Quantization in Control Systems and Forward Error Analysis of Iterative Numerical Algorithms

The use of control theory to study iterative algorithms, which can be considered as dynamical systems, opens many opportunities to find new tools for analysis of algorithms. In this paper we show that results from the study of quantization effects in control systems can be used to find systematic ways for forward error analysis of iterative algorithms. The proposed schemes are applied to the classical iterative methods for solving a system of linear equations. The obtained bounds are compared with bounds given in the numerical analysis literature

A Warm-start Interior-point Method for Predictive Control

In predictive control, a quadratic program (QP) needs to be solved at each sampling instant. We present a new warm-start strategy to solve a QP with an interior-point method whose data is slightly perturbed from the previous QP. In this strategy, an initial guess of the unknown variables in the perturbed problem is determined from the computed solution of the previous problem. We demonstrate the effectiveness of our warm-start strategy to a number of online benchmark problems. Numerical results indicate that the proposed technique depends upon the size of perturbation and it leads to a reduction of 30–74% in floating point operations compared to a cold-start interior-point method

Fast word-level power models for synthesis of FPGA-based arithmetic

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Perturbation analysis for word-length optimization

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A novel 2D filter design methodology

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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

A heuristic approach for multiple restricted multiplication

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