Uncertainty Analysis

# Uncertain Systems

* Bounding the Dynamic Behavior of an Uncertain System via Polynomial Chaos-based Simulation

# Interval Arithmetic

* Should we be concerned about round-off error?: There are several extensions or libraries in the FORTRAN, Pascal, and the C++ programming languages with facilities for ¯xed or arbitrary precision interval arithmetic [1,2]. In addition, Maple V version 2 and Mathematica version 2.2 provide for "range" and "Interval" arithmetic, respectively. However, the "range" arithmetic in **Maple V version 2 (using evalr()) does not seem to do outward rounding (or outward-rounded I/O) properly**. For this reason, Maple was not able to provide guaranteed "traps" for several interval computations tested.
* Some disasters attributable to bad numerical computing
* Tutorial on Verified Scientific Computing
* Lecture Notes
* Interval Computations
* Interval arithmetic - Wikipedia, the free encyclopedia
* Computer Algebra Systems
* ROA Using Interval Arithmetic
* Interval and Related Software
* Interval analysis and computation
* www.interval-methods.de
* Applied Interval Analysis
* Interval Arithmetic: Interval type and basic interval operations-V1.2 - Application Center
* Applied Interval Analysis
* Interval Computations: Introduction, Uses, and Resources
* interval.louisiana.edu

## Global Optimization

* Global Optimization Methods based on Interval analysis

## Books

* Applied Interval Analysis
* Interval Methods for Circuit Analysis
* Introduction to Interval Analysis - lnk
* Robust Model-Based Fault Diagnosis For Dynamic Systems
* Applications of Interval Computations
* Interval Methods for Systems of Equations
* Global Optimization Using Interval Analysis

## Theses

* GPU Accelerated Parameter Estimation by Global Optimization using Interval Analysis

## Interval Transfer Function

* Model Order Reduction of Interval Systems

## Inequalities

* SOLVABILITY OF SYSTEMS OF LINEAR INTERVAL EQUATIONS
* Checking solvability of systems of interval linear equations and inequalities via mixed integer programming
* Sergiy Butenko

## Implementation

* Interval Arithmetic: Python Implementation and Applications

## Applications

* Interval Computations Introduction Uses and Resources

### Circuit Analysis

* Interval methods for analog circuits
* A comparison of interval methods in symbolic circuit analysis applications
* Interval methods for analog circuits
* Interval Methods for Circuit Analysis
* Interval Analysis of Linear Analog Circuits
* Combination of Symbolic and Interval-Numeric Methods for Analysis of Analog Circuits
* AUTOMATIC COMPUTATION OF A LINEAR INTERVAL ENCLOSURE
* Lubomir V. Kolev
* Reliable Computing
* Interval Arithmetic for the Design of Electrical Engineering Systems

# Affine Arithmetic

* Affine Arithmetic
* Libaffa
* ASOL

# Taylor Models

* Approximations polynomiales rigoureuses et applications
* Modern Map Methods in Particle Beam Physics (Chapter 2)
* Taylor models and other validated functional inclusion methods
* Using Taylor Models in Reachability Analysis of Nonlinear Hybrid Systems
* Taylor Model Based Integration
* Computation and Application of Taylor Polynomials with Interval Remainder Bounds
* Benoit's Slides 2011
* Rigorous Polynomial Approximations and Applications
* Verified Computations with Taylor Model Using Symbolic Algebra
* Verified Methods
* Introduction to Validated Computing

## Implementation

* TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS
* Bounding roundoff errors in Taylor models arithmetic
* Bernstein Polynomials

## Algebraic Equations

* Veriﬁcation of Invertibility of Complicated Functions over Large Domains
* Rigorous Numerical Analysis with High-Order Taylor Models
* Uncertainties in the solutions to boundary element method: an interval approach
* Modified-accelerated Krawczyk’s algorithm
* INTERVAL OPERATORS AND FIXED IKTERVALS
* Algorithms for Verified Linear Algebra Verifying the solution of a linear system
* The Krawczyk Algorithm: Rigorous Bounds for Linear Equation Solution on an FPGA
* Real Root Isolation of Polynomial Equations Based on Hybrid Computation

## Inverse Taylor Model

* Verified High-Order Inversion of Functional Dependencies and Interval Newton Methods
* Verification of Invertibility of Complicated Functions over Large Domains

## Solving ODE

* Interval Methods and Taylor Model Methods for ODEs
* Taylor Model Flowpipe Construction
* Rigorous Integration of Flows and ODEs using Taylor Models
* Suppression of the Wrapping Effect by Taylor Model-based Verified Integrators: Long-term Stabilization by Shrink Wrapping
* Suppression of the Wrapping Effect by Taylor Model-based Verified Integrators: Long-term Stabilization by Preconditioning
* Suppression of the Wrapping Effect by Taylor Model-based Verified Integrators: The Single Step
* Rigorous Analysis of Nonlinear Motion in Particle Accelerators
* Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models
* A Rigorous ODE Solver and Smale’s 14th Problem

## Implicit DAEs

* Computing Validated Solutions of Implicit Differential Equations
* Eﬃcient high-order methods for ODEs and DAEs
* Rigorous Numerical Analysis with High-Order Taylor Models

## Applications

* Graphical Algorithms
* Continuous Collision Detection

## Workshops

* SWIM 2013
* SWIM 2012
* SWIM 2011
* SWIM 2010
* SWIM 2009
* SWIM 2008

# Chebychev Models

* Rigorous Computation with Function Enclosures in Chebyshev Basis
* VERICOMP
* Chebychev Interpolations Model

# Ellipsoids

* Trajectory Enclosures for Nonlinear Systems with Uncertain Initial Conditions and Parameters

# The Inverse Problem for ODEs

* Solving the Inverse Problem for ODES Using the Picard Contraction Mapping
* Picard Operator

# Books

* Modeling, Design, and Simulation of Systems with Uncertainties
* Applied Interval Analysis
* Introduction to Interval Analysis

# People

* Martin Berz and Kyoko Makino
* Nedialkov
* Sriram Sankaranarayanan
* Bruce Krugh(CMU)
* Zhi Han
* Jiri Rohn
* Interval Analysis and its Applications
* Interval Computations
* Alexander Gofen (Analytical Continuation, ODE's and Automatic Differentiation Unified)
* Mark A. Stadtherr
* Stig Skelboe

# Tools

* OptiY
* COSY
* DAETS
* VNODE
* INTLAB
* INTLAB
* C-XSC
* Flow\*
* Sollya
* b4m
* Interval Tools for ODEs and DAEs
* Collection of Tools
* MC++
* intpakX (Slides)
* LdeApprox
* ODEintegrator
* List of Tools
* HyTech

# Formal Verification

* Deliverables - VEDECY
* Parameter Synthesis for Hybrid Systems with an Application to Simulink Models
* Automatic Verification And Analysis of Complex Systems

## People

* Mitra Research Group
* Cyberphysical Systems - Colorado
* Hybrid Systems Group - ETHZ
	+ Software

## Tools

* List of tools
* Sage
	+ Interval Arithmetic
* Python
	+ PyTaylor
* CheckMate
* d/dt
* HSolver
* Hybrid Toolbox
* HyTech
* Toolbox of Level Set Methods
* KeYmaera
* MATISSE
* MPT Toolbox
* PHAVer
* SpaceEx
* Passel
* UPPAAL Academic
* UPPAAL Commercial
* Collection of Tools

# Robust Control

* Industrial Robot Control System Parametric Design on the Base of Methods for Uncertain Systems Robustness
* Robustness of model based simulations