## Selecting Absolute Tolerance and Relative Tolerance for Simulation

SimBiology® uses AbsoluteTolerance and RelativeTolerance to control the accuracy of integration during simulation. Specifically, AbsoluteTolerance is used to control the largest allowable absolute error at any step during simulation. It controls the error when a solution is small. Intuitively, when the solution approaches 0, AbsoluteTolerance is the threshold below which you do not worry about the accuracy of the solution since it is effectively 0. RelativeTolerance controls the relative error of a single step of the integrator. Intuitively, it controls the number of significant digits in a solution, except when it is smaller than the absolute tolerance, and $-{\mathrm{log}}_{10}\left(RelativeTolerance\right)$ is the number of correct digits.

### Algorithm

At each simulation step i, the solver estimates the local error e in the state j of the simulation. The solver reduces the size of time step i until the error of the state satisfies:

$|e\left(i,j\right)|\le \mathrm{max}\left(RelativeTolerance\ast |y\left(i,j\right)|,AbsoluteTolerance\left(i,j\right)\right)$

Thus at state values of larger magnitude, the accuracy is determined by RelativeTolerance. As the state values approach zero, the accuracy is controlled by AbsoluteTolerance.

The correct choice of values for RelativeTolerance and AbsoluteTolerance varies depending on the problem. The default values may work for first trials of the simulation. As you adjust the tolerances, consider that there are trade-offs between speed and accuracy:

• If the simulation takes too long, you can increase (or loosen) the values of RelativeTolerance and AbsoluteTolerance at the cost of some accuracy.

• If the results seem inaccurate, you can decrease (or tighten) the relative tolerance values by dividing with 10N, where N is a real positive number. But this tends to slow down the solver.

• If the magnitude of the state values is high, you can decrease the relative tolerance to get more accurate results.

### Absolute Tolerance Scaling

How SimBiology uses AbsoluteTolerance to determine the error depends on whether the AbsoluteToleranceScaling property is enabled. By default, AbsoluteToleranceScaling is enabled which means each state has its own absolute tolerance that may increase over the course of simulation:

$AbsoluteTolerance\left(i,j\right)=CSAbsTol*Scale\left(i,j\right)$

CSAbsTol is the AbsoluteTolerance property defined in SolverOptions of the active configuration set object.

For a state that has a nonzero initial value, the scale is the maximum magnitude over the state, as seen over the simulation thus far:

$Scale\left(i,j\right)=\mathrm{max}\left(|y\left(1:i,j\right)|\right)$

For a state that has an initial value of zero, the scale is estimated as the state value after taking a trial step of size AbsoluteToleranceStepSize using the Euler method. Let us call this value ye(j). Then:

$Scale\left(i,j\right)=\mathrm{max}\left(|\left[ye\left(j\right);y\left(2:i,j\right)\right]|\right)$

If an initial state is zero and has no dynamic at time = 0, then:

$AbsoluteTolerance\left(i,j\right)=CSAbsTol$

Doses, events, and initial assignment rules at simulation time = 0 are not considered when calculating absolute tolerance scaling.

Get trial now