publications

2024

1. JCP

   Adaptive multi-stage integration schemes for Hamiltonian Monte Carlo

   Lorenzo Nagar , Mario Fernández-Pendás, Jesús María Sanz-Serna , and 1 more author

   Journal of Computational Physics, 2024

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   Hamiltonian Monte Carlo (HMC) is a powerful tool for Bayesian statistical inference due to its potential to rapidly explore high dimensional state space, avoiding the random walk behavior typical of many Markov Chain Monte Carlo samplers. The proper choice of the integrator of the Hamiltonian dynamics is key to the efficiency of HMC. It is becoming increasingly clear that multi-stage splitting integrators are a good alternative to the Verlet method, traditionally used in HMC. Here we propose a principled way of finding optimal, problem-specific integration schemes (in terms of the best conservation of energy for harmonic forces/Gaussian targets) within the families of 2- and 3-stage splitting integrators. The method, which we call Adaptive Integration Approach for statistics, or s-AIA, uses a multivariate Gaussian model and simulation data obtained at the HMC burn-in stage to identify a system-specific dimensional stability interval and assigns the most appropriate 2-/3-stage integrator for any user-chosen simulation step size within that interval. s-AIA has been implemented in the in-house software package HaiCS without introducing computational overheads in the simulations. The efficiency of the s-AIA integrators and their impact on the HMC accuracy, sampling performance and convergence are discussed in comparison with known fixed-parameter multi-stage splitting integrators (including Verlet). Numerical experiments on well-known statistical models show that the adaptive schemes reach the best possible performance within the family of 2-, 3-stage splitting schemes.

   @article{NAGAR2024112800,
     title = {Adaptive multi-stage integration schemes for Hamiltonian Monte Carlo},
     journal = {Journal of Computational Physics},
     volume = {502},
     pages = {112800},
     year = {2024},
     issn = {0021-9991},
     doi = {https://doi.org/10.1016/j.jcp.2024.112800},
     author = {Nagar, Lorenzo and Fernández-Pendás, Mario and Sanz-Serna, Jesús María and Akhmatskaya, Elena},
     keywords = {Hamiltonian Monte Carlo, Multi-stage integrators, Adaptive integration, Bayesian inference, Stability limit, Velocity Verlet}
   }

1967

1. 

   Letters on wave mechanics

   Albert Einstein , Erwin Schrödinger, Max Planck, and 2 more authors

   1967

   Bib

   @book{przibram1967letters,
     title = {Letters on wave mechanics},
     author = {Einstein, Albert and Schrödinger, Erwin and Planck, Max and Lorentz, Hendrik Antoon and Przibram, Karl},
     year = {1967},
     publisher = {Vision},
   }

1956

1. 

   Investigations on the Theory of the Brownian Movement

   Albert Einstein

   1956

   Bib

   @book{einstein1956investigations,
     title = {Investigations on the Theory of the Brownian Movement},
     author = {Einstein, Albert},
     year = {1956},
     publisher = {Courier Corporation},
   }

1950

1. AJP

   The meaning of relativity

   Albert Einstein , and AH Taub

   American Journal of Physics, 1950

   Bib

   @article{einstein1950meaning,
     title = {The meaning of relativity},
     author = {Einstein, Albert and Taub, AH},
     journal = {American Journal of Physics},
     volume = {18},
     number = {6},
     pages = {403--404},
     year = {1950},
     publisher = {American Association of Physics Teachers}
   }

1935

1. PhysRev

   Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

   A. Einstein , B. Podolsky, and N. Rosen

   Phys. Rev., May 1935

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   In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.