restricted
Restrict the growth rates for nested and non-nested rules can be synchronized for consistency.
Specification
Alias: None
Arguments: None
Default: restricted (except for generalized sparse grids)
Description
In the quadrature and sparse grid cases, growth rates for nested and non-nested rules can be synchronized for consistency. For a non-nested Gauss rule used within a sparse grid, linear one-dimensional growth rules of \(m=2l+1\) are used to enforce odd quadrature orders, where l is the grid level and m is the number of points in the rule. The precision of this Gauss rule is then \(i=2m-1=4l+1\) . For nested rules, order growth with level is typically exponential; however, the default behavior is to restrict the number of points to be the lowest order rule that is available that meets the one-dimensional precision requirement implied by either a level l for a sparse grid ( \(i=4l+1\) ) or an order m for a tensor grid ( \(i=2m-1\) ). This behavior is known as “restricted growth” or “delayed sequences.” To override this default behavior in the case of sparse grids, the unrestricted keyword can be used; it cannot be overridden for tensor grids using nested rules since it also provides a mapping to the available nested rule quadrature orders. An exception to the default usage of restricted growth is the dimension_adaptive p_refinement generalized sparse grid case described previously, since the ability to evolve the index sets of a sparse grid in an unstructured manner eliminates the motivation for restricting the exponential growth of nested rules.
