Diffuse-Charge Dynamics in Electrochemical Systems

The response of a model micro-electrochemical system to a time-dependent applied voltage is analyzed. The article begins with a fresh historical review including electrochemistry, colloidal science, and microfluidics. The model problem consists of a symmetric binary electrolyte between parallel-plate, blocking electrodes which suddenly apply a voltage. Compact Stern layers on the electrodes are also taken into account. The Nernst-Planck-Poisson equations are first linearized and solved by Laplace transforms for small voltages, and numerical solutions are obtained for large voltages. The ``weakly nonlinear'' limit of thin double layers is then analyzed by matched asymptotic expansions in the small parameter $\epsilon = \lambda_D/L$, where $\lambda_D$ is the screening length and $L$ the electrode separation. At leading order, the system initially behaves like an RC circuit with a response time of $\lambda_D L / D$ (not $\lambda_D^2/D$), where $D$ is the ionic diffusivity, but nonlinearity violates this common picture and introduce multiple time scales. The charging process slows down, and neutral-salt adsorption by the diffuse part of the double layer couples to bulk diffusion at the time scale, $L^2/D$. In the ``strongly nonlinear'' regime (controlled by a dimensionless parameter resembling the Dukhin number), this effect produces bulk concentration gradients, and, at very large voltages, transient space charge. The article concludes with an overview of more general situations involving surface conduction, multi-component electrolytes, and Faradaic processes.Comment: 10 figs, 26 pages (double-column), 141 reference

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

Piecewise-polynomial associated transform macromodeling algorithm for fast nonlinear circuit simulation

We present a piecewise-polynomial based associated transform algorithm (PWPAT) for macromodeling nonlinear circuits in system-level circuit design. The generated reduced model can provide both global and local accuracies with the most compact dimension. Numerical examples compare it with existing algorithms and verify its superior accuracy in higher order harmonics simulation over traditional Trajectory Piecewise-Linear (TPWL) approach. © 2013 IEEE.published_or_final_versio

Fast nonlinear model order reduction via associated transforms of high-order volterra transfer functions

We present a new and fast way of computing the projection matrices serving high-order Volterra transfer functions in the context of (weakly and strongly) nonlinear model order reduction. The novelty is to perform, for the first time, the association of multivariate (Laplace) variables in high-order multiple-input multiple-output (MIMO) transfer functions to generate the standard single-s transfer functions. The consequence is obvious: instead of finding projection subspaces about every s i, only that about a single s is required. This translates into drastic saving in computation and memory, and much more compact reduced-order nonlinear models, without compromising any accuracy. © 2012 ACM.published_or_final_versio

Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction

Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal. Here we study the classic model of weakly activated cascades and obtain analytical solutions for a variety of inputs. We show that in the special but important case of optimal-gain cascades (i.e., when the deactivation rates are identical) the downstream output of the cascade can be represented exactly as a lumped nonlinear module containing an incomplete gamma function with real parameters that depend on the rates and length of the cascade, as well as parameters of the input signal. The expressions obtained can be applied to the non-identical case when the deactivation rates are random to capture the variability in the cascade outputs. We also show that cascades can be rearranged so that blocks with similar rates can be lumped and represented through our nonlinear modules. Our results can be used both to represent cascades in computational models of differential equations and to fit data efficiently, by reducing the number of equations and parameters involved. In particular, the length of the cascade appears as a real-valued parameter and can thus be fitted in the same manner as Hill coefficients. Finally, we show how the obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction.Comment: 18 pages, 7 figure

Properties of recoverable region and semi-global stabilization in recoverable region for linear systems subject to constraints

This paper investigates time-invariant linear systems subject to input and state constraints. It is shown that the recoverable region (which is the largest domain of attraction that is theoretically achievable) can be semiglobally stabilized by continuous nonlinear feedbacks while satisfying the constraints. Moreover, a reduction technique is presented which shows, when trying to compute the recoverable region, that we only need to compute the recoverable region for a system of lower dimension which generally leads to a considerable simplification in the computational effort

Special solutions to a compact equation for deep-water gravity waves

Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. Further, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of singular traveling waves, we consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.Comment: 17 pages, 14 figures, 41 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh