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Methods of Numerical Integration

In this book the authors describe the principles and methods behind probabilistic forecasting and Bayesian data assimilation. Instead of focusing on particular application areas, the authors adopt a general dynamical systems approach, with a profusion of low-dimensional, discrete-time numerical examples designed to build intuition about the subject. Part I explains the mathematical framework of ensemble-based probabilistic forecasting and uncertainty quantification. Part II is devoted to Bayesian filtering algorithms, from classical data assimilation algorithms such as the Kalman filter, variational techniques, and sequential Monte Carlo methods, through to more recent developments such as the ensemble Kalman filter and ensemble transform filters. The McKean approach to sequential filtering in combination with coupling of measures serves as a unifying mathematical framework throughout Part II. Assuming only some basic familiarity with probability, this book is an ideal introduction for graduate students in applied mathematics, computer science, engineering, geoscience and other emerging application areas.

Probabilistic forecasting and Bayesian data assimilation

https://www.dam.brown.edu/AFOSR_MURI_pub/Ma_Zabaras_JCP.pdf

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations

Satellite observations identify the Mongolian steppes as a hotspot of global biomass reduction, the extent of which is comparable with tropical rainforest deforestation. To conserve or restore these grasslands, the relative contributions of climate and human activities to degradation need to be understood. Here we use a recently developed 21-year (1988–2008) record of satellite based vegetation optical depth (VOD, a proxy for vegetation water content and aboveground biomass), to show that nearly all steppe grasslands in Mongolia experienced significant decreases in VOD. Approximately 60% of the VOD declines can be directly explained by variations in rainfall and surface temperature. After removing these climate induced influences, a significant decreasing trend still persists in the VOD residuals across regions of Mongolia. Correlations in spatial patterns and temporal trends suggest that a marked increase in goat density with associated grazing pressures and wild fires are the most likely non-climatic factors behind grassland degradation.

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Changing Climate and Overgrazing Are Decimating Mongolian Steppes

Disclaimer: This pdf version differs slightly from the printed version. Few typos have been corrected! Acknowledgments This thesis would not have been possible without the direct and indirect contributions of several colleagues and friends to which I owe my greatest gratitude. Foremost, I am heartily thankful to my supervisor Hans-Joachim Bungartz for " panem et circenses " , enabling me to work on the topic of sparse grids in a positive and collaborative atmosphere at his chair. I have much appreciated his unconditional support, the open atmosphere, and the full confidence he always offered. His encouragement to look at the bigger picture led to exciting collaborations with other groups and to the challenging applications I was able to deal with. During my work I have learned a lot beyond the mere scientific scope. I would like to thank my reviewers Markus Hegland for inspiring discussions and his warm welcome and hospitality in Down Under (even enriched by a funny hunt for a fluffy Australian spider), and Christoph Zenger, the " grand seigneur of sparse grids " , for references to previous works and related problems. I am very grateful to all colleagues and collaborators. Special thanks go to Stefan Zim-mer for all his support, providing everything from a constant source of ideas and excellent fast (and, if necessary, last-minute) feedback, up to jelly beans and cappuccino. I would like to mention especially the sparse grid group at our chair with Janos Benk, Gerrit Buse, Daniel Butnaru, and Stefanie Schraufstetter. I am very pleased to thank Mona Frommert and Torsten Ensslin from the Max Planck Institute for Astrophysics for providing challenging applications and an excellent collaboration, and to Stefan Dirnstorfer and Andreas Grau for input and feedback regarding financial applications. Furthermore, I would like to thank all students who contributed to the SG ++ toolbox and to applications , in particular Finally, and most importantly, I would like to express my deepest gratefulness to my beloved wife Doris and my parents and sisters for all their constant support and patience throughout the last years. Without them, this thesis would have hardly been possible. Furthermore, I am very grateful to Martin who has been best friend, colleague, housemate, support, and much more, all in one. Abstract The curse of dimensionality, i.e. the exponential dependency of the overall computational effort on the number of dimensions, is still a roadblock for the numerical treatment …

Spatially Adaptive Sparse Grids for High-Dimensional Problems.

This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.

Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance

https://www.hcm.uni-bonn.de/fileadmin/user_upload/sparcity_and_computation/Griebel.pdf

Dimension-wise integration of high-dimensional functions with applications to finance

New regulations and a stronger competition have increased the importance of stochastic asset–liability management (ALM) models for insurance companies in recent years. In this paper, we propose a discrete time ALM model for the simulation of simplified balance sheets of life insurance products. The model incorporates the most important life insurance product characteristics, the surrender of contracts, a reserve-dependent bonus declaration, a dynamic asset allocation and a two-factor stochastic capital market. All terms arising in the model can be calculated recursively which allows an easy implementation and efficient simulation. Furthermore, the model is designed to have a modular organization which permits straightforward modifications and extensions to handle specific requirements. In a sensitivity analysis for sample portfolios and parameters, we investigate the impact of the most important product and management parameters on the risk exposure of the insurance company and show that the model captures the main behaviour patterns of the balance sheet development of life insurance products.

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A general asset–liability management model for the efficient simulation of portfolios of life insurance policies

In this chapter, we present various numerical experiments with different applications from finance. We study the performance of the different numerical quadrature methods from Chapter 3 and Chapter 4 and investigate the impact of the different approaches for dimension reduction and smoothing from Chapter 5. Parts of this chapter are taken from [58].

Validation and Applications

This chapter is concerned with sparse grid (SG) quadrature methods These methods are constructed using certain combinations of tensor products of one-dimensional quadrature rules They can exploit the smoothness of f, overcome the curse of dimension to a certain extent and profit from low effective dimensions, see, eg, [16, 44, 45, 57, 116, 146]

Markus Holtz

Papers

Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance

Dimension-wise integration of high-dimensional functions with applications to finance

A general asset–liability management model for the efficient simulation of portfolios of life insurance policies

Validation and Applications

Sparse Grid Quadrature