inis 16(9): e1

Research Article

Split and Merge Strategies for Solving Uncertain Equations Using Affine Arithmetic

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  • @ARTICLE{10.4108/eai.24-8-2015.2260594,
        author={Oliver Scharf and Markus Olbrich and Erich Barke},
        title={Split and Merge Strategies for Solving Uncertain Equations Using Affine Arithmetic},
        journal={EAI Endorsed Transactions on Industrial Networks and Intelligent Systems},
        keywords={split, merge, circuit simulation, implicit equations, uncertain, parametric, non-linear},
  • Oliver Scharf
    Markus Olbrich
    Erich Barke
    Year: 2015
    Split and Merge Strategies for Solving Uncertain Equations Using Affine Arithmetic
    DOI: 10.4108/eai.24-8-2015.2260594
Oliver Scharf1,*, Markus Olbrich1, Erich Barke1
  • 1: Institute of Microelectronic Systems, Leibniz Universit√§t Hannover, Germany
*Contact email:


The behaviour of systems is determined by various parameters. Due to several reasons like e. g. manufacturing tolerances these parameters can have some uncertainties. Corner Case and Monte Carlo simulations are well known approaches to handle uncertain systems. They sample the corners and random points of the parameter space, respectively. Both require many runs and do not guarantee the inclusion of the worst case. As alternatives, range based approaches can be used. They model parameter uncertainties as ranges. The simulation outputs are ranges which include all possible results created by the parameter uncertainties. One type of range arithmetic is the affine arithmetic, which allows to maintain linear correlations to avoid over-approximation. An equation solver based on affine arithmetic has been proposed earlier. Unlike many other range based approaches it can solve implicit non-linear equations. This is necessary for analog circuit simulation. For large uncertainties the solver suffers from convergence problems. To overcome these problems it is possible to split the parameter ranges, calculate the solutions separately and merge them again. For higher dimensional systems this leads to excessive runtimes as each parameter is split. To minimize the additional runtime several split and merge strategies are proposed and compared using two analog circuit examples.