450 research outputs found
Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete non-equally spaced observations on a finite time interval. We
propose a new method for Laplace deconvolution which is based on expansions of
the convolution kernel, the unknown function and the observed signal over
Laguerre functions basis (which acts as a surrogate eigenfunction basis of the
Laplace convolution operator) using regression setting. The expansion results
in a small system of linear equations with the matrix of the system being
triangular and Toeplitz. Due to this triangular structure, there is a common
number of terms in the function expansions to control, which is realized
via complexity penalty. The advantage of this methodology is that it leads to
very fast computations, produces no boundary effects due to extension at zero
and cut-off at and provides an estimator with the risk within a logarithmic
factor of the oracle risk. We emphasize that, in the present paper, we consider
the true observational model with possibly nonequispaced observations which are
available on a finite interval of length which appears in many different
contexts, and account for the bias associated with this model (which is not
present when ). The study is motivated by perfusion imaging
using a short injection of contrast agent, a procedure which is applied for
medical assessment of micro-circulation within tissues such as cancerous
tumors. Presence of a tuning parameter allows to choose the most
advantageous time units, so that both the kernel and the unknown right hand
side of the equation are well represented for the deconvolution. The
methodology is illustrated by an extensive simulation study and a real data
example which confirms that the proposed technique is fast, efficient,
accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1207.223
Laplace deconvolution and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete observations. The study is motivated by Dynamic Contrast
Enhanced imaging using a bolus of contrast agent, a procedure which allows
considerable improvement in {evaluating} the quality of a vascular network and
its permeability and is widely used in medical assessment of brain flows or
cancerous tumors. Although the study is motivated by medical imaging
application, we obtain a solution of a general problem of Laplace deconvolution
based on noisy data which appears in many different contexts. We propose a new
method for Laplace deconvolution which is based on expansions of the
convolution kernel, the unknown function and the observed signal over Laguerre
functions basis. The expansion results in a small system of linear equations
with the matrix of the system being triangular and Toeplitz. The number of
the terms in the expansion of the estimator is controlled via complexity
penalty. The advantage of this methodology is that it leads to very fast
computations, does not require exact knowledge of the kernel and produces no
boundary effects due to extension at zero and cut-off at . The technique
leads to an estimator with the risk within a logarithmic factor of of the
oracle risk under no assumptions on the model and within a constant factor of
the oracle risk under mild assumptions. The methodology is illustrated by a
finite sample simulation study which includes an example of the kernel obtained
in the real life DCE experiments. Simulations confirm that the proposed
technique is fast, efficient, accurate, usable from a practical point of view
and competitive
Review of Inverse Laplace Transform Algorithms for Laplace-Space Numerical Approaches
A boundary element method (BEM) simulation is used to compare the efficiency
of numerical inverse Laplace transform strategies, considering general
requirements of Laplace-space numerical approaches. The two-dimensional BEM
solution is used to solve the Laplace-transformed diffusion equation, producing
a time-domain solution after a numerical Laplace transform inversion. Motivated
by the needs of numerical methods posed in Laplace-transformed space, we
compare five inverse Laplace transform algorithms and discuss implementation
techniques to minimize the number of Laplace-space function evaluations. We
investigate the ability to calculate a sequence of time domain values using the
fewest Laplace-space model evaluations. We find Fourier-series based inversion
algorithms work for common time behaviors, are the most robust with respect to
free parameters, and allow for straightforward image function evaluation re-use
across at least a log cycle of time
Parallel Solution of Diffusion Equations using Laplace Transform Methods with Particular Reference to Black-Scholes Models of Financial Options
Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical
finance.
For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE
techniques.
To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the
design of parallel programs and the parameter values used.
To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference
solutions needed to determine the accuracy of the LTFD solutions.
Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions
and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas
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