Accelerated leap methods for simulating discrete stochastic chemical kinetics

Biologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology - most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods.

Proceedings of Computational Methods for Coupled Problems in Science and Engineering

Computational Methods for Coupled Problems in Science and Engineering

Analysis of computer communication systems by use of digital signal processing methods

Queueing theory is an effective tool in the analysis of canputer camrunication systems. Many results in queueing analysis have teen derived in the form of Laplace and z-transform expressions. Accurate inversion of these transforms is very important in the study of computer systems, but the inversion is very often difficult. In this thesis, methods for solving some of these queueing problems, by use of digital signal processing techniques, are presented. The z-transform of the queue length distribution for the Mj GY jl system is derived. Two numerical methods for the inversion of the transfom, together with the standard numerical technique for solving transforms with multiple queue-state dependence, are presented. Bilinear and Poisson transform sequences are presented as useful ways of representing continuous-time functions in numerical computations.

Numerical modelling of advanced modulation formats and WDM transmission

This thesis presents a theoretical investigation of the application of advanced modelling formats in high-speed fibre lightwave systems. The first part of this work focuses on numerical optimisation of dense wavelength division multiplexing (DWDM) system design. We employ advanced spectral domain filtering techniques and carrier pulse reshaping. We then apply these optimisation methods to investigate spectral and temporal domain characteristics of advanced modulation formats in fibre optic telecommunication systems. Next we investigate numerical methods used in detecting and measuring the system performance of advanced modulation formats. We then numerically study the combination of return-to-zero differential phase-shift keying (RZ-DPSK) with advanced photonic devices. Finally we analyse the dispersion management of Nx40 Gbit/s RZ-DPSK transmission applied to a commercial terrestrial lightwave system.

Computer methods for the heat conduction equation

The merits of various numerical methods for the solution of the one and two dimensional heat conduction equation with a radiation boundary condition have been examined from a practical standpoint in order to determine accuracies and efficiencies. It is found that the use of five increments to approximate the space derivatives gives sufficiently accurate results provided the time step is not too large; further, the implicit backward difference method of Liebmann (27) is found to be the most accurate method. On this basis, a new implicit method is proposed for the solution of the three-dimensional heat conduction equation with radiation boundary conditions. The accuracies of the integral and analogue computer methods are also investigated.

Stochastic Modeling of Reversible Biochemical Reaction-Diffusion Systems and High-Resolution Shock-Capturing Methods for Fluid Interfaces

Thesis (Ph.D.)--University of Washington, 2016-08

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method.

Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools

We explore the recently developed snapshot-based dynamic mode decomposition (DMD) technique, a matrix-free Arnoldi type method, to predict 3D linear global flow instabilities. We apply the DMD technique to flows confined in an L-shaped cavity and compare the resulting modes to their counterparts issued from classic, matrix forming, linear instability analysis (i.e. BiGlobal approach) and direct numerical simulations. Results show that the DMD technique, which uses snapshots generated by a 3D non-linear incompressible discontinuous Galerkin Navier?Stokes solver, provides very similar results to classical linear instability analysis techniques. In addition, we compare DMD results issued from non-linear and linearised Navier?Stokes solvers, showing that linearisation is not necessary (i.e. base flow not required) to obtain linear modes, as long as the analysis is restricted to the exponential growth regime, that is, flow regime governed by the linearised Navier?Stokes equations, and showing the potential of this type of analysis based on snapshots to general purpose CFD codes, without need of modifications. Finally, this work shows that the DMD technique can provide three-dimensional direct and adjoint modes through snapshots provided by the linearised and adjoint linearised Navier?Stokes equations advanced in time. Subsequently, these modes are used to provide structural sensitivity maps and sensitivity to base flow modification information for 3D flows and complex geometries, at an affordable computational cost. The information provided by the sensitivity study is used to modify the L-shaped geometry and control the most unstable 3D mode.

Advanced numerical modeling of masonry vaulted structures based on BIM parametric geometry generation

Vaults are an architectural element which during construction history have been built with a great variety of different materials, shapes, and sizes. The shape of these structural elements was often dependent by the necessity to cover complex spaces, by the needed loading capacity, or by architectural aesthetics. Within this complex scenario masonry patterns generates also different effects on loading capacity, load percolation and stiffness of the structure. These effects were been extensively investigated, both with empirical observations and with modern numerical methods. While most of them focus on analyzing the load bearing capacity or the texture effect on vaulted structures, the aim of this analysis is to investigate on the effects of the variation of a single structural characteristic on the load percolation in the vault. Moreover, an additional purpose of the work is related to the coding of a parametrical model aiming at generating different masonry vaulted structures. Nevertheless, proposed script can generate different typology of vaulted structure basing on some structural characteristics, such as the span and the length to cover and the dimensions of the blocks.

Numerical modelling of axially loaded piled foundations with physical tests validation

L'oggetto della tesi è la modellazione numerica di pali di fondazione per turbine eoliche in ambiente offshore. Il metodo di modellazione comprende l'interpretazione di prove CPT per l'estrapolazione dei parametri geotecnici del terreno, la creazione di un modello dell'interazione tra struttura e terreno tramite il software agli elementi finiti Abaqus, la sua verifica, l'ottimizzazione dell'estensione del modello del terreno e della mesh e la simulazione numerica del comportamento di pali caricati assialmente in compressione. Grazie al confronto con i risultati di test su modelli fisici, eseguiti nel laboratorio dell'istituto Fraunhofer IWES di Hannover, di tre pali aventi la stessa geometria ma installati in punti con diverse condizioni di compattazione della sabbia, è stata possibile la messa a punto di una strategia di simulazione più accurata possibile e la sua validazione. A conclusione del lavoro è stato eseguito un esempio di design di una fondazione jacket per una turbina eolica offshore.

Numerical prediction of the lateral displacement capacity of flexural reinforced concrete shear walls

Previous earthquakes showed that shear wall damage could lead to catastrophic failures of the reinforced concrete building. The lateral load capacity of shear walls needs to be estimated to minimize associated losses during catastrophic events; hence it is necessary to develop and validate reliable and stable numerical methods able to converge to reasonable estimations with minimum computational effort. The beam-column 1-D line element with fiber-type cross-section model is a practical option that yields results in agreement with experimental data. However, shortcomings of using this model to predict the local damage response may come from the fact that the model requires fine calibration of material properties to overcome regularization and size effects. To reduce the mesh-dependency of the numerical model, a regularization method based on the concept of post-yield energy is applied in this work to both the concrete and the steel material constitutive laws to predict the nonlinear cyclic response and failure mechanism of concrete shear walls. Different categories of wall specimens known to produce a different response under in plane cyclic loading for their varied geometric and detailing characteristics are considered in this study, namely: 1) scaled wall specimens designed according to the European seismic design code and 2) unique full-scale wall specimens detailed according to the U.S. design code to develop a ductile behavior under cyclic loading. To test the boundaries of application of the proposed method, two full-scale walls with a mixed shear-flexure response and different values of applied axial load are also considered. The results of this study show that the use of regularized constitutive models considerably enhances the response predictions capabilities of the model with regards to global force-drift response and failure mode. The simulations presented in this thesis demonstrate the proposed model to be a valuable tool for researchers and engineers.

Modelagem matemática e simulação numérica do resfriamento rápido de morango com ar forçado

This work approaches the forced air cooling of strawberry by numerical simulation. The mathematical model that was used describes the process of heat transfer, based on the Fourier's law, in spherical coordinates and simplified to describe the one-dimensional process. For the resolution of the equation expressed for the mathematical model, an algorithm was developed based on the explicit scheme of the numerical method of the finite differences and implemented in the scientific computation program MATLAB 6.1. The validation of the mathematical model was made by the comparison between theoretical and experimental data, where strawberries had been cooled with forced air. The results showed to be possible the determination of the convective heat transfer coefficient by fitting the numerical and experimental data. The methodology of the numerical simulations was showed like a promising tool in the support of the decision to use or to develop equipment in the area of cooling process with forced air of spherical fruits.

Direct Computation of Operational Matrices for Polynomial Bases

Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation.

Looking at the Gregory-Laflamme instability through quasinormal modes

We study evolution of gravitational perturbations of black strings. It is well known that for all wave numbers less than some threshold value, the black string is unstable against the scalar type of gravitational perturbations, which is named the Gregory-Laflamme instability. Using numerical methods, we find the quasinormal modes and time-domain profiles of the black string perturbations in the stable sector and also show the appearance of the Gregory-Laflamme instability in the time domain. The dependence of the black string quasinormal spectrum and late-time tails on such parameters as the wave vector and the number of extra dimensions is discussed. There is numerical evidence that at the threshold point of instability, the static solution of the wave equation is dominant. For wave numbers slightly larger than the threshold value, in the region of stability, we see tiny oscillations with very small damping rate. While, for wave numbers slightly smaller than the threshold value, in the region of the Gregory-Laflamme instability, we observe tiny oscillations with very small growth rate. We also find the level crossing of imaginary part of quasinormal modes between the fundamental mode and the first overtone mode, which accounts for the peculiar time domain profiles.

Shared information in stationary states of stochastic processes

We present four estimators of the shared information (or interdepency) in ground states given that the coefficients appearing in the wave function are all real non-negative numbers and therefore can be interpreted as probabilities of configurations. Such ground states of Hermitian and non-Hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case, the system being a classical not a quantum one. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and nonconformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law with logarithmic corrections when phase transitions take place.