Saddle Point Approximation

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. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.

The integral to be estimated is often of the form
:$\int_Cf(z)e^\,dz,$
where ''C'' is a contour, and λ is large. 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:
# ''C′'' passes through one or more zeros of the derivative ''g''′(''z''),
# the imaginary part of ''g''(''z'') is constant on ''C′''.

The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note by about hypergeometric functions. The contour of steepest descent has a minimax property, see . described some other unpublished notes of Riemann, where he used this method to derive the Riemann–Siegel formula.

Basic idea

The method of steepest descent is a method to approximate a complex integral of the form$I(\lambda) = \int_Cf(z)e^\,\mathrmz$for large $\lambda \rightarrow \infty$, where $f(z)$ and $g(z)$ are analytic functions of $z$. Because the integrand is analytic, the contour $C$ can be deformed into a new contour $C'$ without changing the integral. In particular, one seeks a new contour on which the imaginary part of g(z) = \text (z)+ i \, \text (z)/math> is constant. Then $I(\lambda) = e^ \int_f(z)e^\,\mathrmz,$and the remaining integral can be approximated with other methods like Laplace's method.

Etymology

The method is called the method of steepest descent because for analytic $g(z)$, constant phase contours are equivalent to steepest descent contours.

If $g(z) = X(z) + i Y(z)$ is an analytic function of $z = x + i y$, it satisfies the Cauchy–Riemann equations$\frac = \frac \qquad \text \qquad \frac = - \frac .$Then $\frac \frac + \frac \frac = \nabla X \cdot \nabla Y = 0,$so contours of constant phase are also contours of steepest descent.

A simple estimate

Let and . If

:$M = \sup_ \Re (S(x)) < \infty,$

where $\Re (\cdot)$ denotes the real part, and there exists a positive real number such that

:$\int_ \left, f(x) e^ \ dx < \infty,$

then the following estimate holds:
:$\left, \int_ f(x) e^ dx \ \leqslant \text\cdot e^, \qquad \forall \lambda \in \mathbb, \quad \lambda \geqslant \lambda_0.$
Proof of the simple estimate:

:$\begin \left, \int_ f(x) e^ dx \ &\leqslant \int_C , f(x), \left, e^ \ dx \\ &\equiv \int_ , f(x), e^ \left , e^ e^ \ dx \\ &\leqslant \int_C , f(x), e^ \left, e^ \ dx && \left, e^ \ \leqslant 1 \\ &= \underbrace_ \cdot e^. \end$

The case of a single non-degenerate saddle point

Basic notions and notation

Let be a complex -dimensional vector, and

:$S''_(x) \equiv \left( \frac \right), \qquad 1\leqslant i,\, j\leqslant n,$

denote the Hessian matrix for a function . If

:$\boldsymbol(x) = (\varphi_1(x), \varphi_2(x), \ldots, \varphi_k(x))$

is a vector function, then its Jacobian matrix is defined as

:$\boldsymbol_x' (x) \equiv \left( \frac \right), \qquad 1 \leqslant i \leqslant k, \quad 1 \leqslant j \leqslant n.$

A non-degenerate saddle point, , of a holomorphic function is a critical point of the function (i.e., ) where the function's Hessian matrix has a non-vanishing determinant (i.e., $\det S''_(z^0) \neq 0$).

The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point:

Complex Morse lemma

The Morse lemma for real-valued functions generalizes as follows for holomorphic functions: near a non-degenerate saddle point of a holomorphic function , there exist coordinates in terms of which is exactly quadratic. To make this precise, let be a holomorphic function with domain , and let in be a non-degenerate saddle point of , that is, and $\det S''_(z^0) \neq 0$. Then there exist neighborhoods of and of , and a bijective holomorphic function with such that

:$\forall w \in V: \qquad S(\boldsymbol(w)) = S(z^0) + \frac \sum_^n \mu_j w_j^2, \quad \det\boldsymbol_w'(0) = 1,$

Here, the are the eigenvalues of the matrix $S_''(z^0)$.

The asymptotic expansion in the case of a single non-degenerate saddle point

Assume
# and are holomorphic functions in an open, bounded, and simply connected set such that the is connected;
# $\Re(S(z))$ has a single maximum: $\max_ \Re(S(z)) = \Re(S(x^0))$ for exactly one point ;
# is a non-degenerate saddle point (i.e., and $\det S''_(x^0) \neq 0$).

Then, the following asymptotic holds
e^ \left(f(x^0)+ O\left(\lambda^\right) \right) \prod_^n (-\mu_j)^, \qquad \lambda \to \infty,,
where are eigenvalues of the Hessian $S''_(x^0)$ and $(-\mu_j)^$ are defined with arguments

This statement is a special case of more general results presented in Fedoryuk (1987).
\int_0^ e^ d\xi = \sqrt (-\mu_j)^.

Equation (8) can also be written as

e^ \left ( \det (-S_''(x^0)) \right )^ \left (f(x^0) + O\left(\lambda^\right) \right),,

where the branch of

:$\sqrt$

is selected as follows

:$\begin \left (\det \left (-S_''(x^0) \right ) \right)^ &= \exp\left( -i \text \left (- S_''(x^0) \right ) \right) \prod_^n \left, \mu_j \^, \\ \text \left (-S_''(x^0) \right) &= \tfrac \sum_^n \arg (-\mu_j), && , \arg(-\mu_j), < \tfrac. \end$

Consider important special cases:

* If is real valued for real and in (aka, the multidimensional Laplace method), then $\text \left(-S_''(x^0) \right ) = 0.$
* If is purely imaginary for real (i.e., $\Re(S(x)) = 0$ for all in ) and in (aka, the multidimensional stationary phase method), thenSee equation (2.2.6') on page 186 in $\text \left (-S_''(x^0) \right ) = \frac \textS_''(x_0),$ where $\textS_''(x_0)$ denotes the signature of matrix $S_''(x_0)$, which equals to the number of negative eigenvalues minus the number of positive ones. It is noteworthy that in applications of the stationary phase method to the multidimensional WKB approximation in quantum mechanics (as well as in optics), is related to the Maslov index see, e.g., and .

The case of multiple non-degenerate saddle points

If the function has multiple isolated non-degenerate saddle points, i.e.,

:$\nabla S \left (x^ \right ) = 0, \quad \det S''_ \left (x^ \right ) \neq 0, \quad x^ \in \Omega_x^,$

where

:$\left \_^K$

is an open cover of , 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 such that

:$\begin \sum_^K \rho_k(x) &= 1, && \forall x \in \Omega_x, \\ \rho_k(x) &= 0 && \forall x \in \Omega_x\setminus \Omega_x^. \end$

Whence,

:$\int_ f(x) e^ dx \equiv \sum_^K \int_ \rho_k(x) f(x) e^ dx.$

Therefore as we have:

:$\sum_^K \int_ f(x) e^ dx = \left(\frac\right)^ \sum_^K e^ \left ( \det \left(-S_'' \left (x^ \right )\right) \right)^ f \left (x^ \right ),$

where equation (13) was utilized at the last stage, and the pre-exponential function at least must be continuous.

The other cases

When and $\det S''_(z^0) = 0$, the point is called a degenerate saddle point of a function .

Calculating the asymptotic of

:$\int f(x) e^ dx,$

when is continuous, and 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 into one of the multitude of canonical representations. For further details see, e.g., and .

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., and/or are discontinuous or when an extremum of lies at the integration region's boundary, require special care (see, e.g., and ).

Extensions and generalizations

An extension of the steepest descent method is the so-called ''nonlinear stationary phase/steepest descent method''. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.

Given a contour ''C'' in the complex sphere, a function ''f'' defined on that contour and a special point, say infinity, one seeks a function ''M'' holomorphic away from the contour ''C'', with prescribed jump across ''C'', and with a given normalization at infinity. 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 idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

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.

See also

* Pearcey integral
* Stationary phase approximation
* Laplace's method

Notes

References

*
* English translation in
*.
*.
*.
* n Russian
*.
* (Unpublished note, reproduced in Riemann's collected papers.)
* Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966.
**Translated in .
*.
*
*{{Citation
, last1=Wong
, first1=R.
, title=Asymptotic approximations of integrals
, publisher=Academic Press
, year=1989
.

Asymptotic analysis
Perturbation theory