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6 1. INTRODUCTION equation with FSR model is as follows: ⎧ ⎨ 1 if v ij (n) > 1 x ij (n + 1) = v ij (k) if |v ij (n)| ≤ 1 ⎩ −1 if v ij (n) < −1 ( v ij (n) = (1 − h)x ij (n)+ ) ∑ ∑ +h A ij,kl x kl (n)+ B ij,kl u kl (n) + z ij C(kl)∈N r(i,j) C(kl)∈N r(i,j) (1.4) Now the x and y variables are combined by introducing a truncation, which is simple in the digital world from computational aspect. In addition, the h and (1-h) terms are included into the A and B template matrices resulting templates Â, ˆB. By using these modified template matrices, the iteration scheme is simplified to a 3×3 convolution plus an extra addition: ∑ v ij (n + 1) = Â ij,kl x kl (n) + g ij (1.5a) g ij = C (kl)∈N r(i,j) ∑ C (kl)∈N r(i,j) ˆB ij,kl u kl + hz ij (1.5b) If the input is constant or changing slowly, g ij can be treated as a constant and should be computed only once at the beginning of the computation. 1.1.2 Nonlinear templates The implementation of nonlinear templates are very difficult on analog VLSI and quite simple on emulated digital CNN. In some interesting spatio-temporal problems (Navier-Stokes equations) the nonlinear templates (nonlinear interactions) play key role. In general the nonlinear CNN template values are defined by an arbitrary nonlinear function of input variables (nonlinear B template), output variables (nonlinear A template) or state variables and may involve some time delays. The survey of the nonlinear templates shows that in many cases the nonlinear template values depend on the difference of the value of the currently processed cell (C ij ) and the value of the neighboring cell (C kl ). The Cellular Wave Computing Library [21] contains zero- and first-order nonlinear templates.
1.2 Cellular Neural/Nonlinear Network - Universal Machine 7 b 0.5 a 0.25 -0.125 -0.18 -0.5 0.18 v uij -v ukl -2 -1 v yij -v ykl (a) (b) Figure 1.3: Zero- (a) and first-order (b) nonlinearity In case of the zero-order nonlinear templates, the nonlinear functions of the template contains horizontal segments only as shown in Figure 1.3(a). This kind of nonlinearity can be used, e.g., for grayscale contour detection [21]. In case of the first-order nonlinear templates, the nonlinearity of the template contains straight line segments as shown in Figure 1.3(b). This type of nonlinearity is used, e.g., in the global maximum finder template [21]. Naturally, some nonlinear templates exist in which the template elements are defined by two or more nonlinearities, e.g., the grayscale diagonal line detector [21]. 1.2 Cellular Neural/Nonlinear Network - Universal Machine If we consider the CNN template as an instruction, we can make different algorithms, functions from these templates. In order to run these algorithms efficiently, the original CNN cell has to be extended (see Figure 1.4) [22]. extended architecture is the Cellular Neural/Nonlinear Network - Universal Machine (CNN-UM). According to the Turing-Church thesis in case of the algorithms, which are defined on integers or on a finite set of symbols, the Turing Machine, the grammar and the µ - recursive functions are equivalent. The CNN- UM is universal in Turing sense because every µ - recursive function can be computed on this architecture. The
Page 1: An Optimal Mapping of Numerical Sim
Page 5 and 6: Acknowledgements It is not so hard
Page 7 and 8: Contents 1 Introduction 1 1.1 Cellu
Page 9: CONTENTS iii 4.3.3 Operating Steps
Page 12 and 13: vi LIST OF FIGURES 2.4 Performance
Page 15: List of Tables 2.1 Comparison of di
Page 18 and 19: xii LIST OF ABBREVIATIONS FPE Falco
Page 21 and 22: Chapter 1 Introduction Due to the r
Page 23 and 24: 3 can be represented in arbitrary t
Page 25: 1.1 Cellular Neural/Nonlinear Netwo
Page 29 and 30: 1.3 CNN-UM Implementations 9 the Lo
Page 31 and 32: 1.3 CNN-UM Implementations 11 modif
Page 33 and 34: 1.4 Field Programmable Gate Arrays
Page 47 and 48: 1.5 IBM Cell Broadband Engine Archi
Page 55 and 56: Chapter 2 Mapping the Numerical Sim
Page 57 and 58: 2.1 Introduction 37 18 Load instruc
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Page 61 and 62: 2.1 Introduction 41 2E+07 1E+07 9E+
Page 63 and 64: 2.1 Introduction 43 Main memory 8 t
Page 65 and 66: 2.1 Introduction 45 6 4.5 3 Speedup
Page 67 and 68: 2.1 Introduction 47 2.50E+07 1.88E+
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2.3 Computational Fluid Flow Simula
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Chapter 3 Investigating the Precisi
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3.2 Numerical Solutions of the PDEs
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3.3 Testing Methodology 73 In fixed
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3.4 Properties of the Arithmetic Un
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3.4 Properties of the Arithmetic Un
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3.5 Results 79 in the L2 cache of t
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3.5 Results 81 1E-02 1E-03 Error 1E
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3.5 Results 83 arithmetic unit requ
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3.6 Conclusion 85 point and the 40
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4. IMPLEMENTING A GLOBAL ANALOGIC P
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110 5. SUMMARY OF NEW SCIENTIFIC RE
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References The author’s journal p
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5.2 Application of the results 123
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