Model reduction of multi-scale chemical Langevin equations

This paper addresses the model reduction problem for a class of stiff chemical Langevin equations that arise as models of biomolecular networks with fast and slow reactions and can be described as continuous Markov processes. Initially, a coordinate transformation is sought that allows the decoupling of fast and slow variables in the model equations. Necessary and sufficient conditions are derived for such a linear transformation to exist, along with an explicit change of variables which achieves the desired decoupling. For the systems for which this step is applicable, the method of adiabatic elimination is applied to determine a representation of the slow dynamics. Theoretical concepts and results are illustrated with simple examples.     

On the Approximation of Stochastic Concurrent Constraint Programming by Master Equation

We explore the relation between the stochastic semantic associated to stochastic Concurrent Constrain Programming (sCCP) and its fluid-flow approximation. Writing the master equation for a sCCP model, we can show that the fluid flow equation is a first-order approximation of the true equation for the average. Moreover, we introduce a second-order correction and first-order equations for the variance and the covariance.     

Stochastic approximation with ‘controlled Markov’ noise

Stochastic approximation algorithms with additional noise that can be modelled as a controlled Markov process are analyzed and shown to track the solutions of a differential inclusion defined in terms of the ergodic occupation measures associated with the controlled Markov process.     

Building Blocks for Computer Vision with Stochastic Partial Differential Equations

We discuss the basic concepts of computer vision with stochastic partial differential equations (SPDEs). In typical approaches based on partial differential equations (PDEs), the end result in the best case is usually one value per pixel, the “expected” value. Error estimates or even full probability density functions PDFs are usually not available. This paper provides a framework allowing one to derive such PDFs, rendering computer vision approaches into measurements fulfilling scientific standards due to full error propagation. We identify the image data with random fields in order to model images and image sequences which carry uncertainty in their gray values, e.g. due to noise in the acquisition process. The noisy behaviors of gray values is modeled as stochastic processes which are approximated with the method of generalized polynomial chaos (Wiener-Askey-Chaos). The Wiener-Askey polynomial chaos is combined with a standard spatial approximation based upon piecewise multi-linear finite elements. We present the basic building blocks needed for computer vision and image processing in this stochastic setting, i.e. we discuss the computation of stochastic moments, projections, gradient magnitudes, edge indicators, structure tensors, etc. Finally we show applications of our framework to derive stochastic analogs of well known PDEs for de-noising and optical flow extraction. These models are discretized with the stochastic Galerkin method. Our selection of SPDE models allows us to draw connections to the classical deterministic models as well as to stochastic image processing not based on PDEs. Several examples guide the reader through the presentation and show the usefulness of the framework.     

A note on the two disk mixed sensitivity design for control system with multiple output time delays

This note presents a tuning design method for output time delayed control systems that satisfies the two disk mixed sensitivity control design specification. The key features of this method are that it focuses on the two disk mixed sensitivity problem; it handles multiple time delays and allows the majority of the calculations to be carried out as in the time delay free case. A numerical example is given to show how the procedure works.     

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.     

Discrete time credibilistic processes: Construction and convergences

In this paper, we discuss a method of constructing a credibilistic process which is a family of fuzzy variables on credibility space. Applying the extension theorem in credibility theory, the finite or infinite horizon credibilistic process is made up from a family of credibilistic kernels. Also, for the Markov case, convergence theorems are given and credibilistic risk models for reward processes are considered, whose risk is completed by the recursive equation.     

Root mean square gain of discrete-time switched linear systems under dwell time constraints

This paper deals with discrete-time switched linear systems and considers the problem of computing an upper bound to the dwell time ensuring a pre-specified root mean square (RMS) gain. As a natural consequence of treating general systems of this class in terms of the order and the number of subsystems, only sufficient conditions are worked out. They depend on the complete separation of the stabilizing and anti-stabilizing solutions of the algebraic Riccati equation associated to each subsystem. Moreover, as positive features, it is shown that the dwell time preserving the specification can be calculated through linear matrix inequalities (LMIs) and line search, being thus numerically solvable in polynomial time, and this allows the treatment of stable switched linear systems which do not admit a common Lyapunov function. The case of a guaranteed RMS gain for arbitrary switching signals is also addressed. A simple academic example constituted by three subsystems of third order is included for illustration.     

Uncertain jumping systems with strong and weak functional delays

This paper provides complete results on the stability behavior of a class of uncertain dynamical systems with jumping parameters and functional time-delays. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. The parametric uncertainties are norm-bounded appearing in all system matrices and the delay factor depends on the mode of operation. Notions of weak and strong stochastic stability for the jumping system are developed depending on the available information using a prescribed -performance. Memoryless and delayed-state feedback are considered to guarantee the closed-loop stability. All the results are cast into linear matrix inequalities format. A numerical example is given to illustrate the developed results.     

Learning to pour with a robot arm combining goal and shape learning for dynamic movement primitives

When describing robot motion with dynamic movement primitives (DMPs), goal (trajectory endpoint), shape and temporal scaling parameters are used. In reinforcement learning with DMPs, usually goals and temporal scaling parameters are predefined and only the weights for shaping a DMP are learned. Many tasks, however, exist where the best goal position is not a priori known, requiring to learn it. Thus, here we specifically address the question of how to simultaneously combine goal and shape parameter learning. This is a difficult problem because learning of both parameters could easily interfere in a destructive way. We apply value function approximation techniques for goal learning and direct policy search methods for shape learning. Specifically, we use “policy improvement with path integrals” and “natural actor critic” for the policy search. We solve a learning-to-pour-liquid task in simulations as well as using a Pa10 robot arm. Results for learning from scratch, learning initialized by human demonstration, as well as for modifying the tool for the learned DMPs are presented. We observe that the combination of goal and shape learning is stable and robust within large parameter regimes. Learning converges quickly even in the presence of disturbances, which makes this combined method suitable for robotic applications.