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# Efficient Method for Computing Gradient, Analytic and Pseudo Analytic Sensitivities for Computer Aided Design and Optimization

IP.com Disclosure Number: IPCOM000049169D
Original Publication Date: 1982-May-01
Included in the Prior Art Database: 2005-Feb-09
Document File: 3 page(s) / 59K

IBM

## Related People

Chang, CS: AUTHOR [+2]

## Abstract

Two methods are described here for Computer-Aided Design (CAD) programs that compute sensitivities and perform optimization: (a) Computing sensitivities with respect to arbitrary type or design parameters, especially those that occur in complicated expressions in integrated circuitry. (b) An efficient technique to compute the gradient of an objective function in time-domain optimization. where this technique avoids the excessive storage requirement of the adjoint method. Sensitivity Computations:

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Efficient Method for Computing Gradient, Analytic and Pseudo Analytic Sensitivities for Computer Aided Design and Optimization

Two methods are described here for Computer-Aided Design (CAD) programs that compute sensitivities and perform optimization:
(a) Computing sensitivities with respect to arbitrary type

or design

parameters, especially those that occur in complicated

expressions

in integrated circuitry.
(b) An efficient technique to compute the gradient of an

objective

function in time-domain optimization. where this

technique avoids

the excessive storage requirement of the adjoint method. Sensitivity Computations:

A CAD program solves an algebraic equation:

f (x,p) = 0 the steady-state (DC) mode of analysis, and a differential equation: g (x, (see original), p, t) (2) in the transient mode.

For the DC case. the sensitivity equations are:

J/T/ y = (see original) (3) and

(see original) where J is the Jacobian (circuit) matrix that is used in solving (1), using the Newton formula, (see original) = (0, 0, ..., 1, 0. ...
0)/T/

(see original)

The superscript, T, indicates transpose. The matrix B is characterized uniquely by the table on the next page for the network parameters indicated, where in the table, B.j is the jth column of B corresponding to the jth sensitivity parameter.

Using (2) the basic sensitivity relation is obtained by differentiating (2) with respect to p, i.e., (see original) (see original) (see original) where . = d/dt and g(r) = (see original) for every g and r.

If we approximate the derivative by a backward Euler, then (5) becomes: (see original) where n is the integration step number' The right-hand side (RHS) of (6) consists of two terms, the DC component.
i.e., the matrix

B = -f(p) and a dynamic component. For a wide class of

1

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circuits the DC component predominates and, therefore, if the component is neglected, (6) becomes n+l

J . x(p) = B (7) (Note the J and B are the same as those of (3) and (4), and (3) and (4) can be derived from (7) in e straightforward manner.) Computing Sensitivities with Respect to Arbitrary Design Parameters

If the sensitivity parameter is a network element belonging to the set of parameters in the first column of the table, then using (3), (4) and the table, the sensitivities computed are analytic, since each parameter of p(j) contributes a unique column, B.j, to the matrix B, assuming modified nodal formulation.

Now, assume that p is not in that set of parameters but an independent var...