Inverse Problem Theory

Introduction to inverse problem theory:

In maths the inverse problem is one branch with theory. It get the data from observation of model and it is considered as input. We can formulate the inverse problem as Data---> Model parameters. The jargon of inverse problem theory is define the model constitutes and predict the observations. Having problem with Interesting Data Sets keep reading my upcoming posts, i will try to help you.


Explanation for inverse problem theory


Theory of inverse problem:

The result for model requires volume and probability of system by investigation. The parameters are measured by using physical theory links.In inverse problem, parameters are characterized by investigation and use the actual observations.

We have two reasons for solving inverse problem because it is difficult to solve. They are,

The data consisted by model parameter values.
A huge parameter space is available for find the model parameters.
The optimization problem directly set with inverse problem formulation. The volume notion with manifold is V (A) = `int` dV.

The function f has volumetric probability as P(A) = ?dV f. Here A is associates with M. The volume is changed when metric manifold is included and coordinates { x1, …….xn}.

dV = `sqrt(det g)`


More about inverse problem


In inverse problem theory, the volumetric probabilities product extended as,

The first manifold M is define the volumetric probability.
Here second volumetric probability also defined on second manifold N.
P à Q = Q (P) from M into N is application.
It gives the basic operation as g ( P ) = `(1)/(v)` f( P ) f (Q (P) ).

A typical inverse problem theory contains some parameters.  They are,

A model parameter set { m^1, m^2, …….mn }.
An observable parameter set { o^1, o^2, ……., on }.
The possible observations predict the relation as oi = oi ( m^1, m^2, …….., mn).
The inverse problem theory establish the basic elements:

The M is define the volumetric probability of model parameters and it is represented as ?prior (M).
An observable parameters are represent the volumetric  probability over M as sobs(O).
We can see the relation of forward modeling as M `|->` O = O(M).