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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 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.
(Ax, y) = (x, By). Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices; Hermitian adjoint (adjoint of a linear operator) in functional analysis; Adjoint endomorphism of a Lie algebra; Adjoint representation of a Lie group; Adjoint functors in category theory; Adjunction (field theory)
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
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The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators. Named after Bernard Helffer and Johannes Sjöstrand , this formula provides a way to calculate functions of operators without requiring the operator to have a simple or explicitly known spectrum.