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Discrete processes and their continuous limits

Supported in part by NSERC Discovery Grant 84306
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  • The possibility that a discrete process can be fruitfully approximated by a continuous one, with the latter involving a differential system, is fascinating. Important theoretical insights, as well as significant computational efficiency gains may lie in store. A great success story in this regard are the Navier-Stokes equations, which model many phenomena in fluid flow rather well. Recent years saw many attempts to formulate more such continuous limits, and thus harvest theoretical and practical advantages, in diverse areas including mathematical biology, economics, finance, computational optimization, image processing, game theory, and machine learning.

    Caution must be applied as well, however. In fact, it is often the case that the given discrete process is richer in possibilities than its continuous differential system limit, and that a further study of the discrete process is practically rewarding. Furthermore, there are situations where the continuous limit process may provide important qualitative, but not quantitative, information about the actual discrete process. This paper considers several case studies of such continuous limits and demonstrates success as well as cause for caution. Consequences are discussed.

    Mathematics Subject Classification: Primary: 65K05, 65L04; Secondary: 68U10.


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  • Figure 1.  Exponential filter $ \omega (s) = 1 - e^{-ts} $ and Tikhonov filter $ \omega_{\rm T} (s) = \frac {s}{s + \beta} $ for $ t\beta = 1/2 $

    Figure 2.  Gradient descent with step sizes by (26) for the discretized heat equation with $N = 63^2 $. The Calculated Step Sizes Are Displayed Vs Iteration Counter $K $. The stability limit for a constant step size gives the straight blue line

    Figure 3.  Convergence behaviour of LSD for the model Poisson problem with $ n = 63^2 $. The errors $ \| {\bf r}_k \| $ are displayed as a function of iteration counter $ k $

    Figure 4.  Convergence behaviour of LSD for the model Poisson problem with $ n = 63^2 $. The errors $ f( {\bf x}_k) - f( {\bf x}^*) $ are displayed as a function of iteration counter $ k $

    Figure 5.  Convergence behaviour of monotonized LSD (LSDm) as well as SD, conjugate gradient (CG) and Nesterov's (Nes) for the model Poisson problem with $ n = 63^2 $. The errors $ \| {\bf r}_k \| $ are displayed as a function of iteration counter $ k $

    Figure 6.  Convergence behaviour of monotonized LSD (LSDm) as well as SD, conjugate gradient (CG) and Nesterov's (Nes) for the model Poisson problem with $ n = 63^2 $. The errors $ f( {\bf x}_k) - f( {\bf x}^*) $ are displayed as a function of iteration counter $ k $

    Table 1.  Maximum time step $ h $ for the heat-to-Poisson process as a function of spatial step $ \xi $

    $ \xi $ $ 2^{-5} $ $ 2^{-6} $ $ 2^{-7} $ $ 2^{-8} $
    $ h $ .05 .039 .043 .035
     | Show Table
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