In the last two sections, we have used *Monte Carlo Sampling* and *Latin Hypercube Sampling* on the Rosenbrock function. Both of those algorithms suffered from some common
problems; they were slow to converge on a correct answer and their results
were non-deterministic, varying from run to run due to the inherent randomness.

The Smolyak algorithm is a significant improvement. It uses a smaller number of runs to generate a response surface.

To move from Monte Carlo to Latin Hypercube sampling, we had to change a single line in our control script. To use Smolyak, we simply change that same line:

```
uq = LHS([x,y], num=num)
```

to:

```
uq = Smolyak([x,y], level=2)
```

Or use ‘rosen.py’ in puq/examples/rosen.:

```
~/puq/examples/rosen> puq start rosen
Saving run to sweep_140682290.hdf5
Processing <HDF5 dataset "z": shape (13,), type "<f8">
Surface = 301.0*x**2 - 2.0*x + 300.0*y**2 - 266.667*y - 345.667
RMSE = 7.40e+02 (2.05e+01 %)
SENSITIVITY:
Var u* dev
-----------------------------
x 3.9402e+03 5.2374e+03
y 2.0828e+03 1.8510e+03
```

The puq program first created a file using the unique id it generated. That file is sweep_140682290.hdf5. All information about this run is placed in there. The puq program next ran 13 jobs, using the parameters calculated by the Smolyak method. Then it calculated a response surface.

For the Smolyak method, the response surface should ge a perfect fit if the polynomial degree of the function is less than or equal to the Smolyak level parameter. For our example, we used a level 2 Smolyak run to approzimate a The Rosenbrock function, which is 4th degree. Because of this, the response surface does not exactly fit the data points. The RMSE is 20.5%.

PUQ also calculates the sensitivity of the parameters using the Elementary Effects Method

We can plot the response surface and pdf:

```
~/puq/examples/rosen> puq plot -r sweep_140682290.hdf5
plotting PDF for z
```

And if you look at the response surface, you see the actual data as blue dots. Many of them are not on the response surface, indicating a bad fit.

For comparison, here is the expected rosenbrock function.

Next, if we bring up an editor and change the Smolyak level first to 3 then to 4, or do ‘puq extend’ twice, we get the following.

```
~/puq/examples/rosen> puq extend sweep_140682290.hdf5
Extending sweep_140682290.hdf5 Using Smolyak
Extending Smolyak to level 3
Processing <HDF5 dataset "z": shape (29,), type "<f8">
Surface = -200.0*x**2*y + 343.857*x**2 - 2.0*x + 100.0*y**2 - 136.143
RMSE = 2.40e+02 (6.66e+00 %)
SENSITIVITY:
Var u* dev
-----------------------------
x 4.8307e+03 6.5106e+03
y 2.0022e+03 1.7623e+03
```

```
~/puq/examples/rosen> puq extend sweep_140682290_A.hdf5
Extending sweep_140682290.hdf5 using Smolyak
Extending Smolyak to level 4
Processing <HDF5 dataset "z": shape (65,), type "<f8">
Surface = 100.0*x**4 - 200.0*x**2*y + 1.0*x**2 - 2.0*x + 100.0*y**2 + 1.0
RMSE = 3.96e-09 (1.10e-10 %)
SENSITIVITY:
Var u* dev
-----------------------------
x 5.0835e+03 6.9637e+03
y 1.9661e+03 1.7441e+03
```

With 65 samples Smolyak outperforms Monte Carlo with 1000 samples.

Note

Smolyak is only suitable for cases where the response function can be approximated
as a polynomial, at least over the input range with which we are concerned. For functions
with abrupt changes or discontinuities, *Latin Hypercube Sampling* is probably a better choice.