Linear Analysis of Stochastic Verlet-Type Integrators for Langevin Equations

One of the most basic struggles in simulations of statistical and dynamical systems is how to appropriately balance the desire for simulation accuracy, obtained in the small time step limit, with simulation efficiency, obtained for large time steps. Obviously, we would like to have the benefits of both limits of accuracy and efficiency. Thus, understanding the influence of discrete time on the behavior of equations of motion is crucial for the understanding and optimization of numerical simulations in physical science and engineering. There are two different types of crucial discretization errors. One is that the simulated, discrete-time trajectory increasingly deviates from the true, continuous-time trajectory as the time step is increased. Another is that the simulated discrete-time velocity increasingly deviates from the expected velocity of the corresponding simulated trajectory as the time step is increased. While the former error is a signature that the simulated trajectory is not exactly the intended one, the simulation may still be representative of the true system. In contrast, the latter error is a signature of a fundamental internal inconsistency in the simulation, which may greatly complicate physical interpretation of the simulation results, especially when both configurational and kinetic measures are required. It is therefore crucially important to understand the relationship between a discrete-time coordinate and its corresponding velocity.It has previously been shown that, unlike in continuous-time, discrete-time configurational dynamics does not need a discrete-time velocity variable. This realization allowed for the derivation of the complete configurational GJ set of stochastic Verlet-type algorithms for simulating the Langevin equation. This set has the key feature of simultaneously giving exact, time-step independent statistical sampling of damped, noisy harmonic oscillators, and giving exact spatial transport values, drift and Einstein diffusion, on flat surfaces. The apparent fact that a discrete-time velocity is not required to simulate a coordinate trajectory implies that discrete-time velocity is an ambiguous concept, which, in turn, implies that multiple velocity definitions can be entertained in order to avoid internal simulation inconsistencies between configurational and kinetic measures. From the derivation of the stochastic GJ methods it was shown that a statistically correct velocity obtained at the same time as the coordinate (an on-site velocity) cannot be obtained. In contrast, a velocity at the half-step between consecutive configurational time steps (a half-step velocity) was identified to produce exact statistical kinetic response for harmonic oscillators regardless of the applied time step. However, only for a single GJ method does this velocity also give the correct drift value.We here derive new, practical velocity definitions that are optimized for statistical kinetic measures, such that, e.g., measures of kinetic energy and correlations with the configurational coordinate are exact for the noisy harmonic oscillator, while also producing the correct kinetic measure of transport on flat surfaces. We identify and analyze three half-step velocity definitions that fulfill these criteria, and we outline simple and practical Verlet-type algorithms that incorporate these velocities within the statistically desirable GJ format that provides the optimized configurational statistics. The application of the methods are demonstrated through characteristic Molecular Dynamics simulations of both simple and complex systems, illustrating how the resulting methods are able to efficiently and simultaneously simulate configurational and kinetic statistical ensembles in different situations using very large time steps.

Data mining with a multitude of methodologies is a good basis for the integration of intelligent systems. Small, specialised systems have a large number of feasible solutions, but developing truly adaptive, and still understandable, systems for highly complex systems require domain expertise and more compact approaches at the basic level. This paper focuses on the integration of methodologies in the smart adaptive applications. Statistical methods and arti cial neural networks form a good basis for the data-driven analysis of interactions and fuzzy logic introduces solutions for knowledge-based understanding the system behaviour and the meaning of variable levels. Efficient normalisation, scaling and decomposition approaches are the key methodologies in developing large-scale applications. Linguistic equation (LE) approach originating from fuzzy logic is an efficient technique for these problems. The nonlinear scaling methodology based on advanced statistical analysis is the corner stone in representing the variable meanings in a compact way to introduce intelligent indices for control and diagnostics. The new constraint handling together with generalised norms and moments facilitates recursive parameter estimation approaches for the adaptive scaling. Well-known linear methodologies are used for the steady state, dynamic and case-based modelling in connection with the cascade and interactive structures in building complex large scale applications. To achieve insight and robustness the parameters are de ned separately for the scaling and the interactions. Introduction EK Juuso Intelligent Methods in Modelling and Simulation of Complex Systems 2 SNE 24(1) – 4/2014 ON 1 Steady-State Modelling multiple input, multiple output MIMO response surface methodology RSM multiple input, single output MISO Fuzzy set theory Extension principle Type-2 fuzzy Figure 1: Methodologies for modelling of complex system. EK Juuso Intelligent Method of Modelling and Simulation in Complex Systems SNE 24(1) – 4/2014 3 O N Linguistic fuzzy models Takagi-Sugeno (TS) fuzzy models Singleton models Fuzzy relational models Figure 2: Combined fuzzy modelling. Artificial neural networks ANN Linear networks multilayer perceptron MLP backpropagation learning Neurofuzzy systems Figure 3: A fuzzy neuron. function expansion Approximate reasoning 2 Decomposition Methodologies EK Juuso Intelligent Methods in Modelling and Simulation of Complex Systems 4 SNE 24(1) – 4/2014 ON Decomposition Figure 4: Decomposition for modelling. Hierarchical clustering Partitioning-based clustering algorithms fuzzy clustering Fuzzy c-means (FCM) Subtractive clustering Neural clustering Robust clustering number of clusters Composite local model Linear parameter varying (LPV) models Piecewise affine (PWA) systems Fuzzy models Multiple neural network systems mixed approach EK Juuso Intelligent Method of Modelling and Simulation in Complex Systems SNE 24(1) – 4/2014 5 O N 3 Adaptive Nonlinear Scaling Membership 3.1 Working point and feasible ranges Figure 5: Nonlinear scaling [28] EK Juuso Intelligent Methods in Modelling and Simulation of Complex Systems 6 SNE 24(1) – 4/2014 ON 3.2 Membership definitions linguistic range linguistic values • the corner points (Figure 5) are good for visualisation; • the parameters suit for tuning; • the coefficients are used in the calculations. 3.3 Adaptation of nonlinear scaling EK Juuso Intelligent Method of Modelling and Simulation in Complex Systems SNE 24(1) – 4/2014 7 O N 4 Intelligent Systems 4.1 LE models 4.2 Hybrid LE systems EK Juuso Intelligent Methods in Modelling and Simulation of Complex Systems 8 SNE 24(1) – 4/2014 ON extension principle linguistification

The context of our work is the complex systems simulation. Complex systems modelling and simulation require to take into account the interactions between the different components. For instance, an airport ground traffic simulation requires to model the planes with the pilots, the control tower, the

This study examines the performance of integration methods for hybrid simulation of large and complex structural systems in the context of structural collapse due to seismic excitations. The target application is not necessarily for real-time testing, but rather for models that involve large-scale physical sub-structures and highly nonlinear numerical models. Four case studies are presented and discussed. In the first case study, the accuracy of integration schemes including two widely used methods, namely, modified version of the implicit Newmark with fixed-number of iteration (iterative) and the operator-splitting (non-iterative) is examined through pure numerical simulations. The second case study presents the results of 10 hybrid simulations repeated with the two aforementioned integration methods considering various time steps and fixed-number of iterations for the iterative integration method. The physical sub-structure in these tests consists of a single-degree-of-freedom (SDOF) cantilever column with replaceable steel coupons that provides repeatable highlynonlinear behavior including fracture-type strength and stiffness degradations. In case study three, the implicit Newmark with fixed-number of iterations is applied for hybrid simulations of a 1:2 scale steel moment frame that includes a relatively complex nonlinear numerical substructure. Lastly, a more complex numerical substructure is considered by constructing a nonlinear computational model of a moment frame coupled to a hybrid model of a 1:2 scale steel gravity frame. The last two case studies are conducted on the same porotype structure and the selection of time steps and fixed number of iterations are closely examined in pre-test simulations. The generated unbalance forces is used as an index to track the equilibrium error and predict the accuracy and stability of the simulations.

In this study, we present a new molecular dynamics program for simulation of complex molecular systems. The program, named ORAC, combines . state-of-the-art molecular dynamics MD algorithms with flexibility in handling different types and sizes of molecules. ORAC is intended for simulations of molecular systems and is specifically designed to treat biomolecules efficiently and effectively in solution or in a crystalline environment. Among its unique . features are: i implementation of reversible and symplectic multiple time step . algorithms or r-RESPA, reversible reference system propagation algorithm specifically designed and tuned for biological systems with periodic boundary . conditions; ii availability for simulations with multiple or single time steps of . standard Ewald or smooth particle mesh Ewald SPME for computation of . electrostatic interactions; and iii possibility of simulating molecular systems in a variety of thermodynamic ensembles. We believe that the combination of these algorithms makes ORAC more advanced than other MD programs using standard simulation algorithms. Q 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1848)1862, 1997

Complexity science offers many theories such as chaos theory and coevolutionary theory. These theories illustrate a large set of real life systems and help decipher their nonlinear and unpredictable behaviours. Categorizing an observed Complex System among these theories depends on the aspect that we intend to study, and it can help better understand the phenomena that occur within the system. This article aims to give an overview on Complex Systems and their modelling. Therefore, we compare these theories based on their main behavioural characteristics, e.g. emergence, adaptability, and dynamism. Then we compare the methods used in the literature to model and simulate Complex Systems, and we propose and discuss simple guidelines to help understand one's Complex System and choose the most adequate model to simulate it.

Controllability of linear and nonlinear systems governed by Stieltjes differential equations

The translational motion of anisotropic or self-propelled colloidal particles is closely linked with the particle’s orientation and its rotational Brownian motion. In the overdamped limit, the stochastic evolution of the orientation vector follows a diffusion process on the unit sphere and is characterized by an orientation-dependent (“multiplicative”) noise. As a consequence, the corresponding Langevin equation attains different forms depending on whether Itō’s or Stratonovich’s stochastic calculus is used. We clarify that both forms are equivalent and derive them in a top-down approach from a geometric construction of Brownian motion on the unit sphere, based on infinitesimal random rotations. Our approach suggests further a geometric integration scheme for rotational Brownian motion, which preserves the normalization constraint of the orientation vector exactly. We show that a simple implementation of the scheme, using Gaussian random rotations, converges weakly at order 1 of the integration time step, and we outline an advanced variant of the scheme that is weakly exact for an arbitrarily large time step. Due to a favorable prefactor of the discretization error, already the Gaussian scheme allows for integration time steps that are one order of magnitude larger compared to a commonly used algorithm for rotational Brownian dynamics simulations based on projection on the constraining manifold. For torques originating from constant external fields, we prove by virtue of the Fokker-Planck equation that the constructed diffusion process satisfies detailed balance and converges to the correct equilibrium distribution. The analysis is restricted to time-homogeneous rotational Brownian motion (i.e., a single rotational diffusion constant), which is relevant for axisymmetric particles and also chemically anisotropic spheres, such as self-propelled Janus particles.

We have been working for several years on the representation, study and simulation of complex systems by means of formal methods. APL2 and other programming languages have been used to develop our tools and experiments. This paper summarizes our APL2 works on fractal sets, automatic programming of cellular automata and simulation of multi agent systems.

There has been a long histroy of the System Dynamics developed by J.Forrester and its analysis of complex system behaviors. It is necessary to know the detailed system structure and a large amount statistical data to build the flow diagram when using this method. However, our quantitative analysis show that the integrated model can be seen as a combination of many abstracting grey system modeles which may be decomposed from complex system. In this way, the integrated model can be simulated on a computer. The approach of combining quantitative analysis with simulation has the following features, (1) A continuous differential model is established using grey generating numbers. (2) The integrated model can be run on a computer to simulate the system behavior using DYNAMO language. In this paper, the approach of modelling the available abstracting grey dynamic systems and of simulating the integrated systems is described. The system analysis of experimental example show that grey system modelling is very useful in System Dynamics model.

An Approach of the Grey System Modelling and Simulation for Complex Systems was published in Systems Analysis and Simulation 1988, I: Theory and Foundations. Proceedings of the International Symposium held in Berlin (GDR), September 12–16, 1988 on page 117.

Modelling and simulation is of crucial importance for the understanding of system dynamics. In aircraft, simulation has been strong in the area of flight control. Modelling and simulation of the hydraulic systems has also a long tradition. The rapid increase in computational power has now come to a point where complete modelling and simulation of all the sub systems in an aircraft is not far away. This means new challenges in dealing with very complex multi domain systems. Of all the systems in an aircraft fluid power systems is one of the most difficult to handle from a numerical point of view. They are characterised by difficulties such as discontinuities, very strong non-linearities, stiff differential equations and a high degree of complexity. A considerable effort has therefore been made to develop methods suitable for simulation of such systems. This means that it is possible to simulate fluid power systems much more efficiently than before, and that a lot of systems can even be simulated in real time. The distributed modelling technique used, also makes it possible to partition the system for parallel processing. This means that it is possible to simulate much larger systems within reasonable time. It also is a valid paradigm for multi domain system. The rapid development in simulation methods and the general increase in hardware performance imply that simulations can be run thousands of times faster than just a few years ago. As a consequence, there is no longer such need to use very economic, more or less analytical methods. This means that design methods based on different kinds of numerical optimisation for system design, are becoming much more important.

We introduce PyCX, an online repository of simple, crude, easy-to-understand sample codes for various complex systems simulation, including iterative maps, cellular automata, dynamical networks and agent-based models. All the sample codes were written in plain Python, a general-purpose programming language widely used in industry as well as in academia, so that students can gain practical skills for both complex systems simulation and computer programming simultaneously. The core philosophy of PyCX is on the simplicity, readability, generalizability and pedagogical values of simulation codes. PyCX has been used in instructions of complex systems modeling at several places with successful outcomes.

Current simulation tools for multibody dynamics are not problem dependent, they use the same modeling process to all cases regardless of their characteristics. In addition, real-time simulation of small multibody systems is achievable by existing simulation tools, however, real-time simulation of large and complex systems is not possible with existing methods. This is a challenge that needs to be addressed before further advances in mechanical simulation with hardware-in-the-loop and man-in-the-loop, as well as virtual prototyping are made possible. This paper addresses the issue of how the modeling process-dependent versus independent coordinates, and descriptor form versus state-space form of the equations of motion-affects the dynamic simulation of multibody systems and how it may be taken into account and added to the concept of intelligent simulation. With this new concept all the factors involved in the simulation process — modeling, equations, solution, etc. — are chosen and combined depending upon the characteristics of the system to be simulated. It is envisioned that this concept will lead to faster and more robust real-time simulators.

Over the course of three to four decades, most well-established companies lose their dominating position in the market or fail entirely. Their failure occurs even though they have resources for sensing shifting market trends, skills and assets to develop next-generation technologies, and the financial means to fill skill gaps and afford risky investments. Nevertheless, incumbents obviously find it very difficult to invest in innovation that takes attention and resources away from a highly successful core business. A solution to this "innovator's dilemma" is the concept of "organizational ambidexterity," which has garnered considerable attention among researchers in organization and innovation. According to empirical findings and emergent theory, companies can improve their financial performance and ensure their long-term survival by balancing their innovation activities, so that they are equally focused on exploratory (discontinuous) and exploitative (incremental, continuous) innovations. But how can such a balance be achieved? The literature on the organizational theory and related fields (product innovation, knowledge management, creativity, etc.) identifies more than 300 contributing factors to innovation and ambidexterity: many are interdependent so that their impacts compound or cancel each other. Moreover, for many factors, there is limited empirical data and the size of impacts is unknown. To understand which managerial actions lead to ambidexterity, this dissertation develops a novel approach to the study and analysis of complex casual systems with high uncertainty: exploratory fuzzy cognitive mapping. Fuzzy Cognitive Mapping (FCM) is a semi-quantitative system modeling and simulation technique. It is used to represent qualitative information about complex systems as networks of casual relationships that can be studied computationally. Exploratory modeling and analysis (EMA) is a new approach to modeling and simulation of complex systems when there is high uncertainty about the structural properties of the system. This work is the first to combine both approaches. The work makes several contributions: First, it shows that only a small fraction of management interventions will actually lead to ambidexterity while most will, at best, improve one type of innovation at the expense of the other. Second, it provides a simulation tool to management researchers and practitioners that allows them to test ideas for improving ambidexterity against a model that reflects our current collective knowledge about innovation. And third, it develops a range of techniques (and software code) for exploratory FCM modeling, such as methods for transforming qualitative data to FCM, for exploratory simulation of large and complex FCM models, and for data