Data Decoding with Inverse Interpolation
Inverse Interpolation — a strong numerical numerical weapon
“Mathematics possesses not only truth but supreme beauty-a beauty cold and austere, like that of a sculpture." -Bertrand Russel
Being a math postgrad dealing with computational mathematics can be quite rewarding as it allows me the flexibility to dive into numerous topics, suck the juice, and interpret it in my mathematical intuition. My research interest is primarily numerical algorithms and computations in astronomy and imaging, and I’ve heard many quoting numerical analyses that are very tiresome and bulky. But trust me, it is not! So today I thought, why not introduce my audience to another one of my favorite numerical techniques? Inverse interpolation is a numerical technique that allows us to determine the input value corresponding to a given output value based on a given set of data points. Inverse interpolation plays a crucial role in various fields, including cosmology. I can't demonstrate such an important technique in one post, so I've highlighted a few points and shown a numerical illustration. I hope it helps!
Inverse interpolation is a technique for finding an estimate of the value of an independent variable x corresponding to a given value of a dependent variable y within the range of observed values of y.
We can achieve inverse interpolation by interchanging the role of independent variable x and dependent variable y and interpolating x on data points {(y_i,x_i),i=0,1,2,3…n}
The particular approach for implementing inverse interpolation can be used on the processes for the distinctness of values of x. This ensures the distinctness of corresponding values of y, making it a favorable mechanism for understanding the correlation between different variables.
Inverse interpolation is important in our day-to-day lives, as it is dominant in the fields of:
- Medical imaging facilitates enhancing resolution, helps in understanding the radiation dose required for certain types of medical scans, and hence configuring the reduction of dose, to name a few.
- Financial Modelling: Inverse interpolation finds its significance in financial modeling as it helps fill the gaps of data for the completion and accuracy of the financial model. It also helps in performing sensitivity analysis.
- Scientific Research: Inverse interpolation can be used to analyze data from scientific experiments by finding the underlying relationships between the different variables. For example, a scientist might use inverse interpolation to find the relationship between the temperature and pressure of a gas, given a set of experimental data. Inverse interpolation can be used to study the universe's evolution by finding the cosmological parameters, such as the Hubble constant and the dark matter density, given the observed properties of the universe, such as the cosmic microwave background radiation and the universe's large-scale structure. It can be useful for a variety of astronomical applications, such as:
- Stellar evolution: Inverse interpolation can be used to study the evolution of stars by finding their stellar properties at different stages of their lives. This can be used to understand how stars form, how they evolve, and how they die.
- Planetary formation: Inverse interpolation can be used to study the formation of planets by finding the physical conditions in protoplanetary disks that are conducive to planet formation. This can be used to understand how planets form and evolve.
- For cosmological events: Inverse interpolation plays a crucial role in cosmology as it allows us to uncover the history and structure of the universe. This is particularly relevant for mapping the Milky Way halo with RR Lyrae stars, where inverse interpolation techniques developed in a cosmological context can be applied to extract meaningful information about the distribution and dynamics of these stars in the halo. Inverse interpolation is essential in cosmology for a variety of reasons. First, it allows us to fill in missing or incomplete data, common in astronomical observations.
Impact of Machine Learning
Machine learning algorithms have gained significant attention in cosmology as they offer powerful tools for analyzing complex and large-scale datasets. Inverse interpolation is particularly relevant in the field of cosmology because it provides a way to analyze and interpret a wide variety of data about the universe. This data often contains uncertainties and missing information, making it challenging to extract meaningful insights. Inverse interpolation becomes necessary in cosmology because it allows us to fill in gaps in our understanding and extract valuable information from incomplete or uncertain data.
In certain cases, when y is a single-valued function of x, we can interchange the roles of x and y, and hence inverse interpolation can be implemented.
Below, I will demonstrate a problem that requires inverse interpolation.
In the given problem, we are using Lagrange inverse interpolation,, but one can also use Newton divided and complete based on the perspective or sometimes by the requirements of the problem.
The beauty of this method lies in its ability to comprehend data. We can bridge the gap between the data and understand it better. In layman's terms, we want to backtrace the data from the functional value and reveal hidden patterns in the process. Traditional interpolation ties one simply to existing trends, while inverse interpolation helps uncover previously unknown information. By filling in the gaps between data points, it can reveal new patterns and relationships that would otherwise be invisible. This creativity is akin to scientific discovery, where new knowledge is generated from incomplete information. It also allows us to predict outcomes and glimpses of the unseen.
For reference, I follow Jain, Iyengar: Numerical Methods
Jaan Kiusalaas: Numerical Methods in Engineering With MATLAB
Leveque—Numerical Methods for Conservation Laws. and a few more.
