In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

https://www3.nd.edu/~zxu2/acms60790S13/Shu-WENO-notes.pdf

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

/pdf/inverse-problems-a-bayesian-perspective-1dknxttxjb.pdf

Inverse problems: A Bayesian perspective

: This thesis applies neural network feature selection techniques to multivariate time series data to improve prediction of a target time series. Two approaches to feature selection are used. First, a subset enumeration method is used to determine which financial indicators are most useful for aiding in prediction of the S&P 500 futures daily price. The candidate indicators evaluated include RSI, Stochastics and several moving averages. Results indicate that the Stochastics and RSI indicators result in better prediction results than the moving averages. The second approach to feature selection is calculation of individual saliency metrics. A new decision boundary-based individual saliency metric, and a classifier independent saliency metric are developed and tested. Ruck's saliency metric, the decision boundary based saliency metric, and the classifier independent saliency metric are compared for a data set consisting of the RSI and Stochastics indicators as well as delayed closing price values. The decision based metric and the Ruck metric results are similar, but the classifier independent metric agrees with neither of the other metrics. The nine most salient features, determined by the decision boundary based metric, are used to train a neural network and the results are presented and compared to other published results. (AN)

/pdf/nonlinear-time-series-analysis-1j3a35n4y3.pdf

Nonlinear Time Series Analysis.

Statistical shape models for 3D medical image segmentation: a review.

Computational geometry

Multistep scattered data interpolation using compactly supported radial basis functions

This application-oriented work concerns the design of efficient, robust and reliable algorithms for the numerical simulation of multiscale phenomena. To this end, various modern techniques from scattered data modelling, such as splines over triangulations and radial basis functions, are combined with customized adaptive strategies, which are developed individually in this work. The resulting multiresolution methods include thinning algorithms, multi- levelapproximation schemes, and meshfree discretizations for transport equa- tions. The utility of the proposed computational methods is supported by their wide range of applications, such as image compression, hierarchical sur- face visualization, and multiscale flow simulation. Special emphasis is placed on comparisons between the various numerical algorithms developed in this work and comparable state-of-the-art methods. To this end, extensive numerical examples, mainly arising from real-world applications, are provided. This research monograph is arranged in six chapters: 1. Introduction; 2. Algorithms and Data Structures; 3. Radial Basis Functions; 4. Thinning Algorithms; 5. Multilevel Approximation Schemes; 6. Meshfree Methods for Transport Equations. Chapter 1 provides a preliminary discussion on basic concepts, tools and principles of multiresolution methods, scattered data modelling, multilevel methods and adaptive irregular sampling. Relevant algorithms and data structures, such as triangulation methods, heaps, and quadtrees, are then introduced in Chapter 2.

Multiresolution Methods in Scattered Data Modelling

Image compression by linear splines over adaptive triangulations

https://www.math.uni-hamburg.de/home/iske/papers/ader.pdf

ADER schemes on adaptive triangular meshes for scalar conservation laws

An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline weighted essentially non-oscillatory (WENO) reconstruction with high order flux evaluation. Polyharmonic splines are utilized in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible, and optimal in the associated Sobolev (Beppo-Levi) space. The flux evaluation is accomplished by solving generalized Riemann problems across cell interfaces. The mesh adaptation is performed through an a posteriori error indicator, which relies on the polyharmonic spline reconstruction scheme. The performance of the proposed method is illustrated by a series of numerical experiments, including linear advection, Burgers's equation, Smolarkiewicz's deformational flow test, and the five-spot problem.

/pdf/adaptive-ader-methods-using-kernel-based-polyharmonic-spline-2ysx3c3g2b.pdf

Armin Iske

Papers

Multistep scattered data interpolation using compactly supported radial basis functions

Multiresolution Methods in Scattered Data Modelling

Image compression by linear splines over adaptive triangulations

ADER schemes on adaptive triangular meshes for scalar conservation laws

Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction