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

Picard Operator

 * A good example

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