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Sign in[Submit an Article](https://write.insightmediagroup.io) [Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/) * * * [Write For TDS](https://towardsdatascience.com/submissions/) [LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search [Data Science](https://towardsdatascience.com/category/data-science/) # 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](https://towardsdatascience.com/author/tirthajyoti/) October 27, 20216 min read ![Image source: Created by the author](../images/000.png) **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**](https://medium.com/productive-data-science/how-to-interpolate-data-with-scipy-d314143285bc) 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](https://roadmap.sh/ai-data-scientist) and [Computer Science](https://roadmap.sh/computer-science) roadmaps. * run a limited set of well-spaced experiments (if you have knowledge about design-of-experiment that will help immensely) * create an interpolation function using linear, quadratic, or cubic splines * run the interpolation function to evaluate the experiment/simulation results on a dense grid i.e. thousands of points * simply pick the maximum or minimum and the corresponding index to know which input setting caused the min/max value as per your requirement **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](../images/001.png) **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](../images/002.png) **Image source:** Created by the author So, here it is, ![Image source: Created by the author](../images/003.png) **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](../images/004.png) **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](../images/005.png) **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](../images/006.png) **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**](https://doepy.readthedocs.io/en/latest/), see my article here. > [**Design your engineering experiment plan with a simple Python command**](https://towardsdatascience.com/design-your-engineering-experiment-plan-with-a-simple-python-command-35a6ba52fa35) ## 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 \(from another article by the same author\)](../images/007.png) **Image source** : Author created (from [another article](https://medium.com/productive-data-science/how-to-interpolate-data-with-scipy-d314143285bc) 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. * * * _Loved the article? Become a_[ _**Medium member**_](https://medium.com/@tirthajyoti/membership) _to continue_ _**learning without limits**_ _. I'll receive a portion of your membership fee if you use the following link,__**with no extra cost to you**_ _._ ## Register Now to Continue Reading ### Unlock Towards Data Science with a Free Account All articles on Towards Data Science are available to registered users. Create a free account today to access this article and thousands more. Register for freeSign in to your account * * * ![](../images/body-662aba1723e2e3d49fcc.jpeg) Written By ## [Tirthajyoti Sarkar](https://towardsdatascience.com/author/tirthajyoti/) [See all from Tirthajyoti Sarkar](https://towardsdatascience.com/author/tirthajyoti/) [Data Science](https://towardsdatascience.com/tag/data-science/)[Programming](https://towardsdatascience.com/tag/programming/)[Python](https://towardsdatascience.com/tag/python/)[Science](https://towardsdatascience.com/tag/science/)[Technology](https://towardsdatascience.com/tag/technology/) Share This Article * [](https://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Ftowardsdatascience.com%2Foptimizing-complex-simulations-use-scipy-interpolation-dc782c27dcd2%2F&title=Optimizing%20complex%20simulations%3F%20Use%20Scipy%20interpolation) * [](https://www.linkedin.com/shareArticle?mini=true&url=https%3A%2F%2Ftowardsdatascience.com%2Foptimizing-complex-simulations-use-scipy-interpolation-dc782c27dcd2%2F&title=Optimizing%20complex%20simulations%3F%20Use%20Scipy%20interpolation) * [](https://x.com/share?url=https%3A%2F%2Ftowardsdatascience.com%2Foptimizing-complex-simulations-use-scipy-interpolation-dc782c27dcd2%2F&text=Optimizing%20complex%20simulations%3F%20Use%20Scipy%20interpolation) Towards Data Science is a community publication. 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