Parallel Implementations of the Jacobi Linear Algebraic Systems Solve

Data Rates for Network Linear Equations

“…Theorem 2 (Low Data Rate) Suppose A1, A2, and A3 hold, with F d and M (α, h) as defined in (9) and (10). Then the following hold.…”

“…Remark 2 Theorem 1 shows that by using a scaling function decaying exponentially and a uniform quantizer, Algorithm 1 can ensure asymptotic convergence to the unique solution. It is worth pointing out that for any given α, h, the obtained quantization level K(α, h) is conservative, while (10) gives us some intuition on the relationship between the number of bits required and the control gains and the scaling factor. In addition, Theorem 1 gives an estimate of the rate of convergence: the smaller the scaling factor α, the faster the convergence rate from (13) but more bits have to be communicated by (10), and, if α → ρ h , the required number of bits goes to infinity.…”

“…It is worth pointing out that for any given α, h, the obtained quantization level K(α, h) is conservative, while (10) gives us some intuition on the relationship between the number of bits required and the control gains and the scaling factor. In addition, Theorem 1 gives an estimate of the rate of convergence: the smaller the scaling factor α, the faster the convergence rate from (13) but more bits have to be communicated by (10), and, if α → ρ h , the required number of bits goes to infinity. Thus, an appropriate selection of α amounts to a tradeoff between the rate of convergence and the communication overhead.…”

“…Then α − ρ h = ǫhλ min (F d ) > 0 by ρ h = 1 − hλ min (F d ), and α ∈ (1 − hλ min (F d ), 1). Also, by the definition M (α, h) in (10), we obtain the following:…”

“…Network linear equations also arise from resource allocation problems when node cost functions are quadratic, see [7,8,9]. In the context of parallel computation, computer scientists aimed to develop algorithms that eventually compute part entries of the solutions [10,11,12,13,14]. On the other hand, in view of distributed gradient optimization [15,16,17], distributed algorithms that compute the entire solution vector at each node were also proposed for both discrete-time and continuous-time node dynamics [18,6,19,20,21,22,23,24,25].…”