C++ Header-Only Method of Weighted Residuals for n-D PDEs
This library allows you to numerically solve problems stated as [linear operator + inhomogeneity] acting on a function in some domain. Solutions are represented as a linear combination of a basis set of functions and computed using the method of weighted residuals.
This is explained in more detail below.
The method of weighted lets you numerically solve problems of the form P[f] = 0, where f is a function in some domain D and P is an operator.
Defining the operator P[f] = Lf - r (linear + inhomogeneity), we try to find an approximate solution f_N.
We express f_N as a weighted sum of K basis functions f_k with an imposed inhomogeneity f_R:
f_N = f_R + sum(k < K, w_k*f_k)
where w_k are the weights. The "residual" R is the given by P[f_N] = R.
Integrating the residual R over the domain D and finding the weights w_k which minimize the expression results in a linear system which can be solved for the weights.
Note that the number of basis functions K and the basis functions themselves are chosen in advance.
This library allows you to choose P, D, f_k and K to solve for w_k.
Some example applications are:
- Fit data points in n-dimensions to some basis function representation
- Solve differential equations
- Least squares energy minimization for complex formulated problems
- Perform a discrete fourier / cosine transform on a non-regular grid
- Interpolate data
The library currently implements three weighted residual solvers:
- Least Squares Method Weighted Residual
- Galerkin
- Collocation Method (Residuals Disappear at Sample Points)
These methods are expressed as a linear system and solved using Eigen3.
The library currently includes the following basis sets:
- cosine functions, i.e. cosine transform
- taylor polynomials (i.e. regular polynomials)
- chebyshev polynomials
- complex fourier modes
- Arbitrary maps between R^n -> R^m, i.e. multidimensional problems
- Operator class for choosing the problem formulation
- Domain class for transforming coordinates / testing validity
- Better utilization of class inheritance to clean the code
- Mean squared error output
- Better templating
Data is currently being visualized using TinyEngine. This dependency will be separated out later.