Updating the hamiltonian problem
The HMC uses the molecular dynamics (MD) steps as the global Monte Carlo (MC) moves to reach areas of high probability where the gradient of the log-density of the Posterior acts as a guide during the search process.However, the acceptance rate of HMC is sensitive to the system size as well as the time step used to evaluate the MD trajectory.
Recent research in the field of finite element model updating (FEM) advocates the adoption of Bayesian analysis techniques to dealing with the uncertainties associated with these models.To send this article to your Kindle, first ensure [email protected] added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Note you can select to send to either the @free.or @variations.Then enter the ‘name’ part of your Kindle email address below. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.Let's have a slightly more general starting expression: $H= a \frac b m c^2$, where $a$ and $b$ are \times2$ matrices whose properties are to be determined.Square $H$ to get: $H^2 = a^2 \left(\frac\right)^2 \ \frac mc^2 b^2 m^2c^4$.However, Bayesian formulations require the evaluation of the Posterior Distribution Function which may not be available in analytical form. In such cases sampling methods can provide good approximations of the Posterior distribution when implemented in the Bayesian context.Markov Chain Monte Carlo (MCMC) algorithms are the most popular sampling tools used to sample probability distributions.If we can find $a$ and $b$ such to have $a^2=0$ $\=2\cdot$ (here $\:\equiv A B B A$) and $b^2=$ (and $$ is the unit matrix), then we would get the original dispersion equation for a free relativistic particle: $H^2 = p^2c^2 m^2c^4$.One such a choice is $a= \tau_3 i\tau_2$ and $b=\tau_3$, but there are many other choices, too.This is done by sampling from a modified Hamiltonian function instead of the normal Hamiltonian function.In this paper, the efficiency and accuracy of the SHMC method is tested on the updating of two real structures; an unsymmetrical H-shaped beam structure and a GARTEUR SM-AG19 structure and is compared to the application of the HMC algorithm on the same structures.