## Speed Up Parameter Estimation Using Parallel Computing

### When to Use Parallel Computing for Parameter Estimation

You can use Simulink^{®}
Design Optimization™ software with Parallel Computing Toolbox™ software to speed up parameter estimation of Simulink models. Using parallel computing may reduce the estimation time in the
following cases:

The model contains a large number parameters to estimate, and the estimation method is specified as either

`Nonlinear least squares`

or`Gradient descent`

.The

`Pattern search`

method is selected as the estimation method.The model is complex and takes a long time to simulate.

When you use parallel computing, the software distributes independent simulations
to run them in parallel on multiple MATLAB^{®} sessions, also known as *workers*. The time
required to simulate the model dominates the total estimation time. Therefore,
distributing the simulations significantly reduces the estimation time.

For information on how the software distributes the simulations and the expected speedup, see How Parallel Computing Speeds Up Estimation.

For information on configuring your system and using parallel computing, see Use Parallel Computing for Parameter Estimation.

### How Parallel Computing Speeds Up Estimation

You can enable parallel computing with the ```
Nonlinear least
squares
```

, `Gradient descent`

and ```
Pattern
search
```

estimation methods.

#### Parallel Computing with Nonlinear least squares and Gradient descent Methods

When you select `Gradient descent`

as the estimation method,
the model is simulated during the following computations:

Objective value computation — One simulation per iteration

Objective gradient computations — Two simulations for every tuned parameter per iteration

Line search computations — Multiple simulations per iteration

The total time, $$Ttotal$$, taken per iteration to perform these simulations is given by the following equation:

$$Ttotal=T+(Np\times 2\times T)+(Nls\times T)=T\times (1+(2\times Np)+Nls)$$

where $$T$$ is the time taken to simulate the model and is assumed to be equal for all simulations, $$Np$$ is the number of parameters to estimate, and $$Nls$$ is the number of line searches. $$Nls$$ is difficult to estimate and you generally assume it to be equal to one, two, or three.

When you use parallel computing, the software distributes the simulations required for objective gradient computations. The simulation time taken per iteration when the gradient computations are performed in parallel, $$TtotalP$$, is approximately given by the following equation:

$$TtotalP=T+(ceil\left(\frac{Np}{Nw}\right)\times 2\times T)+(Nls\times T)=T\times (1+2\times ceil\left(\frac{Np}{Nw}\right)+Nls)$$

where $$Nw$$ is the number of MATLAB workers.

**Note**

The equation does not include the time overheads associated with configuring the system for parallel computing and loading Simulink software on the remote MATLAB workers.

The expected reduction of the total estimation time is given by the following equation:

$$\frac{TtotalP}{Ttotal}=\frac{1+2\times ceil\left(\frac{Np}{Nw}\right)+Nls}{1+(2\times Np)+Nls}$$

For example, for a model with `N`

,
_{p}=3`N`

, and
_{w}=4`N`

, the expected reduction
of the total estimation time equals $$\frac{1+2\times ceil\left(\frac{3}{4}\right)+3}{1+(2\times 3)+3}=0.6$$._{ls}=3

#### Parallel Computing with the Pattern search Method

The `Pattern search`

method uses search and poll sets to
create and compute a set of candidate solutions at each estimation iteration.

The total time, $$Ttotal$$, taken per iteration to perform these simulations, is given by the following equation:

$$Ttotal=(T\times Np\times Nss)+(T\times Np\times Nps)=T\times Np\times (Nss+Nps)$$

where $$T$$ is the time taken to simulate the model and is assumed to be equal for all simulations, $$Np$$ is the number of parameters to estimate, $$Nss$$ is a factor for the search set size, and $$Nps$$ is a factor for the poll set size. $$Nss$$ and $$Nps$$ are typically proportional to $$Np$$.

When you use parallel computing, Simulink
Design Optimization software distributes the simulations required for the search and
poll set computations, which are evaluated in separate `parfor`

(Parallel Computing Toolbox) loops. The simulation
time taken per iteration when the search and poll sets are computed in parallel, $$TtotalP$$, is given by the following equation:

$$\begin{array}{c}TtotalP=(T\times ceil(Np\times \frac{Nss}{Nw}))+(T\times ceil(Np\times \frac{Nps}{Nw}))\\ =T\times (ceil(Np\times \frac{Nss}{Nw})+ceil(Np\times \frac{Nps}{Nw}))\end{array}$$

where $$Nw$$ is the number of MATLAB workers.

**Note**

The equation does not include the time overheads associated with configuring the system for parallel computing and loading Simulink software on the remote MATLAB workers.

The expected speed up for the total estimation time is given by the following equation:

$$\frac{TtotalP}{Ttotal}=\frac{ceil(Np\times \frac{Nss}{Nw})+ceil(Np\times \frac{Nps}{Nw})}{Np\times (Nss+Nps)}$$

For example, for a model with `N`

,
_{p}=3`N`

,
_{w}=4`N`

, and
_{ss}=15`N`

, the expected speedup
equals $$\frac{ceil(3\times \frac{15}{4})+ceil(3\times \frac{2}{4})}{3\times (15+2)}=0.27$$._{ps}=2

Using the `Pattern search`

method with parallel computing may
not speed up the estimation time. When you do not use parallel computing, the
method stops searching for a candidate solution at each iteration as soon as it
finds a solution better than the current solution. When you use parallel
computing, the candidate solution search is more comprehensive. Although the
number of iterations may be larger, the estimation without using parallel
computing may be faster.