8,996 research outputs found
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
Certified Roundoff Error Bounds Using Semidefinite Programming.
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementation. This problem becomes challenging when the program does not employ solely linear operations as non-linearities are inherent to many interesting computational problems in real-world applications. Existing solutions to reasoning are limited in the presence of nonlinear correlations between variables, leading to either imprecise bounds or high analysis time. Furthermore, while it is easy to implement a straightforward method such as interval arithmetic, sophisticated techniques are less straightforward to implement in a formal setting. Thus there is a need for methods which output certificates that can be formally validated inside a proof assistant. We present a framework to provide upper bounds on absolute roundoff errors. This framework is based on optimization techniques employing semidefinite programming and sums of squares certificates, which can be formally checked inside the Coq theorem prover. Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization and space control. Our tool produces more precise error bounds for 37 percent of all programs and yields better performance in 73 percent of all programs
Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
Approximating the set of reachable states of a dynamical system is an
algorithmic yet mathematically rigorous way to reason about its safety.
Although progress has been made in the development of efficient algorithms for
affine dynamical systems, available algorithms still lack scalability to ensure
their wide adoption in the industrial setting. While modern linear algebra
packages are efficient for matrices with tens of thousands of dimensions,
set-based image computations are limited to a few hundred. We propose to
decompose reach set computations such that set operations are performed in low
dimensions, while matrix operations like exponentiation are carried out in the
full dimension. Our method is applicable both in dense- and discrete-time
settings. For a set of standard benchmarks, it shows a speed-up of up to two
orders of magnitude compared to the respective state-of-the art tools, with
only modest losses in accuracy. For the dense-time case, we show an experiment
with more than 10.000 variables, roughly two orders of magnitude higher than
possible with previous approaches
Asymptotic Derivation and Numerical Investigation of Time-Dependent Simplified Pn Equations
The steady-state simplified Pn (SPn) approximations to the linear Boltzmann
equation have been proven to be asymptotically higher-order corrections to the
diffusion equation in certain physical systems. In this paper, we present an
asymptotic analysis for the time-dependent simplified Pn equations up to n = 3.
Additionally, SPn equations of arbitrary order are derived in an ad hoc way.
The resulting SPn equations are hyperbolic and differ from those investigated
in a previous work by some of the authors. In two space dimensions, numerical
calculations for the Pn and SPn equations are performed. We simulate neutron
distributions of a moving rod and present results for a benchmark problem,
known as the checkerboard problem. The SPn equations are demonstrated to yield
significantly more accurate results than diffusion approximations. In addition,
for sufficiently low values of n, they are shown to be more efficient than Pn
models of comparable cost.Comment: 32 pages, 7 figure
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