A laplace duality for integration
Résumé
We consider the integral v(y) = Ky f (x)dx on a domain Ky = {x ∈ R d : g(x) ≤ y}, where g is nonnegative and Ky is compact for all y ∈ [0, +∞). Under some assumptions, we show that for every y ∈ (0, ∞) there exists a distinguished scalar λy ∈ (0, +∞) such that
which is the counterpart analogue for integration of Lagrangian duality for optimization. A crucial ingredient is the Laplace transform, the analogue for integration of Legendre-Fenchel transform in optimization. In particular, if both f and g are positively homogeneous then λy is a simple explicitly rational function of y. In addition if g is quadratic form then computing v(y) reduces to computing the integral of f with respect to a specific Gaussian measure for which exact and approximate numerical methods (e.g. cubatures) are available.
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