Theoretical and Computational Aspects of the Optimized Effective Potential Approach within Density Functional TheoryHeaton-Burgess, Tim January 2009 (has links)
<p>The computational success of density functional theory relies on the construction of suitable approximations to the exchange-correlation energy functional. Use of functional approximations depending explicitly upon the density alone appear unable to address all aspects of many-body interactions, such as the fundamental constraint that the ground state energy is a piecewise linear function of the total number of electrons, and the ability to model nonlocal effects. Functionals depending explicitly upon occupied and unoccupied Kohn–Sham orbitals are considered necessary to address these and other issues. This dissertation considers certain issues relevant to the successful implementation of explicitly orbital-dependent functionals through the optimized effective potential (OEP) approach, as well as extending the potential functional formalism that provides the formal basis for the OEP approach to systems in the presence of noncollinear magnetic fields.</p><p>The self-consistent implementation of orbital-dependent energy functionals is correctly done through the optimized effective potential approach—minimization of the ground state energy with respect to the Kohn–Sham potential that generates the set of orbitals employed in the energy evaluation. The focus on the potential can be problematic in finite basis set approaches as determining the exchange-correlation potential in this manner is an inverse problem which, depending upon the combination of orbital and potential basis sets employed, is often ill-posed. The ill-posed nature manifests itself as nonphysical exchange-correlation potentials and total energies. We address the problem of determining meaningful exchange-correlation potentials for arbitrary combinations of orbital and potential basis sets through an L-curve regularization approach based on biasing towards smooth potentials in the energy minimization. This approach generates physically reasonable potentials for any combination of basis sets as shown by comparisons with grid-based OEP calculations on atoms, and through direct comparison with DFT calculations employing functionals not depending on orbitals for which OEP can also be performed. This work ensures that the OEP methodology can be considered a viable many-body computational methodology.</p><p>A separate issue of our OEP implementation is that it can suffer from a lack of size-extensivity—the total energy of a system of infinitely separated monomers may not scale linearly with the total number of monomers depending upon how we construct the Kohn–Sham potential. Typically, a fixed reference potential is employed to aid in the convergence of a finite basis set expansion of the Kohn–Sham potential. This reference potential can be utilized to ensure other desirable properties of the resulting potential. In particular, it can enforce the correct asymptotic behavior. The Fermi–Amaldi potential is often used for this purpose but suffers from size-nonextensivity owing to the explicit dependence of the potential on the total number of electrons. This error is examined and shown to be rather small and rapidly approaches a limiting linear behavior. A size-extensive reference potential with the correct asymptotic behavior is suggested and examined.</p><p>We also consider a formal aspect of the potential-based approach that provides the underlying justification of the OEP methodology. The potential functional formalism of Yang, Ayers, and Wu is extended to include systems in the presence of noncollinear magnetic fields. In doing so, a solution to the nonuniqueness issue associated with mapping between potentials and wave functions in such systems is provided, and a computational implementation of the OEP in noncollinear systems is suggested.</p><p>Finally, as an example of an issue for which orbital-dependent functionals seem necessary to obtain a correct description, we consider the ground state structures of C<sub>4<italic>N</italic> + 2</sub> rings which are believed to exhibit a geometric transition from angle-alternation (<italic>N</italic> ≤ 2) to bond-alternation (<italic>N</italic> > 2). So far, no published DFT approach has been able to reproduce this behavior owing to the tendency of common density functional approximations to bias towards delocalized electron densities. Calculations are presented with the rCAM-B3LYP exchange-correlation functional that correctly predict the structural evolution of this system. This is rationalized in terms of the recently proposed delocalization error for which rCAM-B3LYP explicitly attempts to address.</p> / Dissertation
01 November 2004
(has links) (PDF)
In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Problem III and IV) or on a time interval (in Problem I and II). If solution of inverse problem exists, then the temperature can be defined everywhere on the domain at all instants. The thesis consists of six chapters. In the first chapter, there is the introduction where the definition and applications of inverse problems are given and definition of the four inverse problems, that we will analyze in this thesis, are stated. In the second chapter, some definitions and theorems which we will use to obtain some conclusions about the corresponding direct problem of our four inverse problems are stated, and the conclusions about direct problem are obtained. In the third, fourth, fifth and sixth chapters we have the analysis of inverse problems I, II, III and IV, respectively.
25 April 2007
Advanced nuclear magnetic resonance (NMR) imaging methodologies have been developed to determine porous media properties associated with fluid flow processes. This dissertation presents the development of NMR experimental and analysis methodologies, called NMR probes, particularly for determination of porosity, permeability, and pore-size distributions of porous media while the developed methodologies can be used for other properties. The NMR relaxation distribution can provide various information about porous systems having NMR active nuclei. The determination of the distribution from NMR relaxation data is an ill-posed inverse problem that requires special care, but conventionally the problem has been solved by ad-hoc methods. We have developed a new method based on sound statistical theory that suitably implements smoothness and equality/inequality constraints. This method is used for determination of porosity distributions. A Carr-Purcell-Meiboom-Gill (CPMG) NMR experiment is designed to measure spatially resolved NMR relaxation data. The determined relaxation distribution provides the estimate of intrinsic magnetization which, in turn, is scaled to porosity. A pulsed-field-gradient stimulated-echo (PFGSTE) NMR velocity imaging experiment is designed to measure the superficial average velocity at each volume element. This experiment measures velocity number distributions as opposed to the average phase shift, which is conventionally measured, to suitably quantify the velocities within heterogeneous porous media. The permeability distributions are determined by solving the inverse problem formulated in terms of flow models and the velocity data. We present new experimental designs associated with flow conditions to enhance the accuracy of the estimates. Efforts have been put forth to further improve the accuracy by introducing and evaluating global optimization methods. The NMR relaxation distribution can be scaled to a pore-size distribution once the surface relaxivity is known. We have developed a new method, which avoids limitations on the range of time for which data may be used, to determine surface relaxivity by the PFGSTE NMR diffusion experiment.
High-quality factor oscillators are often used in measurements of verysmall force since they exhibit an enhanced sensitivity in the narrow frequencyband around resonance. Forces containing frequencies outside this frequencyband are often not detectable and the total force acting on the oscillatorremains unknown. In this thesis we present methods to eciently use theavailable bandwidth around resonance to reconstruct the force from partialspectral information.We apply the methods to dynamic atomic force microscopy (AFM) wherea tip at the end of a small micro-cantilever oscillates close to a sample surface.By reconstructing the force between the tip and the surface we can deducedierent properties of the surface. In contrast, in conventional AFM only oneof the many frequency components of the time-dependent tip-surface forceallowing for only qualitative conclusions about the tip-surface force.To increase the number of measurable frequency components we developed Intermodulation AFM (ImAFM). ImAFM utilizes frequency mixing ofa multifrequency drive scheme which generates many frequencies in the response to the nonlinear character of the tip-surface interaction. ImAFM,amplitude-modulated AFM and frequency-modulated AFM can be considered as special cases of narrow-band AFM, where the tip motion can bedescribed by a rapidly oscillating part and a slowly-varying envelope function. Using the concept of force quadratures, each rapid oscillation cycle canbe analyzed individually and ImAFM measurements can be interpreted as arapid measurement of the dependence of the force quadratures on the oscillation amplitude or frequency. To explore the limits of the force quadraturesdescription we introduce the force disk which is a complete description of thetip-surface force in narrow-band AFM at xed static probe height.We present a polynomial force reconstruction method for multifrequencyAFM data. The polynomial force reconstruction is a linear approximativeforce reconstruction method which is based on nding the parameters of amodel force which best approximates the tip-surface force. Another classof reconstruction methods are integral techniques which aim to invert theintegral relation between the tip-surface force and the measured spectraldata. We present an integral method, amplitude-dependence force spectroscopy (ADFS), which reconstructs the conservative tip-surface force fromthe amplitude-dependence of the force quadratures. Together with ImAFMwe use ADFS to combine high-resolution AFM imaging at high speeds withhighly accurate force measurements in each point of an image. For the measurement of dissipative forces we discuss how methods from tomography canbe used to reconstruct forces that are a function of both tip position andvelocity.The methods developed in this thesis are not limited to dynamic AFM andwe describe them in the general context of a harmonic oscillator subject to anexternal force. We hope that theses methods contribute to the transformationof AFM from a qualitative imaging modality into quantitative microscopy andwe hope that they nd application in other measurements which exploit theenhanced sensitivity of a high-quality factor oscillator. / <p>QC 20130527</p>
北, 英輔, KITA, Eisuke, 池田, 洋一, IKEDA, Yoichi, 神谷, 紀生, KAMIYA, Norio
No description available.
Atefi Monfared, Kamelia
It has long been known that any activity that results in changes in subsurface pressure, such as hydrocarbon production or waste or water reinjection, also causes underground deformations and movement, which can be described in terms of volumetric changes. Such deformations induce surface movement, which has a significant environmental impact. Induced surface deformations are measurable as vertical displacements; horizontal displacements; and tilts, which are the gradient of the surface deformation. The initial component of this study is a numerical model developed in C++ to predict and calculate surface deformations based on assumed subsurface volumetric changes occurring in a reservoir. The model is based on the unidirectional expansion technique using equations from Okada’s theory of dislocations (Okada, 1985). A second numerical model calculates subsurface volumetric changes based on surface deformation measurements, commonly referred to as solving for the inverse case. The inverse case is an ill-posed problem because the input is comprised of measured values that contain error. A regularization technique was therefore developed to help solve the ill-posed problem. A variety of surface deformation data sets were analyzed in order to determine the surface deformation input data that would produce the best solution and the optimum reconstruction of the initial subsurface volumetric changes. Tilt measurements, although very small, were found to be much better input than vertical displacement data for finding the inverse solution. Even in an ideal case with 0 % error, tilts result in a smaller RMSE (about 12 % smaller in the case studied) and thus a better resolution. In realistic cases with error, adding only 0.55 % of the maximum random error in the surface displacement data affects the back-calculated results to a significant extent: the RMSE increased by more than 13 times in the case studied. However, in an identical case using tilt measurements as input, adding 20 % of the maximum surface tilt value as random error increased the RMSE by 7 times, and remodelling the initial distribution of the volumetric changes in the subsurface was still possible. The required area of observation can also be reduced if tilt measurements are used. The optimal input includes tilt measurements in both directions: dz/dx and dz/dy. iv With respect to the number of observation points chosen, when tilts are used with an error of 0 %, very good resolution is obtainable using only 0.4 % of the unknowns as the number of benchmarks. For example, using only 10 observation points for a reservoir with 2500 elements, or unknowns resulted in an acceptable reconstruction. With respect to the sensitivity of the inverse solution to the depth of the reservoir and to the geometry of the observation grid, the deeper the reservoir, the more ill-posed the problem. The geometry of the benchmarks also has a significant effect on the solution of the inverse problem.
Rendon Restrepo, Cesar Augusto
28 July 2008
Photodynamic therapy (PDT) derives its tumour selectivity from preferential photosensitizer accumulation and short light penetration in tissue. However, additional strategies are needed to improve the therapeutic index of PDT in oncological applications where light is delivered interstitially to large volumes (e.g. prostate), or when adjacent normal tissue is extremely sensitive (e.g. brain). Much research to improve PDT's selectivity is directed towards developing targeted photosensitizers. Here, I present two alternative strategies to improve PDT's selectivity, without compromising its efficacy. For interstitial delivery, I investigated whether customizable cylindrical diffusers can be used to deliver light doses that conform better to target geometries, specifically the prostate. Additionally, I examined whether the neuroprotectant erythropoietin, used as an adjuvant to PDT for brain tumours, can reduce the sensitivity of normal tissue, thereby improving treatment selectivity. To determine if tailored diffusers constitute an improvement over conventional ones, I introduce a novel optimization algorithm for treatment planning. I also analyze the sensitivity of the resulting plans to changes in the optical properties and diffuser placement. These results are contextualized by a mathematical formalism to characterize the light dose distributions arising from tailored diffusers. In parallel, I investigate the neuroprotective effects of erythropoietin in PDT of primary cortical neurons in culture and normal rat brain in vivo. I show that the most important parameter determining prostate coverage is the number of diffusers employed. Moreover, while tailored diffusers do offer an improvement over conventional ones, the improvement is likely masked by perturbations introduced by the uncertainties of light delivery. Although these results largely discard the use of tailored diffusers in prostate PDT, significant insight has been gained into PDT treatment planning, and tailored diffusers may still be advantageous in more complicated geometries. Additionally, I show that erythropoietin does not improve survival of PDT-treated neurons PDT, nor reduces the volume of necrosis in vivo, for the ranges of conditions and doses studied. To our knowledge, this is the first time this strategy has been tested in brain PDT and deserves to be investigated further, by using later time-points, functional outcomes, and other neuroprotectants.
Mair, Angus MacDonald
The aim of this study was to investigate the composition of an acoustic scattering layer in the North Sea that is particularly strong at 38 kHz. A full definition of the biological composition of the layer, along with its acoustic properties, would allow for it to be confidently removed from data collected during acoustic fish surveys, where it presents a potential source of bias. The layer, traditionally and informally referred to as consisting of zooplankton, appears similar to others observed internationally. The methodology utilised in this study consisted of biological and acoustic sampling, followed by application of forward and inverse acoustic modelling techniques. Acoustic data was collected at 38, 120 and 200 kHz in July 2003, with the addition of 18 kHz in July 2004. Net samples were collected in layers of relatively strong 38 kHz acoustic scattering using a U-tow vehicle (2003) and a MIKT net (2004). Acoustic data were scrutinised to determine actual backscattering, expressed as mean volume backscattering strength (MVBS) (dB). This observed MVBS (MVBSobs) was compared with backscattering predicted by applying the forward problem solution (MVBSpred) to sampled animal densities in order to determine whether those animals were responsible for the enhanced 38 kHz scattering. In most instances, MVBSobs > MVBSpred, more pronounced at 38 kHz. It was found that MVBSpred approached MVBSobs more closely with MIKT than with U-tow samples, but that the 38 kHz mismatch was present in both. Inversion of candidate acoustic models predicted gas-bearing scatterers, which are strong at 38 kHz, as most likely to be responsible for this. Potential sources of inconsistencies between MVBSpred and MVBSobs were identified. The presented forward and inverse solutions infer that although the layer often contains large numbers of common zooplankton types, such as copepods and euphausiids, these are not the dominant acoustic scatterer at 38 kHz. Rather, there remains an unidentified, probably gas-bearing scatterer that contributes significantly to observed scattering levels at this frequency. This study identifies and considerably narrows the list of candidates that are most likely to be responsible for enhanced 38 kHz scattering in the North Sea layer, and recommendations are made for potential future studies.
Digital Image Elasto-Tomography: Mechanical Property Reconstruction from Surface Measured Displacement DataPeters, Ashton January 2007 (has links)
Interest in elastographic techniques for soft tissue imaging has grown as relevant research continues to indicate a correlation between tissue histology and mechanical stiffness. Digital Image Elasto-Tomography (DIET) presents a novel method for identifying cancerous lesions via a three-dimensional image of elastic properties. Stiffness reconstruction with DIET takes steady-state motion captured with a digital camera array as the input to an elastic property reconstruction algorithm, where finite element methods allow simulation of phantom motion at a range of internal stiffness distributions. The low cost and high image contrast achievable with a DIET system may be particularly suited to breast cancer screening, where traditional modalities such as mammography have issues with limited sensitivity and patient discomfort. Proof of concept studies performed on simulated data sets confirmed the potential of the DIET technique, leading to the development of an experimental apparatus for surface motion capture from a range of soft tissue approximating phantoms. Error studies performed on experimental data from these phantoms using a limited number of shape and modulus parameters indicated that accurate measurements of surface motion provide sufficient information to identify a stiffness distribution in both homogeneous and heterogeneous cases. The elastic reconstruction performed on simulated and experimental data considered both deterministic and stochastic algorithms, with a combination of the two approaches found to give the most accurate results, for a realistic increase in computational cost. The reconstruction algorithm developed has the ability to successfully resolve a hard spherical inclusion within a soft phantom, and in addition demonstrated promise in reconstructing the correct stiffness distribution when no inclusion is present.
Shape optimization is an important step in many design processes. With the growing use of Computer Aided Engineering in the design chain, it has become very important to develop robust and efficient shape optimization algorithms. The field of Computer Aided Optimal Shape Design has grown substantially over the recent past. In the early days of its development, the method based on small shape perturbation to probe the parameter space and identify an optimal shape was routinely used. This method is nothing but an educated trial and error method. A key development in the pursuit of good shape optimization algorithms has been the advent of the adjoint method to compute the shape sensitivities more formally and efficiently. While undoubtedly, very attractive, this method relies on very sophisticated and advanced mathematical tools which are an impediment to its wider use in the engineering community. It that spirit, it is the purpose of this thesis to propose a new shape optimization algorithm based on more intuitive engineering principles and numerical procedures. In this thesis, the new shape optimization procedure which is proposed is based on the generation of a body-fitted mesh. This process maps the physical domain into a regular computational domain. Based on simple arguments relating to the use of the chain rule in the mapped domain, it is shown that an explicit expression for the shape sensitivity can be derived. This enables the computation of the shape sensitivity in one single solve, a performance analogous to the adjoint method, the current state-of-the art. The discretization is based on the Finite Difference method, a method chosen for its simplicity and ease of implementation. This algorithm is applied to the Laplace equation in the context of heat transfer problems and potential flows. The applicability of the proposed algorithm is demonstrated on a number of benchmark problems which clearly confirm the validity of the sensitivity analysis, the most important aspect of any shape optimization problem. This thesis also explores the relative merits of different minimization algorithms and proposes a technique to “fix” meshes when inverted element arises as part of the optimization process. While the problems treated are still elementary when compared to complex multiphysics engineering problems, the new methodology presented in this thesis could apply in principle to arbitrary Partial Differential Equations.
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