Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms

We present a mathematical framework for constructing and analyzing parallel algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. The algorithms can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms are controlled-error approximations of kinetic Monte Carlo algorithms, departing from the predominant paradigm of creating parallel KMC algorithms with exactly the same master equation as the serial one. Our methodology relies on a spatial decomposition of the Markov operator underlying the KMC algorithm into a hierarchy of operators corresponding to the processors' structure in the parallel architecture. Based on this operator decomposition, we formulate Fractional Step Approximation schemes by employing the Trotter Theorem and its random variants; these schemes, (a) determine the communication schedule} between processors, and (b) are run independently on each processor through a serial KMC simulation, called a kernel, on each fractional step time-window. Furthermore, the proposed mathematical framework allows us to rigorously justify the numerical and statistical consistency of the proposed algorithms, showing the convergence of our approximating schemes to the original serial KMC. The approach also provides a systematic evaluation of different processor communicating schedules.Comment: 34 pages, 9 figure

Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting

The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity

Part 2: pushing the envelope. A process perspective for architecture, engineering and construction

In this article, I am building on an emerging 'process view of nature' and how biological membranes emerge through the combined action of (locally) autonomous construction agents. In Part 1, we considered the simultaneous aggregation and disaggregation of matter around embedded processes, used to create, sustain and regulate matter, energy and information gradients from which 'work' is derived for the benefit of the agents or organisms present in the system. In Part 2, I intend to demonstrate that emerging digital design, simulation and fabrication techniques, when linked to sensory and effector feedback, memory and actions, directed by pre-encoded objectives (as rules or algorithms), produce the same fundamental unit of 'agency' as biological agents possess. By understanding how biological membranes emerge in nature, as the outcome of 'negotiated agency', to regulate matter, energy and information exchange between adjacent spaces, we can begin to consider the building envelope as a biological interface or membrane from which 'work' can be derived from the environment we inhabit, as a physiological extension of ourselves

Part 1: a process view of nature. Multifunctional integration and the role of the construction agent

This is the first of two linked articles which draw s on emerging understanding in the field of biology and seeks to communicate it to those of construction, engineering and design. Its insight is that nature 'works' at the process level, where neither function nor form are distinctions, and materialisation is both the act of negotiating limited resource and encoding matter as 'memory', to sustain and integrate processes through time. It explores how biological agents derive work by creating 'interfaces' between adjacent locations as membranes, through feedback. Through the tension between simultaneous aggregation and disaggregation of matter by agents with opposing objectives, many functions are integrated into an interface as it unfolds. Significantly, biological agents induce flow and counterflow conditions within biological interfaces, by inducing phase transition responses in the matte r or energy passing through them, driving steep gradients from weak potentials (i.e. shorter distances and larger surfaces). As with biological agents, computing, programming and, increasingly digital sensor and effector technologies share the same 'agency' and are thus convergent