Inhalt

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   Introduction
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   Notations and the basics of interval analysis
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  Algorithms, numerical results and fonts
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   Notation of algorithms
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   Numerical results
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   Conventions
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   Intervals
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   Vectors of real numbers and intervals
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   Matrices of real numbers and intervals
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   Functions of interval vectors and matrices
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   Interval arithmetic
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  Properties of interval arithmetic
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   Dependency problem
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   Algebraic rules
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   Wrapping effect
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   Cluster effect
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   Function evaluation and enclosures
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   Natural function enclosure
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   Centered forms
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   Taylor forms
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   Taylor models
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   Derivatives and slopes
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   Contractors
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   Expression optimization
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   Interval arithmetic and computers
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   Nonlinear systems of equations
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   Basic approach: branch-and-bound
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   Branching / Subdivision
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   Subdivision using gaps
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   Handling of unbounded variables
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   Determining a direction for subdivision
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   Bisection
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   Multisection
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   Non-equidistant subdivision
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   Iterated subdivision
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   Subdivision strategy in SONIC
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   Numerical studies
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  Bounding / Contracting
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   Constraint propagation
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   Taylor refinement
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   Contraction by Taylor models
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   Interval Newton method
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   Preconditioning
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   Application scheme for contractors
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   Application of contractors in SONIC
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   Operating sequence for the branch-and-bound algorithm
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   Extended systems
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  Basics
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   Introduction of extended systems
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   Contractors on extended systems
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   Extended systems in SONIC
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   Separate analysis phase
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   A box-intern hierarchy controlling extended systems (BIH)
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   Studies for selecting extended systems
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   Skipping extended systems
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   Predicting contraction success
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   A new, adaptive strategy (AS)
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   Implementation details
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  Comparison of the strategies BIH and AS
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   Comparison in computation time and box number
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   Modifications of the adaptive strategy
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   Restarting
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   Varying parameter
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   Use all extended systems on designated recursion levels
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   Reducing E if system E is not applied
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   Averaging E depending on the recursion level
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   Changing the averaging for the gauges E
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   Volume computation
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   Using all extended systems
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   Using other extended systems
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   Survey of the utilization of the extended systems
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   Choosing the variables considered by the Newton method
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   Improvements for preconditioners
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   Interaction of branching and bounding strategies
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   Verification
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  Definitions
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   Preconditioning
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   Epsilon-inflation
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   Facets and subfacets
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  Verification for non-square systems
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   Underdetermined systems
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   Overdetermined systems
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  Verification for square systems
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   Newton test
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   Miranda test
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   Borsuk test
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   A test based on the topological degree
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   Intersecting boxes
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   Verification scheme
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   Practical details concerning verification
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   Lists of facets and subfacets
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   When to apply verification tests
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   Segregation of computation and verification
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   Row-wise verification
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   Subdividing facets
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   Subdivision of symmetrical facets
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   Variations of the degree test
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  Comparison of the verification tests
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   Theoretical comparisons
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   Numerical comparisons using SONIC
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   SONIC
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   Features and concept
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   Install and run SONIC
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   Interval libraries
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  Parallelization of the nonlinear solver
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   Fundamental concepts
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   OpenMP
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   MPI
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   Optimization
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   General approach
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   Unconstrained optimization
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   Constrained optimization
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   Parallelization of the optimizer
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   Other nonlinear solvers and optimizers
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   Conclusion
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   List of symbols
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   Test problems