Construct a symbolic expression for an approximation of the mapping
PDEs of this form show up frequently in control theory as they are associated to invariant manifolds. This can be observed by considering the dynamic system given by the interconnection of the systems
While there are many approaches to determining the solution to a PDE numerically, sometimes it is useful to have a symbolic solution to the PDE (for example, the solution must be embedded into a model in model reduction, and having a lookup table instead could cause the model to act piecewise continuous rather than smooth if care is not taken). One approach is to numerically solve the PDE and then try to fit a linear combination of basis functions to the numerical data. In this package, the approach is simply to use symbolic computation to determine the coefficients of a Taylor series representation of the PDE solution.
Assuming that the mappings
Now we define a new operator
The approach to building the solution in the symbolic toolbox is as follows:
- build a symbolic series representation of
$\hat{\pi}(\cdot)$ up to the desired total order,$d$ - compose symbolic Taylor series representations of the mappings
$\hat{\pi}(\cdot)$ ,$s(\cdot)$ ,$\ell(\cdot)$ , and$f(\cdot,\cdot)$ to get a symbolic Taylor series representation of$e(\hat{\pi}(\cdot),\cdot)$ - take derivatives of
$e(\hat{\pi}(\cdot),\cdot)$ to isolate a symbolic equation for each Taylor series coefficient of$e(\hat{\pi}(\cdot),\cdot)$ up to the desired total order to form a system of equations, denoted$C(\hat{\pi},s,\ell,f)$ - the exact solution has Taylor series coefficients such that each obtained equation evaluates to zero, hence to obtain an approximate solution up to the desired total order
$d$ solve$C(\hat{\pi},s,\ell,f) = 0$ where the solution variables are the series coefficients of$\hat{\pi}(\cdot)$ - the resulting mappings have the property that
$$\pi(\omega) - \hat{\pi}(\omega) = \sum_{i = d+1}^{\infty} \sum_{|m| = i} \Pi_m \omega^m,$$ so the solution approximation error is$\theta(|\omega|^{d+1})$ and can be improved with each increment in the desired total order of the approximation
Proof that, under certain nonresonance conditions, the system of equations
- generating a reperesentation of all combinations of multi-set indices for a particular total order
- generating a vector containing symbolic representations of all monomials of a particular total order
- isolating each coefficient of a total order by taking high-order multivariable partial derivatives and substituting zeros
- the procedure of loading a known solution approximation of total order
$d$ , using it to determine the approximation coefficients of total order$d+1$ , combining them together, and saving the result as the new known solution approximation of total order$d+1$ - the structure of the solution properties (.csv), solution coefficients (.csv), and solution symfuns (.mat) files
- checking for consistency of the mappings given by the user; checking that the mappings form a well-posed and solvable PDE
It can require large amounts of memory for handling big systems of PDEs with many independent variables when determining the coefficients for higher order monomial terms (a particular issue for my work in nonlinear model reduction). To avoid having wasted computational effort if a memory issue arises, the package in its current form saves the solved coefficients to a file every time a family of monomials of particular degree has had their coefficients determined. That is, for example, if coefficients of monomials of the form x^d have been determined, but MATLAB runs out of memory in the middle of determining coefficients for monomials of the form x^{d+1}, then the coefficients for monomials up to and including x^d are not lost.