Consistent Dynamic Mode Decomposition

We propose a new method for computing Dynamic Mode Decomposition (DMD) evolution matrices, which we use to analyze dynamical systems. Unlike the majority of existing methods, our approach is based on a variational formulation consisting of data alignment penalty terms and constitutive orthogonality constraints. Our method does not make any assumptions on the structure of the data or their size, and thus it is applicable to a wide range of problems including non-linear scenarios or extremely small observation sets. In addition, our technique is robust to noise that is independent of the dynamics and it does not require input data to be sequential. Our key idea is to introduce a regularization term for the forward and backward dynamics. The obtained minimization problem is solved efficiently using the Alternating Method of Multipliers (ADMM) which requires two Sylvester equation solves per iteration. Our numerical scheme converges empirically and is similar to a provably convergent ADMM scheme. We compare our approach to various state-of-the-art methods on several benchmark dynamical systems

Soft computing applications in dynamic model identification of polymer extrusion process

This paper proposes the application of soft computing to deal with the constraints in conventional modelling techniques of the dynamic extrusion process. The proposed technique increases the efficiency in utilising the available information during the model identification. The resultant model can be classified as a ‘grey-box model’ or has been termed as a ‘semi-physical model’ in the context. The extrusion process contains a number of parameters that are sensitive to the operating environment. Fuzzy ruled-based system is introduced into the analytical model of the extrusion by means of sub-models to approximate those operational-sensitive parameters. In drawing the optimal structure for the sub-models, a hybrid algorithm of genetic algorithm with fuzzy system (GA-Fuzzy) has been implemented. The sub-models obtained show advantages such as linguistic interpretability, simpler rule-base and less membership functions. The developed model is adaptive with its learning ability through the steepest decent error back-propagation algorithm. This ability might help to minimise the deviation of the model prediction when the operational-sensitive parameters adapt to the changing operating environment in the real situation. The model is first evaluated through simulations on the consistency of model prediction to the theoretical analysis. Then, the effectiveness of adaptive sub-models in approximating the operational-sensitive parameters during the operation is further investigated

Blazes: Coordination Analysis for Distributed Programs

Distributed consistency is perhaps the most discussed topic in distributed systems today. Coordination protocols can ensure consistency, but in practice they cause undesirable performance unless used judiciously. Scalable distributed architectures avoid coordination whenever possible, but under-coordinated systems can exhibit behavioral anomalies under fault, which are often extremely difficult to debug. This raises significant challenges for distributed system architects and developers. In this paper we present Blazes, a cross-platform program analysis framework that (a) identifies program locations that require coordination to ensure consistent executions, and (b) automatically synthesizes application-specific coordination code that can significantly outperform general-purpose techniques. We present two case studies, one using annotated programs in the Twitter Storm system, and another using the Bloom declarative language.Comment: Updated to include additional materials from the original technical report: derivation rules, output stream label

The Consistency of Partial Observability for PDEs

In this paper, a new definition of observability is introduced for PDEs. It is a quantitative measure of partial observability. The quantity is proved to be consistent if approximated using well posed approximation schemes. A first order approximation of an unobservability index using empirical gramian is introduced. For linear systems with full state observability, the empirical gramian is equivalent to the observability gramian in control theory. The consistency of the defined observability is exemplified using a Burgers' equation.Comment: 28 pages, 3 figure

On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations

In this note we study the convergence of monotone P1 finite element methods on unstructured meshes for fully non-linear Hamilton-Jacobi-Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretisations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under non-degeneracy assumptions, strong L2 convergence of the gradients.Comment: Keywords: Bellman equations, finite element methods, viscosity solutions, fully nonlinear operators; 18 pages, 1 figur

Revisiting Kawasaki dynamics in one dimension

Critical exponents of the Kawasaki dynamics in the Ising chain are re-examined numerically through the spectrum gap of evolution operators constructed both in spin and domain wall representations. At low temperature regimes the latter provides a rapid finite-size convergence to these exponents, which tend to $z \simeq 3.11$ for instant quenches under ferromagnetic couplings, while approaching to $z \simeq 2$ in the antiferro case. The spin representation complements the evaluation of dynamic exponents at higher temperature scales, where the kinetics still remains slow.Comment: 11 pages, 8 figure

Statistical exponential formulas for homogeneous diffusion

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \left\{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. is given by the exponential formula $$u(t) ~ := ~ \lim_{n \to \infty}{ \left( M^{t/n}_{p} \right)^{n} u_{0} } \,$$ where the statistical operator $M^{h}_{p} \colon BUC( \mathbb{R}^{N} ) \to BUC( \mathbb{R}^{N} )$ is defined by $$\left(M^{h}_{p} \varphi \right)(x) := (1-q) \operatorname{median}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, when $1 \leq p \leq 2$ and by $$\left(M^{h}_{p} \varphi \right)(x) := ( 1 - q ) \operatorname{midrange}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q = \frac{ N }{ N + p - 2 }$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly