DSpace Collection:
https://dspace.lboro.ac.uk/2134/4642
2016-09-25T19:29:15ZThe use of differential equations in optimization
https://dspace.lboro.ac.uk/2134/22089
Title: The use of differential equations in optimization
Authors: Zghier, Abbas K.
Abstract: A new approach for unconstrained optimization of a function f(x)
has been investigated. The method is based on solving the differential
equation dx/dt = ± ∇f(x) which defines orthogonal trajectories in Rⁿ-space.
A number of numerical integration techniques have been used for solving
the above differential equation, the most powerful one which gives rise to
a very efficient optimization algorithm is the generalization of the
Trapezoidal rule. The interaction between the parameters which appear
as a result of using the numerical integration has been investigated.
In the above approach factorization of the positive definite matrix
(θG + λI), allowing some control over the diagonal elements of the matrix.
is presented.
A Liapunov function approach has been used in constructing a number
of different differential equations of the above form. It is well known
that if a Liapunov function which satisfies certain conditions can be
found for a given system of differential equations then the origin of
the system is stable. Pursuing this idea further we constructed a Liapunov
function and then the corresponding differential equation. Application
of this differential equation to the problem of finding a minimum of a
function f is shown to yield a vector that converges to a point where
∇f = 0.
The use of differential equations is also extended to the optimal
control problem. The technique is only applicable to unconstrained optimal
control problems. If a terminal condition and inequality constraints are
presented, the problem should be converted to unconstrained form, e.g. by
the use of penalty functions. The method tends to converge, even from a
poor approximation point to the minimum without using line searches.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.1981-01-01T00:00:00ZSafety system design optimisation
https://dspace.lboro.ac.uk/2134/22019
Title: Safety system design optimisation
Authors: Pattison, Rachel L.
Abstract: This thesis investigates the efficiency of a design optimisation scheme that is
appropriate for systems which require a high likelihood of functioning on demand.
Traditional approaches to the design of safety critical systems follow the preliminary
design, analysis, appraisal and redesign stages until what is regarded as an acceptable
design is achieved. For safety systems whose failure could result in loss of life it is
imperative that the best use of the available resources is made and a system which is
optimal, not just adequate, is produced.
The object of the design optimisation problem is to minimise system unavailability
through manipulation of the design variables, such that limitations placed on them by
constraints are not violated.
Commonly, with mathematical optimisation problem; there will be an explicit
objective function which defines how the characteristic to be minimised is related to
the variables. As regards the safety system problem, an explicit objective function
cannot be formulated, and as such, system performance is assessed using the fault tree
method. By the use of house events a single fault tree is constructed to represent the
failure causes of each potential design to overcome the time consuming task of
constructing a fault tree for each design investigated during the optimisation
procedure. Once the fault tree has been constructed for the design in question it is
converted to a BDD for analysis.
A genetic algorithm is first employed to perform the system optimisation, where the
practicality of this approach is demonstrated initially through application to a High-Integrity
Protection System (HIPS) and subsequently a more complex Firewater
Deluge System (FDS).
An alternative optimisation scheme achieves the final design specification by solving
a sequence of optimisation problems. Each of these problems are defined by
assuming some form of the objective function and specifying a sub-region of the
design space over which this function will be representative of the system
unavailability.
The thesis concludes with attention to various optimisation techniques, which possess
features able to address difficulties in the optimisation of safety critical systems.
Specifically, consideration is given to the use of a statistically designed experiment
and a logical search approach.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2000-01-01T00:00:00ZNumerical and analytical study of the convective Cahn-Hilliard equation
https://dspace.lboro.ac.uk/2134/21774
Title: Numerical and analytical study of the convective Cahn-Hilliard equation
Authors: Alesemi, Meshari
Abstract: We consider the convective Cahn-Hilliard equation that is used as a model of coarsening dynamics
in driven systems and that in two spatial dimensions (x; y) has the form
ut + Duux + r2(u u3 + r2u) = 0:
Here t denotes time, u = u(x; y; t) is the order parameter and D is the parameter measuring
the strength of driving. We primarily consider the case of one spatial dimension, when there is
no y-dependence. For the case of no driving, when D = 0, the standard Cahn-Hilliard equation
is recovered, and it is known that solutions to this equation are characterised by an initial
stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters
or droplets/holes or islands are obtained) followed by the coarsening process, where the
average size of the clusters grows in time and the number of the clusters decreases. Moreover,
two main coarsening mechanisms have been identified in the literature, namely, coarsening due
to volume and translational modes. On the other hand, for the case of strong driving, when
D ! 1, the well-known Kuramoto-Sivashinsky equation is recovered, solutions of which are
characterised by complicated chaotic oscillations in both space and time. The primary aim of
the present thesis is to perform a detailed and systematic investigation of the transitions in the
solutions of the convective Cahn-Hilliard equation for a wide range of parameter values as the
driving-force parameter is increased, and, in particular, to understand in detail how the coarsening
dynamics is affected by driving. We find that one of the coarsening modes is stabilised
at relatively small values of D, and the type of the unstable coarsening mode may change as
D increases. In addition, we find that there may be intervals in the driving-force parameter D
where coarsening is completely stabilised. On the other hand, there may be intervals where twomode
solutions are unstable and the solutions can evolve, for example, into one-droplet/hole
solutions, symmetry-broken two-droplet/hole solutions or time-periodic solutions. We present
detailed stability diagrams for 2-mode solutions in the parameter planes and corroborate our
findings by time-dependent simulations. Finally, we present preliminary results for the case of
the (convective) Cahn-Hilliard equation in two spatial dimensions.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZComputation of scattering matrices and resonances for waveguides
https://dspace.lboro.ac.uk/2134/21762
Title: Computation of scattering matrices and resonances for waveguides
Authors: Roddick, Greg
Abstract: Waveguides in Euclidian space are piecewise path connected subsets of R^n that can be written as the union of a compact domain with boundary and their cylindrical ends. The compact and non-compact parts share a common boundary. This boundary is assumed to
be Lipschitz, piecewise smooth and piecewise path connected. The ends can be thought of as the cartesian product of the boundary with the positive real half-line. A notable feature of Euclidian waveguides is that the scattering matrix admits a meromorphic continuation to a certain Riemann surface with a countably infinite number of leaves [2], which we will
describe in detail and deal with. In order to construct this meromorphic continuation,
one usually first constructs a meromorphic continuation of the resolvent for the Laplace
operator. In order to do this, we will use a well known glueing construction (see for example [5]), which we adapt to waveguides. The construction makes use of the meromorphic Fredholm theorem and the fact that the resolvent for the Neumann Laplace operator on the ends of the waveguide can be easily computed as an integral kernel. The resolvent can then be used to construct generalised eigenfunctions and, from them, the scattering matrix.Being in possession of the scattering matrix allows us to calculate resonances; poles of
the scattering matrix. We are able to do this using a combination of numerical contour integration and Newton s method.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00Z