Error Estimation of Prompt Jump Approximation by Singular Perturbation Theory

Singular perturbation theory

When we apply the steady-state approximation (SSA) in chemical kinetics, we typically argue that some of the intermediates are highly reactive, so that they are removed as fast as they are made. We then set the corresponding rates of change to zero. What we are saying is not that these rates are identically zero, of course, but that they are much smaller than the other rates of reaction. The st...

Geometric Singular Perturbation Theory

Singular Perturbation Approximation of Balanced Systems

This paper relates the singular perturbation approximation technique for model reduction to the direct truncation technique if the system model to be reduced is stable, minimal, and internally balanced. It shows that these two methods constitute two fully compatible model reduction techniques for a continuous-time system, and both methods yield a stable, minimal, and internally balanced reduced...

Excess Error , Approximation Error , and Estimation Error

Another question of interest concerns the behaviour of a learning algorithm in the infinite sample limit: as it receives more and more data, does the algorithm converge to an optimal prediction rule, i.e. does the generalization error of the learned function approach the optimal error? Recall that for a distribution D on X × Y and a loss ` : Y × Y→[0,∞), the optimal error w.r.t. D and ` is the ...

Perturbation Theory and Backward Error

Because of the special structure of the equations AX XB = C the usual relation for linear equations "backward error = relative residual" does not hold, and application of the standard perturbation result for Ax = b yields a perturbation bound involving sep (A, B)x that is not always attainable. An expression is derived for the backward error of an approximate solution Y; it shows that the backw...