ShaneLogic/chebfun-resolution-strategies

Chebfun Resolution Strategies

Splitting Mode

Chebfuns can represent functions as either:

• Splitting OFF: Single global polynomial (one smooth piece)
• Splitting ON: Concatenation of smooth pieces (for piecewise smooth functions)

When to Use Each Mode

Splitting OFF (recommended for smooth functions):

• More efficient for smooth functions over large intervals
• Example: sin(x) over [0, 1000] as single polynomial
• Set: chebfunpref.setDefaults('splitting', 'off')

Splitting ON (for piecewise smooth or non-smooth functions):

• Handles functions with discontinuities, corners, or varying behavior
• Automatically subdivides intervals at detected features
• Example: Functions with absolute values or piecewise definitions

Sampling Preferences

minSamples Parameter

Factory default: 17

Defines minimum starting points for adaptive sampling:

• Sequence: 17, 33, 65... (values are 2^k + 1)
• Non-power-of-2 values are rounded UP to next such value

Process:

1. Sample function at minSamples points
2. Compute Chebyshev expansion coefficients
3. If tail is negligible, stop
4. Otherwise, double grid size (minus 1) and repeat

When to Increase:

• Functions with narrow spikes that might be missed (e.g., exp(-30*(x-0.47)^2))
• Need to capture sharp features between grid points

resampling Parameter

Factory default: 'off'

• 'off': Reuses values from nested Chebyshev grids (more efficient)
• 'on': Forces recomputation at every grid step

When to Use 'on':

• Functions that depend on grid size itself (e.g., length(x)*sin(15*x))
• Working with Chebops (where matrices change with grid size)

Length Limits

splitLength (Splitting ON Mode)

Factory default: 129 (configured as 160)

• If polynomial segment of splitLength is insufficient, interval is subdivided
• Can increase to allow longer segments before splitting (up to 513)

maxLength (Splitting OFF Mode)

Factory default: 2^16 + 1 = 65537

• Defines "giving-up point" for global polynomial representation
• If function cannot be resolved within maxLength points, constructor fails with warning

Adjustment Guidelines:

• Lower (e.g., 50): Examine low-degree interpolants
• Raise significantly (millions): Resolve high-degree smooth functions (e.g., |x|^(5/4))

Piecewise Smooth Functions

Many functions of interest are not globally smooth but piecewise smooth:

• Chebfun represents these as concatenations of smooth pieces
• Each smooth piece has its own polynomial representation
• Structure allows handling non-smooth functions by breaking them into manageable smooth segments

Use When

• Constructing chebfuns for functions with complex behavior
• Dealing with piecewise smooth functions
• Capturing narrow spikes or sharp features
• Optimizing performance for smooth vs. non-smooth functions
• Troubleshooting construction failures or poor resolution