Method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase.
One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that the following conditions hold: The method of steepest descent was first published by Debye (1909), who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note by Riemann (1863) about hypergeometric functions.
The contour of steepest descent has a minimax property, see Fedoryuk (2001).
Siegel (1932) described some other unpublished notes of Riemann, where he used this method to derive the Riemann–Siegel formula.
denotes the real part, and there exists a positive real number λ0 such that then the following estimate holds:[2] Proof of the simple estimate: Let x be a complex n-dimensional vector, and denote the Hessian matrix for a function S(x).
The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point: The Morse lemma for real-valued functions generalizes as follows[3] for holomorphic functions: near a non-degenerate saddle point z0 of a holomorphic function S(z), there exist coordinates in terms of which S(z) − S(z0) is exactly quadratic.
To make this precise, let S be a holomorphic function with domain W ⊂ Cn, and let z0 in W be a non-degenerate saddle point of S, that is, ∇S(z0) = 0 and
[4] We begin by demonstrating From the identity we conclude that and Without loss of generality, we translate the origin to z0, such that z0 = 0 and S(0) = 0.
Using the Auxiliary Statement, we have Since the origin is a saddle point, we can also apply the Auxiliary Statement to the functions gi(z) and obtain Recall that an arbitrary matrix A can be represented as a sum of symmetric A(s) and anti-symmetric A(a) matrices, The contraction of any symmetric matrix B with an arbitrary matrix A is i.e., the anti-symmetric component of A does not contribute because Thus, hij(z) in equation (1) can be assumed to be symmetric with respect to the interchange of the indices i and j.
Note that hence, det(hij(0)) ≠ 0 because the origin is a non-degenerate saddle point.
Let us show by induction that there are local coordinates u = (u1, ... un), z = ψ(u), 0 = ψ(0), such that First, assume that there exist local coordinates y = (y1, ... yn), z = φ(y), 0 = φ(0), such that where Hij is symmetric due to equation (2).
By a linear change of the variables (yr, ... yn), we can assure that Hrr(0) ≠ 0.
, we write Motivated by the last expression, we introduce new coordinates z = η(x), 0 = η(0), The change of the variables y ↔ x is locally invertible since the corresponding Jacobian is non-zero, Therefore, Comparing equations (4) and (5), we conclude that equation (3) is verified.
are defined with arguments This statement is a special case of more general results presented in Fedoryuk (1987).
We employ the Complex Morse Lemma to change the variables of integration.
into a Taylor series and keep just the leading zero-order term Here, we have substituted the integration region Iw by Rn because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term.
of equation (11) can be expressed as From this representation, we conclude that condition (9) must be satisfied in order for the r.h.s.
, which is readily calculated Equation (8) can also be written as where the branch of is selected as follows Consider important special cases: If the function S(x) has multiple isolated non-degenerate saddle points, i.e., where is an open cover of Ωx, then the calculation of the integral asymptotic is reduced to the case of a single saddle point by employing the partition of unity.
The partition of unity allows us to construct a set of continuous functions ρk(x) : Ωx → [0, 1], 1 ≤ k ≤ K, such that Whence, Therefore as λ → ∞ we have: where equation (13) was utilized at the last stage, and the pre-exponential function f (x) at least must be continuous.
Calculating the asymptotic of when λ → ∞, f (x) is continuous, and S(z) has a degenerate saddle point, is a very rich problem, whose solution heavily relies on the catastrophe theory.
Here, the catastrophe theory replaces the Morse lemma, valid only in the non-degenerate case, to transform the function S(z) into one of the multitude of canonical representations.
Integrals with degenerate saddle points naturally appear in many applications including optical caustics and the multidimensional WKB approximation in quantum mechanics.
The other cases such as, e.g., f (x) and/or S(x) are discontinuous or when an extremum of S(x) lies at the integration region's boundary, require special care (see, e.g., Fedoryuk (1987) and Wong (1989)).
If f and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.
An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method.
The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of the Russian mathematician Alexander Its.
A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou.
As in the linear case, steepest descent contours solve a min-max problem.
In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).
The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.
