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An adjoint state equation is introduced, including a new unknown variable. The adjoint method formulates the gradient of a function towards its parameters in a constraint optimization form. By using the dual form of this constraint optimization problem, it can be used to calculate the gradient very fast. A nice property is that the number of ...
[1] [2] It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order differential equations ˙ = where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables.
An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts.Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation.
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type (Ax, y) = (x, By). Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices
If (,,) is a probability space, (,) is a measurable space, and : is a (,)-valued random variable, then the probability distribution of is the pushforward measure of by onto (,). A natural " Lebesgue measure " on the unit circle S 1 (here thought of as a subset of the complex plane C ) may be defined using a push-forward construction and ...
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
The Kolmogorov forward equation in the notation is just =, where is the probability density function, and is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that.
In this case, the adjoint map is given by Ad g (x) = gxg −1. If G is SL(2, R) (real 2×2 matrices with determinant 1), the Lie algebra of G consists of real 2×2 matrices with trace 0. The representation is equivalent to that given by the action of G by linear substitution on the space of binary (i.e., 2 variable) quadratic forms.