Generalizing Parallel Replica Dynamics: Trajectory Fragments, Asynchronous Computing, and PDMPs

Abstract

We study the Parallel Replica Dynamics in a general setting. We introduce a trajectory fragment framework that can be used to design and prove consistency of Parallel Replica algorithms for generic Markov processes. We use our framework to formulate a novel condition that guarantees an asynchronous algorithm is consistent. Exploiting this condition and our trajectory fragment framework, we present new synchronous and asynchronous Parallel Replica algorithms for piecewise deterministic Markov processes. ) 2. Notation. Throughout, X(t) t≥0 is a time homogeneous Markov process, either discrete or continuous in time, with values in a standard Borel state space; U is a subset of state space; and g is a real-valued function defined on state space. Without explicit mention we assume all sets are measurable and all functions are bounded and measurable. We write X(t) to refer to the process X(t) t≥0 at time t. We denote various expectations and probabilities by E and P, with the precise meaning being clear from context. We write L for the probability law of a random object, with L above an equals sign indicating equality in law. We say a random object is a copy of another random object if it has the same law as that object. When we say a collection of random objects is independent we mean these objects are mutually independent unless otherwise specified. We define a ∧ b = min{a, b} and a ∨ b = max{a, b}, and write ⌊s⌋ for the greatest integer less than or equal to s. Metastability.Informally, U is a metastable set for X(t) t≥0 if X(t) t≥0 tends to reach a local equilibrium in U much faster than it escapes from U . Local equilibrium can be understood in terms of quasistationary distributions (QSDs):Definition 3.1. Fix a subset U of state space, and considerfor every t ≥ 0 and A ⊆ U .Note that ρ is supported in U . Equation (3.1) states that if X(0) is distributed as ρ and X(t) t≥0 does not escape from U by time t, then X(t) is distributed as ρ. Throughout, we will assume the QSD of X(t) t≥0 in U exists, is unique, and is the long time distribution of X(t) conditioned to never escape U . That is, we assume that for any initial distribution of X(0) supported in U , (3.2) ρ(A) = lim t→∞ P(X(t) ∈ A|X(s) ∈ U for s ∈ [0, t]) ∀ A ⊆ U.The QSD ρ can then be sampled as follows: choose a time T ρ corr (U ) for relaxation to ρ. Start X(t) t≥0 in U , and if it escapes from U before time t = T ρ corr (U ), restart it

Exploiting model uncertainty to improve the scalability of long-time simulations using Parallel Trajectory Splicing

“…With this conceptual picture of ParSplice in mind, we can clearly distinguish the respective sources of accuracy and efficiency. The ParSplice formalism guarantees that splicing any independently-generated set of segments produces a statistically correct state-to-state trajectory so long as τ c is chosen properly and that the order in which segments are spliced is proper [18]. Many different segment orderings are possible in practice, as discussed in reference [18].…”

“…The ParSplice formalism guarantees that splicing any independently-generated set of segments produces a statistically correct state-to-state trajectory so long as τ c is chosen properly and that the order in which segments are spliced is proper [18]. Many different segment orderings are possible in practice, as discussed in reference [18]. In short, the order can be arbitrary so long as it is independent of the content of the segment (e.g., of its length, of whether or not it contains a transition, etc).…”

Resource allocation for task-level speculative scientific applications: a proof of concept using Parallel Trajectory Splicing

“…These segments are then returned to a database where they are stored until they can be spliced. Due to the specially-designed protocol by which segments are produced and stored 47 , any segment in the database can be spliced onto any other so long as it began in the same state that the other finished (see Figure 3). This allows for a single state-to-state trajectory to be formed by extracting individual segments from the database and splicing them onto the end of the trajectory.…”