Derivative-Based Sensitivity

Derivative-based global sensitivity measures average squared gradients over the input distribution, cheaply bounding Sobol total effects.

Averaging local gradients globally

Derivative-based global sensitivity measures (DGSM) compute the expectation of squared partial derivatives over the full input distribution: nu_i = E[(dY/dX_i)^2]. Unlike a single local derivative, this averages over the whole uncertain range, giving a global measure while retaining the low cost of gradient evaluation.

Link to Sobol indices

Kronos motion — uncertainty

A key result is that DGSM upper-bound the Sobol total-effect indices, up to a constant depending on the input distribution (a Poincare constant). An input with a small DGSM therefore provably has a small total effect and can be screened out with confidence, even though DGSM does not give the exact index.

Computation

Sample inputs from their distribution
Evaluate gradients via adjoint, automatic differentiation, or finite differences
Average the squared partial derivatives per input

Advantages

When gradients are cheap, especially through an adjoint solver, DGSM are far less expensive than variance-based indices and scale to many inputs. They connect directly to active-subspace analysis, which uses the same averaged gradient outer-product matrix. This makes DGSM a natural first pass in gradient-rich workflows.

Cautions

DGSM require differentiability and can miss discontinuous or threshold behavior between sample points. They provide an upper bound rather than an exact apportionment, so a large DGSM does not by itself prove a large effect. Use them to eliminate unimportant inputs and to seed dimension reduction, then confirm the survivors with variance-based analysis if precision is needed.