Volterra expansion

As in the theory of functions, a calculus exists for functionals. This calculus provides the tools necessary to develop and implement density functional theory. We begin with the discussion of expansions of functionals, which plays an important role in developing models within DFT and in deriving perturbation expansions. Analogous to the Taylor Series expansion for a function, a functional can be expanded about a reference function. This expansion, called a Volterra expansion, exists provided the functional has functional derivatives to any order and provided the last term in the infinite expansion has limit zero. Assuming these conditions, the Volterra expansion of ft[p] about a reference function, po, is given by... 

The Volterra Series. For a time invariant system defined by equation 4.25, it is possible to form a Taylor series expansion of the non-linear function to give [Priestley, 1988] ... 

The Volterra series is an expansion of the response signal into multi-dimensional convolution integrals,... 

They differ from the kernels it (ti, ..., r ) of the Volterra series only by a faster signal decay with increasing time arguments [Bliil]. For coinciding time arguments the crosscorrelation function is the sum of the n-dimensional impulse-response function h with the impulse-response functions hm of lower orders m < n. The stochastic impulse-response functions h are the kernels of an expansion of the system response y(t) similar to the Volterra series (4.2.4) but with functionals orthogonalized for white-noise excitation x t) [Bliil, Marl, Leel, Schl], This expansion is known by the name Wiener series, and the h are referred to as Wiener kernels. 

In the nonparametric approach, the input-output relation is represented either analytically (in convolutional form through Volterra-Wiener expansions where the unknown quantities are kernel functions). 

The inability to estimate the Volterra kernels in the general case of an infinite series prompted Wiener to suggest the orthogonalization of the Volterra series when a GWN test input is used. The functional terms of the Wiener series are constructed on the basis of a Gram-Schmidt orthogonalization procedure requiring that the covariance between any two Wiener functionals be zero. The resulting Wiener series expansion takes the form ... 

The Wiener kernels depend on the GWN input power level P (because they correspond to an orthogonal expansion), whereas the Volterra kernels are independent of any input characteristics. This situation can be likened to the coefficients of an orthogonal expansion of an analytic function being dependent on the interval of expansion. It is therefore imperative that Wiener kernel estimates be reported in the literature with reference to the GWN input power level that they were estimated with. When a complete set of Wiener kernels is obtained, then the complete set of Volterra kernels can be evaluated. Approximations of Volterra kernels can be obtained from Wiener kernels of the same order estimated with various input power levels. Complete Wiener or Volterra models can predict the system output to any given input. 

To reduce the requirements of long experimental data records and improve the kernel estimation accuracy, least-squares methods also can be used to solve the classical linear inverse problem described earlier in Equation 13.6, where the parameter vector 9 includes all discrete kernel values of the finite Volterra model of Equation 13.17, which is Knear in these unknown parameters (i.e., kernel values). Least-squares methods also can be used in connection with orthogonal expansions of the kernels to reduce the number of unknown parameters, as outlined below. Note that solution of this inverse problem via OLS requires inversion of a large square matrix with dimensions [(M -I- f -I- 1) /((M -F 1) / )], where M is... 

The mathematical relations between parametric (NARMAX) and nonparametric (Volterra) models have been explored, and significant benefits are shown to accrue from combined use [Zhao and Marmarehs, 1998]. These studies follow on previous efforts to relate certain classes of nonhnear differential equations with Volterra functional expansions in continuous time [Rugh, 1981 Marmarehs, 1989 Marmarehs, 2004]. 

Similar to Taylor series expansion, a Volterra series of indefinite length is needed for exact representation of a nonlinear system, but for practical applications, finite series can be used. 

We present here the operations of the functional derivative and functional Taylor expansion in a formal fashion, based on an analogy with the discrete case. For more details on the mathematical aspects, the reader is referred to Volterra (1931). 

A numerical tool, sensitivity analysis, which can be used to study the effects of parameter perturbations on systems of dynamical equations is briefly described. A straightforward application of the methods of sensitivity analysis to ordinary differential equation models for oscillating reactions is found to yield results which are difficult to physically interpret. In this work it is shown that the standard sensitivity analysis of equations with periodic solutions yields an expansion that contains secular terms. A Lindstedt-Poincare approach is taken instead, and it is found that physically meaningful sensitivity information can be extracted from the straightforward sensitivity analysis results, in some cases. In the other cases, it is found that structural stability/instability can be assessed with this modification of sensitivity analysis. Illustration is given for the Lotka-Volterra oscillator. 

This is the Neumann series expansion (Appendix 4) for the Volterra integral equation ... 

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