Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Some aspects of the solution of non-linear differential equations

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Resonant gradient force on atom interacting with laser field

The gradient force, as a function of position and velocity, is derived for a two-level atom interacting with a standing-wave laser field. Basing on optical Bloch equations, the numerical solutions for the gradient force f_(|_;n) (n = 0, 1, 2, 3, 4, ...) pointing in the direction of the transverse of the laser beam are given. It is shown the higher order gradient force plays important role at strong intensity (G = 64), the contribution of them can not be neglected.

On the theory of self-similar parabolic optical pulses

Self-similar optical pulses (or “similaritons”) of parabolic intensity profile can be found as asymptotic solutions of the nonlinear Schr¨odinger equation in a gain medium such as a fiber amplifier or laser resonator. These solutions represent a wide-ranging significance example of dissipative nonlinear structures in optics. Here, we address some issues related to the formation and evolution of parabolic pulses in a fiber gain medium by means of semi-analytic approaches. In particular, the effect of the third-order dispersion on the structure of the asymptotic solution is examined. Our analysis is based on the resolution of ordinary differential equations, which enable us to describe the main properties of the pulse propagation and structural characteristics observable through direct numerical simulations of the basic partial differential equation model with sufficient accuracy.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

On the theory of self-similar parabolic optical pulses

Self-similar optical pulses (or “similaritons”) of parabolic intensity profile can be found as asymptotic solutions of the nonlinear Schr¨odinger equation in a gain medium such as a fiber amplifier or laser resonator. These solutions represent a wide-ranging significance example of dissipative nonlinear structures in optics. Here, we address some issues related to the formation and evolution of parabolic pulses in a fiber gain medium by means of semi-analytic approaches. In particular, the effect of the third-order dispersion on the structure of the asymptotic solution is examined. Our analysis is based on the resolution of ordinary differential equations, which enable us to describe the main properties of the pulse propagation and structural characteristics observable through direct numerical simulations of the basic partial differential equation model with sufficient accuracy.

Numerical solution of parabolic Cauchy problems in planar corner domains

An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nyström method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach. © 2014 IMACS.

Nonlinear dynamics of cilia and flagella

Cilia and flagella are hairlike extensions of eukaryotic cells which generate oscillatory beat patterns that can propel micro-organisms and create fluid flows near cellular surfaces. The evolutionary highly conserved core of cilia and flagella consists of a cylindrical arrangement of nine microtubule doublets, called the axoneme. The axoneme is an actively bending structure whose motility results from the action of dynein motor proteins cross-linking microtubule doublets and generating stresses that induce bending deformations. The periodic beat patterns are the result of a mechanical feedback that leads to self-organized bending waves along the axoneme. Using a theoretical framework to describe planar beating motion, we derive a nonlinear wave equation that describes the fundamental Fourier mode of the axonemal beat. We study the role of nonlinearities and investigate how the amplitude of oscillations increases in the vicinity of an oscillatory instability. We furthermore present numerical solutions of the nonlinear wave equation for different boundary conditions. We find that the nonlinear waves are well approximated by the linearly unstable modes for amplitudes of beat patterns similar to those observed experimentally.

Models of gas-liquid solubilities

A recent method for phase equilibria, the AGAPE method, has been used to predict activity coefficients and excess Gibbs energy for binary mixtures with good accuracy. The theory, based on a generalised London potential (GLP), accounts for intermolecular attractive forces. Unlike existing prediction methods, for example UNIFAC, the AGAPE method uses only information derived from accessible experimental data and molecular information for pure components. Presently, the AGAPE method has some limitations, namely that the mixtures must consist of small, non-polar compounds with no hydrogen bonding, at low moderate pressures and at conditions below the critical conditions of the components. Distinction between vapour-liquid equilibria and gas-liquid solubility is rather arbitrary and it seems reasonable to extend these ideas to solubility. The AGAPE model uses a molecular lattice-based mixing rule. By judicious use of computer programs a methodology was created to examine a body of experimental gas-liquid solubility data for gases such as carbon dioxide, propane, n-butane or sulphur hexafluoride which all have critical temperatures a little above 298 K dissolved in benzene, cyclo-hexane and methanol. Within this methodology the value of the GLP as an ab initio combining rule for such solutes in very dilute solutions in a variety of liquids has been tested. Using the GLP as a mixing rule involves the computation of rotationally averaged interactions between the constituent atoms, and new calculations have had to be made to discover the magnitude of the unlike pair interactions. These numbers have been seen as significant in their own right in the context of the behaviour of infinitely-dilute solutions. A method for extending this treatment to "permanent" gases has also been developed. The findings from the GLP method and from the more general AGAPE approach have been examined in the context of other models for gas-liquid solubility, both "classical" and contemporary, in particular those derived from equations-of-state methods and from reference solvent methods.

Optimization of chemical plant simulation using double collocation

A method has been constructed for the solution of a wide range of chemical plant simulation models including differential equations and optimization. Double orthogonal collocation on finite elements is applied to convert the model into an NLP problem that is solved either by the VF 13AD package based on successive quadratic programming, or by the GRG2 package, based on the generalized reduced gradient method. This approach is termed simultaneous optimization and solution strategy. The objective functional can contain integral terms. The state and control variables can have time delays. Equalities and inequalities containing state and control variables can be included into the model as well as algebraic equations and inequalities. The maximum number of independent variables is 2. Problems containing 3 independent variables can be transformed into problems having 2 independent variables using finite differencing. The maximum number of NLP variables and constraints is 1500. The method is also suitable for solving ordinary and partial differential equations. The state functions are approximated by a linear combination of Lagrange interpolation polynomials. The control function can either be approximated by a linear combination of Lagrange interpolation polynomials or by a piecewise constant function over finite elements. The number of internal collocation points can vary by finite elements. The residual error is evaluated at arbitrarily chosen equidistant grid-points, thus enabling the user to check the accuracy of the solution between collocation points, where the solution is exact. The solution functions can be tabulated. There is an option to use control vector parameterization to solve optimization problems containing initial value ordinary differential equations. When there are many differential equations or the upper integration limit should be selected optimally then this approach should be used. The portability of the package has been addressed converting the package from V AX FORTRAN 77 into IBM PC FORTRAN 77 and into SUN SPARC 2000 FORTRAN 77. Computer runs have shown that the method can reproduce optimization problems published in the literature. The GRG2 and the VF I 3AD packages, integrated into the optimization package, proved to be robust and reliable. The package contains an executive module, a module performing control vector parameterization and 2 nonlinear problem solver modules, GRG2 and VF I 3AD. There is a stand-alone module that converts the differential-algebraic optimization problem into a nonlinear programming problem.

Dynamic simulation of chemical plant

This thesis describes the design and implementation of a new dynamic simulator called DASP. It is a computer program package written in standard Fortran 77 for the dynamic analysis and simulation of chemical plants. Its main uses include the investigation of a plant's response to disturbances, the determination of the optimal ranges and sensitivities of controller settings and the simulation of the startup and shutdown of chemical plants. The design and structure of the program and a number of features incorporated into it combine to make DASP an effective tool for dynamic simulation. It is an equation-oriented dynamic simulator but the model equations describing the user's problem are generated from in-built model equation library. A combination of the structuring of the model subroutines, the concept of a unit module, and the use of the connection matrix of the problem given by the user have been exploited to achieve this objective. The Executive program has a structure similar to that of a CSSL-type simulator. DASP solves a system of differential equations coupled to nonlinear algebraic equations using an advanced mixed equation solver. The strategy used in formulating the model equations makes it possible to obtain the steady state solution of the problem using the same model equations. DASP can handle state and time events in an efficient way and this includes the modification of the flowsheet. DASP is highly portable and this has been demonstrated by running it on a number of computers with only trivial modifications. The program runs on a microcomputer with 640 kByte of memory. It is a semi-interactive program, with the bulk of all input data given in pre-prepared data files with communication with the user is via an interactive terminal. Using the features in-built in the package, the user can view or modify the values of any input data, variables and parameters in the model, and modify the structure of the flowsheet of the problem during a simulation session. The program has been demonstrated and verified using a number of example problems.

A three dimensional numerical investigation of hillslope flow processes

Physically based distributed models of catchment hydrology are likely to be made available as engineering tools in the near future. Although these models are based on theoretically acceptable equations of continuity, there are still limitations in the present modelling strategy. Of interest to this thesis are the current modelling assumptions made concerning the effects of soil spatial variability, including formations producing distinct zones of preferential flow. The thesis contains a review of current physically based modelling strategies and a field based assessment of soil spatial variability. In order to investigate the effects of soil nonuniformity a fully three dimensional model of variability saturated flow in porous media is developed. The model is based on a Galerkin finite element approximation to Richards equation. Accessibility to a vector processor permits numerical solutions on grids containing several thousand node points. The model is applied to a single hillslope segment under various degrees of soil spatial variability. Such variability is introduced by generating random fields of saturated hydraulic conductivity using the turning bands method. Similar experiments are performed under conditions of preferred soil moisture movement. The results show that the influence of soil variability on subsurface flow may be less significant than suggested in the literature, due to the integrating effects of three dimensional flow. Under conditions of widespread infiltration excess runoff, the results indicate a greater significance of soil nonuniformity. The recognition of zones of preferential flow is also shown to be an important factor in accurate rainfall-runoff modelling. Using the results of various fields of soil variability, experiments are carried out to assess the validity of the commonly used concept of `effective parameters'. The results of these experiments suggest that such a concept may be valid in modelling subsurface flow. However, the effective parameter is observed to be event dependent when the dominating mechanism is infiltration excess runoff.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

Comments on multiple oscillatory solutions in systems with time-delay feedback

A complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding the disappearance of limit cycle solutions, derive analytical criteria for frequency degeneration, amplitude degeneration, frequency extrema. Furthermore, we discuss the influence of the phase shift parameter and show analytically that the stabilization of the steady state and the decay of all oscillations (amplitude death) cannot happen for global feedback only. Finally, we explain the onset of traveling wave patterns close to the regime of amplitude death.