3,419 research outputs found
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
Toward accurate polynomial evaluation in rounded arithmetic
Given a multivariate real (or complex) polynomial and a domain ,
we would like to decide whether an algorithm exists to evaluate
accurately for all using rounded real (or complex) arithmetic.
Here ``accurately'' means with relative error less than 1, i.e., with some
correct leading digits. The answer depends on the model of rounded arithmetic:
We assume that for any arithmetic operator , for example or , its computed value is , where is bounded by some constant where , but
is otherwise arbitrary. This model is the traditional one used to
analyze the accuracy of floating point algorithms.Our ultimate goal is to
establish a decision procedure that, for any and , either exhibits
an accurate algorithm or proves that none exists. In contrast to the case where
numbers are stored and manipulated as finite bit strings (e.g., as floating
point numbers or rational numbers) we show that some polynomials are
impossible to evaluate accurately. The existence of an accurate algorithm will
depend not just on and , but on which arithmetic operators and
which constants are are available and whether branching is permitted. Toward
this goal, we present necessary conditions on for it to be accurately
evaluable on open real or complex domains . We also give sufficient
conditions, and describe progress toward a complete decision procedure. We do
present a complete decision procedure for homogeneous polynomials with
integer coefficients, {\cal D} = \C^n, and using only the arithmetic
operations , and .Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of
Computational Mathematics: Santander 2005, Cambridge University Press, March
200
Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
The reduced basis method is a model reduction technique yielding substantial
savings of computational time when a solution to a parametrized equation has to
be computed for many values of the parameter. Certification of the
approximation is possible by means of an a posteriori error bound. Under
appropriate assumptions, this error bound is computed with an algorithm of
complexity independent of the size of the full problem. In practice, the
evaluation of the error bound can become very sensitive to round-off errors. We
propose herein an explanation of this fact. A first remedy has been proposed in
[F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced
basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012)
539--542.]. Herein, we improve this remedy by proposing a new approximation of
the error bound using the Empirical Interpolation Method (EIM). This method
achieves higher levels of accuracy and requires potentially less
precomputations than the usual formula. A version of the EIM stabilized with
respect to round-off errors is also derived. The method is illustrated on a
simple one-dimensional diffusion problem and a three-dimensional acoustic
scattering problem solved by a boundary element method.Comment: 26 pages, 10 figures. ESAIM: Mathematical Modelling and Numerical
Analysis, 201
Ray casting implicit fractal surfaces with reduced affine arithmetic
A method is presented for ray casting implicit surfaces defined by fractal combinations of procedural noise functions. The method is robust and uses affine arithmetic to bound the variation of the implicit function along a ray. The method is also efficient due to a modification in the affine arithmetic representation that introduces a condensation step at the end of every non-affine operation. We show that our method is able to retain the tight estimation capabilities of affine arithmetic for ray casting implicit surfaces made from procedural noise functions while being faster to compute and more efficient to store
Accuracy-guaranteed bit-width optimization
Published versio
Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations
Floating point error is an inevitable drawback of embedded systems
implementation. Computing rigorous upper bounds of roundoff errors is
absolutely necessary to the validation of critical software. This problem is
even more challenging when addressing non-linear programs. In this paper, we
propose and compare two new methods based on Bernstein expansions and sparse
Krivine-Stengle representations, adapted from the field of the global
optimization to compute upper bounds of roundoff errors for programs
implementing polynomial functions. We release two related software package
FPBern and FPKiSten, and compare them with state of the art tools. We show that
these two methods achieve competitive performance, while computing accurate
upper bounds by comparison with other tools.Comment: 20 pages, 2 table
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