emergency hospital services (EHS) has increased steadily in Spain in the last decade while the number of immigrants has increased dramatically. Studies show that immigrants use EHS differently than native-born individuals, and this work investigates demographics, diagnoses and utilization rates of EHS in Lleida (Spain).</p> <p>Methods</p> <p>Cross-sectional study of all the 96,916 EHS visits by patients 15 to 64 years old, attended during the years 2004 and 2005 in a public teaching hospital. Demographic data, diagnoses of the EHS visits, frequency of hospital admissions, mortality and diagnoses at hospital discharge were obtained. Utilization rates were estimated by group of origin. Poisson regression was used to estimate the rate ratios of being visited in the EHS with respect to the Spanish-born population.</p> <p>Results</p> <p>Immigrants from low-income countries use EHS services more than the Spanish-born population. Differences in utilization patterns are particularly marked for Maghrebi men and women and sub-Saharan women. Immigrant males are at lower risk of being admitted to the hospital, as compared with Spanish-born males. On the other hand, immigrant women are at higher risk of being admitted. After excluding the visits with gynecologic and obstetric diagnoses, women from sub-Saharan Africa and the Maghreb are still at a higher risk of being admitted than their Spanish-born counterparts.</p> <p>Conclusion</p> <p>In Lleida (Spain), immigrants use more EHS than the Spanish born population. Future research should indicate whether the same pattern is found in other areas of Spain and whether EHS use is attributable to health needs, barriers to access to the primary care services or similarities in the way immigrants access health care in their countries of origin.</p

Split Hamiltonian Monte Carlo revisited

We study Hamiltonian Monte Carlo (HMC) samplers based on splitting the Hamiltonian H as H0(θ , p)+U1(θ ), where H0 is quadratic and U1 small. We show that, in general, such samplers suffer from stepsize stability restrictions similar to those of algorithms based on the standard leapfrog integrator. The restrictions may be circumvented by preconditioning the dynamics. Numerical experiments show that, when the H0(θ , p) + U1(θ ) splitting is combined with preconditioning, it is possible to construct samplers far more efficient than standard leapfrog HMC.Funding for open access charge: CRUE-Universitat Jaume