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Optimizing complex simulations? Use Scipy interpolation

Instead of using fancy optimization algorithms, in many cases, we can use simple interpolation (Scipy) for fast optimization Tirthajyoti Sarkar October 27, 20216 min read Image source: Created by the author Image source: Created by the author

Background

Interpolating experimental or simulation data in complex scientific or business studies is a fairly common practice. In my previous article, I showed how to use simple Scipy code to achieve that for one or more dimensions.

How to Interpolate Data with Scipy One of the key benefits of this approach is that we get the exact function from the Scipy routine and can use it for many purposes - one of them being fast optimization. This is particularly useful for the cases where the individual experiment is costly. In most cases, the cost aspect goes far beyond mere computation. Think of optimization problems related to expensive product design iteration, or healthcare (drug testing). As the designer or researcher, you won't get an unlimited budget to run hundreds of experiments as each experiment can cost thousands of dollars. Instead, you can, Learn this step by step with the interactive AI and Data Scientist and Computer Science roadmaps.

No fancy optimization algorithm. No cost of expensive simulation. Just a little bit of Design-of-experiment knowledge and interpolation routines. This method does not guarantee you the best outcome in all situations. But it works in most practical scenarios. Let's see why.

As the designer or researcher, you won't get an unlimited budget to run hundreds of experiments as each experiment can cost thousands of dollars

Why does it work (in most scenarios)?

Interpolation primarily works on the assumption of smoothness or continuity. A function needs to be smooth and not too jumpy to be interpolated with reasonable accuracy. The idea is illustrated in the figure below with the assumption that we cannot collect a ton of data points (experiments). Image source: Created by the author Image source: Created by the author And, it turns out that most real-life, continuous-data problems are not that jumpy either i.e. they are smooth enough to be handled by interpolation techniques. Therefore, we can use a combination of physical simulation and interpolation to do fast optimization in many of those cases. Image source: Created by the author Image source: Created by the author So, here it is, Image source: Created by the author Image source: Created by the author

This method does not guarantee you the best outcome in all situations. But it works in most practical scenarios.

A simple demo

The boilerplate code is here in my Github repo. Let me show only some pertinent code snippets in this article.

Simulation and experiment functions

Let's suppose we have a complex simulation task that takes some time (0.1 seconds here) to finish. The actual output is just a simple nonlinear function. Here is the code, text

def complex_simulation(x):
    """
    A nonlinear simulation function with some time delay
    """
    time.sleep(0.1)
    y = np.cos(-x**2/9.0)+np.sin(x/6)
    return y

We also define a function run_experiment to run this simulation a fixed number of times. It returns both the experiment domain and results from a given bound (low/high endpoints). text

def run_experiments(experiment_bounds,n=10):
    results = []
    low,high = experiment_bounds
    domain = np.linspace(low, high, num=n, endpoint=True)
    for i in domain:
        y = complex_simulation(i)
        results.append(y)
    return (domain,results)

Let's say, we run it 11 times and get this data, Image source: Created by the author Image source: Created by the author

Interpolation and optimization

A simple cubic interpolation (using Scipy), will give us this nicely fitted result and also a new function that can generate any intermediate result we want. Image source: Created by the author Image source: Created by the author Now, we can write a simple search function with exhaustive simulated results to find the point at which the maximum occurs. text

def optimum_simulation(experiment_bounds,n=11):
    """
    Using exhaustive simulations
    """
    domain,results=np.array(run_experiments(experiment_bounds,n=n))
    imax = np.argmax(results)
    return (domain[imax])

Basically, it runs the experiment a certain number of times (using the previously defined run_experiment function) and search for the argmax of the result array. It's algorithmically simple, but it's not fast because of the nature of the simulation. When we run it for n = 11, it takes ~1.1 seconds, because of the 0.1 seconds time delay for each simulation. It yields the result for the optimum x as, x =8.0. Alternatively, we can define another function that uses the interpolated function instead of the actual simulation function. It, of course, utilizes the init_results obtained from a single experimental run of limited size. text

def optimum_interpolation(domain,init_results,n=101):
    """
    Using interpolation
    """
    low,high = domain.min(),domain.max()
    ip_interpolated = np.linspace(low, high, num=n, endpoint=True)
    f3 = interp1d(domain, init_results, kind='cubic')
    results_interpolated=np.array(f3(ip_interpolated))
    imax = np.argmax(results_interpolated)
    return (ip_interpolated[imax])

When we run it with n = 101, it finishes in ~ 900 microseconds and yields the result of x =7.54. If you look back at the actual function, you will realize that x =7.54 is a much better approximation of the optimum value than x =8.0. This means that the interpolation approach yielded not only a faster but also more accurate result. This may not be true for all cases, but the speed advantage often offsets a small sacrifice in the accuracy.

Design of experiment matters

One must, however, pay attention to the design-of-experiment aspect. Generally, a uniformly sampled experiment space is the most suitable starting point. For example, even for our simple demo, if we did the experiments with uniformly random x values (not linearly spaced), we could have landed with a sub-optimal interpolation function. Image source: Created by the author Image source: Created by the author For more information on design-of-experiment and how you can use a Python package to generate robust experimental plans, see my article here.

Design your engineering experiment plan with a simple Python command

Highly useful in higher dimensions

The Scipy interpolation routines work in two-dimension cases just as easily as they do in a single dimension. Optimization using exhaustive simulations is that much more complex and time-consuming in multiple dimensions because of the curse of dimensionality. An interpolation approach can come in handy in these scenarios. Here is a sample result where we make a smooth interpolated 2D surface out of sparse data using Scipy interpolation. Basically, from 400 actual data points, we created a 4900-data matrix that can point us toward the optimum point with higher accuracy and speed. Image source: Author created from another article by the same author Image source : Author created (from another article by the same author)

Summary

In this article, we showed, how using simple interpolation routines and carefully chosen, well-spaced experimental points, we can create interpolation functions that can yield fast optimization. This approach is particularly suitable for complex, time-consuming scientific or technical simulations that are more or less smooth in their response. Note that, although we showed the demo using a simulated function in Python, the experimental data can come from any source (e.g, a database, a MATLAB connector, or a streaming API) and the only Python portion is the Scipy-based interpolation and the subsequent search routine. Application domains such as semiconductor design, chemical process optimization, production planning, etc., can benefit from this approach.

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