On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

In Markov Chain Monte Carlo (MCMC) simulations, the thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for the MCMC method but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them

Dispersal of larval and juvenile seabream: Implications for Mediterranean marine protected areas

In the marine context, information about dispersal is essential for the design of networks of marine protected areas (MPAs). Generally, most of the dispersal of demersal fishes is thought to be driven by the transport of eggs and larvae in currents, with the potential contribution of dispersal in later life stages relatively minimal.Using otolith chemistry analyses, we estimate dispersal patterns across a spatial scale of approximately 180. km at both propagule (i.e. eggs and larvae) and juvenile (i.e. between settlement and recruitment) stages of a Mediterranean coastal fishery species, the two-banded seabream Diplodus vulgaris. We detected three major natal sources of propagules replenishing local populations in the entire study area, suggesting that propagule dispersal distance extends to at least 90. km. For the juvenile stage, we detected dispersal of up to 165. km. Our work highlights the surprising and significant role of dispersal during the juvenile life stages as an important mechanism connecting populations. Such new insights are crucial for creating effective management strategies (e.g. MPAs and MPA networks) and to gain support from policymakers and stakeholders, highlighting that MPA benefits can extend well beyond MPA borders, and not only via dispersal of eggs and larvae, but also through movement by juveniles

Patterns of variability in early life traits of a Mediterranean coastal fish

Spawning dates and pelagic larval duration (PLD) are early life traits (ELT) crucial for understanding life cycles, properly assessing patterns of connectivity and gathering indications about patchiness or homogeneity of larval pools. Considering that little attention has been paid to spatial variability in these traits, we investigated variability of ELT from the analysis of otolith microstructure in the common two-banded sea bream Diplodus vulgaris. In the southwestern Adriatic Sea, along ~200 km of coast (∼1° in latitude, 41.2° to 40.2°N), variability of ELT was assessed at multiple spatial scales. Overall, PLD (ranging from 25 to 61 d) and spawning dates (October 2009 to February 2010) showed significant variability at small scales (i.e. <6 km), but not at larger scales. These outcomes suggest patchiness of the larval pool at small spatial scales. Multiple causal processes underlying the observed variability are discussed, along with the need to properly consider spatial variability in ELT, for example when delineating patterns of connectivity. Copyright © 2013 Inter-Research

An introduction to a porous shape memory alloy dynamic data driven application system

Shape Memory Alloys are capable of changing their crystallographic structure due to changes of temperature and/or stress. Our research focuses on three points: (1) Iterative Homogenization of Porous SMAs: Development of a Multiscale Model of porous SMAs utilizing iterative homogenization and based on existing knowledge of constitutive modeling of polycrystalline SMAs. (2) DDDAS: Develop tools to turn on and off the sensors and heating unit(s), to monitor on-line data streams, to change scales based on incoming data, and to control what type of data is generated. The application must have the capability to be run and steered remotely. (3) Modeling and applications of porous SMA: Vibration isolation devices with SMA and porous SMA components for aerospace applications will be analyzed and tested. Numerical tools for modeling porous SMAs with a second viscous phase will be developed. The outcome will be a robust, three-dimensional, multiscale model of porous SMA that can be used in complicated, real-life structural analysis of SMA components using a DDDAS framework. © 2012 Published by Elsevier Ltd

Modeling phase-transitions using a high-performance, isogeometric analysis framework

In this paper, we present a high-performance framework for solving partial differential equations using Isogeometric Analysis, called PetIGA, and show how it can be used to solve phase-field problems. We specifically chose the Cahn-Hilliard equation, and the phase-field crystal equation as test cases. These two models allow us to highlight some of the main advantages that we have access to while using PetIGA for scientific computing. © The Authors. Published by Elsevier B.V

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection-diffusion reaction problems

A continuum theory for mineral solid solutions undergoing chemo-mechanical processes

Recent studies on metamorphic petrology as well as microstructural observations suggest the influence of mechanical effects upon chemically active metamorphic minerals. Thus, the understanding of such a coupling is crucial to describe the dynamics of geomaterials. In this effort, we derive a thermodynamically-consistent framework to characterize the evolution of chemically active minerals. We model the metamorphic mineral assemblages as a solid-species solution where the species mass transport and chemical reaction drive the stress generation process. The theoretical foundations of the framework rely on modern continuum mechanics, thermodynamics far from equilibrium, and the phase-field model. We treat the mineral solid solution as a continuum body, and following the Larch\'e and Cahn network model, we define displacement and strain fields. Consequently, we obtain a set of coupled chemo-mechanical equations. We use the aforementioned framework to study single minerals as solid solutions during metamorphism. Furthermore, we emphasise the use of the phase-field framework as a promising tool to model complex multi-physics processes in geoscience. Without loss of generality, we use common physical and chemical parameters found in the geoscience literature to portrait a comprehensive view of the underlying physics. Thereby, we carry out 2D and 3D numerical simulations using material parameters for metamorphic minerals to showcase and verify the chemo-mechanical interactions of mineral solid solutions that undergo spinodal decomposition, chemical reactions, and deformation

A continuum theory for mineral solid solutions undergoing chemo-mechanical processes

Recent studies on metamorphic petrology as well as microstructural observations suggest the influence of mechanical effects upon chemically active metamorphic minerals. Thus, the understanding of such a coupling is crucial to describe the dynamics of geomaterials. In this effort, we derive a thermodynamically consistent framework to characterize the evolution of chemically active minerals. We model the metamorphic mineral assemblages as a solid-species solution where the species mass transport and chemical reaction drive the stress generation process. The theoretical foundations of the framework rely on modern continuum mechanics, thermodynamics far from equilibrium, and the phase-field model. We treat the mineral solid solution as a continuum body, and following the Larché and Cahn network model, we define displacement and strain fields. Consequently, we obtain a set of coupled chemo-mechanical equations. We use the aforementioned framework to study single minerals as solid solutions during metamorphism. Furthermore, we emphasise the use of the phase-field framework as a promising tool to model complex multi-physics processes in geoscience. Without loss of generality, we use common physical and chemical parameters found in the geoscience literature to portrait a comprehensive view of the underlying physics. Thereby, we carry out 2D and 3D numerical simulations using material parameters for mineral solid solutions to showcase and verify the chemo-mechanical interactions of mineral solid solutions that undergo spinodal decomposition, chemical reactions, and deformation.journal articl