This thesis aims to acquire, deepen and promote understanding of computing techniques for high dimensional scattered data approximation with radial basis functions. The main contributions of this thesis include sufficient conditions for the sovability of compactly supported radial basis functions with different shapes, near points preconditioning techniques for high dimensional interpolation systems with compactly supported radial basis functions, a heterogeneous hierarchical radial basis function interpolation scheme, which allows compactly supported radial basis functions of different shapes at the same level, an O(N) algorithm for constructing hierarchical scattered data set andan O(N) algorithm for sparse kernel summation on Cartesian grids. Besides the main contributions, we also investigate the eigenvalue distribution of interpolation matrices related to radial basis functions, and propose a concept of smoothness matching. We look at the problem from different perspectives, giving a systematic and concise description of other relevant theoretical results and numerical techniques. These results are interesting in themselves and become more interesting when placed in the context of the bigger picture. Finally, we solve several real-world problems. Presented applications include 3D implicit surface reconstruction, terrain modelling, high dimensional meteorological data approximation on the earth and scattered spatial environmental data approximation.

Mesh free modeling techniques are a promising alternative to traditional meshed methods for solving computational fluid dynamics problems. These techniques aim to solve for the field variable using solely the values of nodes and therefore do not require the generation of a mesh. This results in a process that can be much more reliably automated and is therefore attractive. Radial basis functions (RBFs) are one type of "meshless" method that has shown considerable growth in the past 50 years. Using these RBFs to directly solve a partial differential equation is known as Kansa's method and has been used to successfully solve many flow problems. The problem with Kansa's method is that there is no formal guarantee that its solution matrix will be non-singular. More recently, an expansion on Kansa's method was proposed that incorporates the boundary and PDE operators into the solution of the field variable. This method, known as Hermitian method, has been shown to be non-singular provided certain nodal criteria are met. This work aims to perform a comparison between Kansa and Hermitian methods to aid in future selection of a method. These two methods were used to solve steady and transient one-dimensional convection-diffusion problems. The methods are compared in terms of accuracy (error) and computational complexity (conditioning number) in order to evaluate overall performance. Results suggest that the Hermitian method does slightly outperform Kansa method at the cost of a more ill-conditioned collocation matrix.

The rapid increase in computational power over the last decade has unlocked the possibility of providing patient-specific healthcare via simulation and data assimilation. In the past 2 decades, computational approaches to simulating cardiovascular flows have advanced significantly due to intense research and adoption of methods in medical device companies. A significant source of friction in porting these tools to the hospital and getting in the hands of surgeons is due to the expertise required to handle the geometry pre-processing and meshing of models. Meshless meth- ods reduce the amount of corner cases which makes it easier to develop robust tools surgeons need. To accurately simulate modifications to a region of vasculature as in surgical planning, the entire system must be modeled. Unfortunately, this is computationally prohibitive even on to- day’s machines. To circumvent this issue, the Radial-Basis Function Finite Difference (RBF-FD) method for solution of the higher-dimensional (2D/3D) region of interest is tightly-coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The incompress- ible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and in- ductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF meshless approach is well-suited for modeling complicated non-Newtonian hemo- dynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of meshless methods and LPMs for fast and accurate hemody- namic simulations. The results show the efficacy of the method to be used in building robust tools to inform surgical decisions and further development is motivated.

We present a unifying framework for the global optimization of functions which are expensive to evaluate. The framework is based on a Bayesian interpretation of radial basis function interpolation which incorporates existing methods such as Kriging, Gaussian process regression and neural networks. This viewpoint enables the application of Bayesian decision theory to derive a sequential global optimization algorithm which can be extended to include existing algorithms of this type in the literature. By posing the optimization problem as a sequence of sampling decisions, we optimize a general cost function at each stage of the algorithm. An extension to multi-stage decision processes is also discussed. The key idea of the framework is to replace the underlying expensive function by a cheap surrogate approximation. This enables the use of existing branch and bound techniques to globally optimize the cost function. We present a rigorous analysis of the canonical branch and bound algorithm in this setting as well as newly developed algorithms for other domains including convex sets. In particular, by making use of Lipschitz continuity of the surrogate approximation, we develop an entirely new algorithm based on overlapping balls. An application of the framework to the integration of expensive functions over rectangular domains and spherical surfaces in low dimensions is also considered. To assess performance of the framework, we apply it to canonical examples from the literature as well as an industrial model problem from oil reservoir simulation.

Nonlinear multivariate statistical techniques on fast computers offer the potential to capture more of the dynamics of the high dimensional, noisy systems underlying financial markets than traditional models, while making fewer restrictive assumptions. This thesis presents a collection of practical techniques to address important estimation and confidence issues for Radial Basis Function networks arising from such a data driven approach, including efficient methods for parameter estimation and pruning, a pointwise prediction error estimator, and a methodology for controlling the "data mining'' problem. Novel applications in the finance area are described, including customized, adaptive option pricing and stock price prediction.

In this work the problem of calibrating Flush Air Data Sensing (FADS) has been addressed. The inverse problem of extracting freestream wind speed and angle of attack from pressure measurements has been solved. The aim of this work was to develop machine learning and statistical tools to optimize design and calibration of FADS systems. Experimental and Computational Fluid Dynamics (EFD and CFD) solve the forward problem of determining the pressure distribution given the wind velocity profile and bluff body geometry. In this work three ways are presented in which machine learning techniques can improve calibration of FADS systems. First, a scattered data approximation scheme, called Sequential Function Approximation (SFA) that successfully solved the current inverse problem was developed. The proposed scheme is a greedy and self-adaptive technique that constructs reliable and robust estimates without any user-interaction. Wind speed and direction prediction algorithms were developed for two FADS problems. One where pressure sensors are installed on a surface vessel and the other where sensors are installed on the Runway Assisted Landing Site (RALS) control tower. Second, a Tikhonov regularization based data-model fusion technique with SFA was developed to fuse low fidelity CFD solutions with noisy and sparse wind tunnel data. The purpose of this data model fusion approach was to obtain high fidelity, smooth and noiseless flow field solutions by using only a few discrete experimental measurements and a low fidelity numerical solution. This physics based regularization technique gave better flow field solutions compared to smoothness based solutions when wind tunnel data is sparse and incomplete. Third, a sequential design strategy was developed with SFA using Active Learning techniques from the machine learning theory and Optimal Design of Experiments from statistics for regression and classification problems. Uncertainty Sampling was used with SFA to demonstrate the effectiveness of active learning versus passive learning on a cavity flow classification problem. A sequential G-optimal design procedure was also developed with SFA for regression problems. The effectiveness of this approach was demonstrated on a simulated problem and the above mentioned FADS problem.

Surface reconstruction of anatomical structures is an integral part of medical modeling. Contour information is extracted from serial cross-sections of tissue data and is stored as "slice" files. Although there are several reasonably efficient triangulation algorithms that reconstruct surfaces from slice data, the models generated from them have a jagged or faceted appearance due to the large inter-slice distance created by the sectioning process. Moreover, inconsistencies in user input aggravate the problem. So, we created a method that reduces inter-slice distance, as well as ignores the inconsistencies in the user input. Our method called the piecewise weighted implicit functions, is based on the approach of weighting smaller implicit functions. It takes only a few slices at a time to construct the implicit function. This method is based on a technique called variational interpolation. <p> Other approaches based on variational interpolation have the disadvantage of becoming unstable when the model is quite large with more than a few thousand constraint points. Furthermore, tracing the intermediate contours becomes expensive for large models. Even though some fast fitting methods handle such instability problems, there is no apparent improvement in contour tracing time, because, the value of each data point on the contour boundary is evaluated using a single large implicit function that essentially uses all constraint points. Our method handles both these problems using a sliding window approach. As our method uses only a local domain to construct each implicit function, it achieves a considerable run-time saving over the other methods. The resulting software produces interpolated models from large data sets in a few minutes on an ordinary desktop computer.

Engineering aircraft systems is a complex task. Therefore models and computer simulations are needed to test functions and behaviors of non existing systems, reduce testing time and cost, reduce the risk involved and to detect problems early which reduce the amount of implementation errors. At the section Vehicle Simulation and Thermal Analysis at Saab Aeronautics in Linköping every basic aircraft system is designed and simulated, for example the fuel system. Currently 2-dimensional rectangular blocks are used in the simulation model to represent the fuel tanks. However, this is too simplistic to allow a more detailed analysis. The model needs to be extended with a more complex description of the tank geometry in order to get a more accurate model. This report explains the different steps in the developed methodology for combining 3-dimensional geometry models of any fuel tank created in CATIA with dynamic simulation of the fuel system in Dymola. The new 3-dimensional representation of the tank in Dymola should be able to calculate fuel surface location during simulation of a maneuvering aircraft.  The first step of the methodology is to create a solid model of the fuel contents in the tank. Then the area of validity for the model has to be specified, in this step all possible orientations of the fuel acceleration vector within the area of validity is generated. All these orientations are used in the automated volume analysis in CATIA. For each orientation CATIA splits the fuel body in a specified number of volumes and records the volume, the location of the fuel surface and the location of the center of gravity. This recorded data is then approximated with the use of radial basis functions implemented in MATLAB. In MATLAB a surrogate model is created which are then implemented in Dymola. In this way any fuel surface location and center of gravity can be calculated in an efficient way based on the orientation of the fuel acceleration vector and the amount of fuel. The new 3-dimensional tank model is simulated in Dymola and the results are compared with measures from the model in CATIA and with the results from the simulation of the old 2-dimensional tank model. The results shows that the 3-dimensional tank gives a better approximation of reality and that there is a big improvement compared with the 2-dimensional tank model. The downside is that it takes approximately 24 hours to develop this model. / Att utveckla ett nytt flygplanssystem är en väldigt komplicerad arbetsuppgift. Därför används modeller och simuleringar för att testa icke befintliga system, minska utvecklingstiden och kostnaderna, begränsa riskerna samt upptäcka problem tidigt och på så sätt minska andelen implementerade fel. Vid sektionen Vehicle Simulation and Thermal Analysis på Saab Aeronautics i Linköping designas och simuleras varje grundflygplanssystem, ett av dessa system är bränslesystemet. För närvarande används 2-dimensionella rätblock i simuleringsmodellen för att representera bränsletankarna, vilket är en väldigt grov approximation. För att kunna utföra mer detaljerade analyser behöver modellerna utökas med en bättre geometrisk beskrivning av bränsletankarna. Denna rapport går igenom de olika stegen i den framtagna metodiken för att kombinera 3- dimensionella tankmodeller skapade i CATIA med dynamisk simulering av bränslesystemet i Dymola. Den nya 3-dimensionella representationen av en tank i Dymola bör kunna beräkna bränsleytans läge under en simulering av ett manövrerande flygplan. Första steget i metodiken är att skapa en solid modell av bränslet som finns i tanken. Därefter specificeras modellens giltighetsområde och alla tänkbara riktningar hos accelerationsvektorn som påverkar bränslet genereras, dessa används sedan i den automatiserade volymanalysen i CATIA.  För varje riktning delar CATIA upp bränslemodellen i ett bestämt antal delar och registrerar volymen, bränsleytans läge samt tyngdpunktens position för varje del. Med hjälp av radiala basfunktioner som har implementerats i MATLAB approximeras dessa data och en surrogatmodell tas fram, denna implementeras sedan i Dymola. På så sätt kan bränsleytans och tyngdpunktens läge beräknas på ett effektivt sätt, baserat på riktningen hos bränslets accelerationsvektor samt mängden bränsle i tanken. Den nya 3-dimensionella tankmodellen simuleras i Dymola och resultaten jämförs med mätningar utförda i CATIA samt med resultaten från den gamla simuleringsmodellen. Resultaten visar att den 3-dimensionella tankmodellen ger en mycket bättre representation av verkligheten och att det är en stor förbättring jämfört med den 2-dimensionella representationen. Nackdelen är att det tar ungefär 24 timmar att få fram denna 3-dimensionella representation.

Surface reconstruction of anatomical structures is an integral part of medical modeling. Contour information is extracted from serial cross-sections of tissue data and is stored as "slice" files. Although there are several reasonably efficient triangulation algorithms that reconstruct surfaces from slice data, the models generated from them have a jagged or faceted appearance due to the large inter-slice distance created by the sectioning process. Moreover, inconsistencies in user input aggravate the problem. So, we created a method that reduces inter-slice distance, as well as ignores the inconsistencies in the user input. Our method called the piecewise weighted implicit functions, is based on the approach of weighting smaller implicit functions. It takes only a few slices at a time to construct the implicit function. This method is based on a technique called variational interpolation. <p> Other approaches based on variational interpolation have the disadvantage of becoming unstable when the model is quite large with more than a few thousand constraint points. Furthermore, tracing the intermediate contours becomes expensive for large models. Even though some fast fitting methods handle such instability problems, there is no apparent improvement in contour tracing time, because, the value of each data point on the contour boundary is evaluated using a single large implicit function that essentially uses all constraint points. Our method handles both these problems using a sliding window approach. As our method uses only a local domain to construct each implicit function, it achieves a considerable run-time saving over the other methods. The resulting software produces interpolated models from large data sets in a few minutes on an ordinary desktop computer.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 95-98). Also available on the Internet.