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^{Markov Chain Monte Carlo Method (MCMC) for an Inverse Problem
(Science by Images)}

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**Inverse Problems** is a *research area* dealing with inversion of models or data.
The background of inverse problems is the *modelling and simulation* of natural phenomena.
When observations are taken of these phenomena, they are used to infer knowledge about
either physical states or underlying quantities. In this case, we talk about the
**“inversion”** of the data, calculating for example an image in computer tomography or a source reconstruction in acoustics.

**Example**

An basic example of simulation and inversion is given by a linear integral equation

$$ f(x) = \int_{a}^{b} k(x,y) \varphi(y) dy $$ with kernel $k(x,y), x,y \in [a,b]$. The*forward problem* is to calculate the integral when $\varphi$ is given. The *inverse problem*
is to reconstruct $\varphi$ when $f$ is provided.

$$ f(x) = \int_{a}^{b} k(x,y) \varphi(y) dy $$ with kernel $k(x,y), x,y \in [a,b]$. The

We can write this as an operator equation $K\varphi = f$ with the *integral operator*
$$
(K\varphi)(x) := \int_{a}^{b} k(x,y) \varphi(y) dy, \;\; x \in \mathbb{R}.
$$

If the kernel $k$ is continous (or weakly singular), then the inverse $K^{-1}$ cannot
be bounded in standard spaces like $C([a,b])$ or $L^2([a,b])$.
In this case, the solution of the integral equation does not depend continuously
on the right-hand side. Problems with this property are called *ill-posed*.

For important applications *dynamical processes* are controlled by using measured data. Here,
*state estimation* is cycled with *propagating* a reconstructed state through time with the help
of a dynamical model. This area of inverse problems is called **data assimilation**. It is the
basis for forcasting, as for example carried out in operational centres for
numerical weather prediction
or in hydrology.

Today, many areas of **remote sensing** employ inverse problems techniques. In
**nondestructive testing** or **medical imaging** either active or passive instruments are used
to infer knowledge about physical or biological states and processes. More and more, these techniques
need to be integrated into modeling of dynamic processes, leading to strong synergy of
the underlying techniques with data assimilation methods.

The goal of this wiki and community platform is to provide news, introductions and links for different areas
and developments in the whole range of **inverse problems** and **data assimilation**, **imaging** and **remote sensing**.

^{Isaak Newton Institute, Cambridge, UK}

The wiki is supervised by Roland Potthast info@inverseproblems.info, see http://www.inverseproblems.info/potthast/. We are very happy to provide a login to you, when you want to contribute to the inverse problems wiki, please send us an email!

start.txt · Last modified: 2014/10/29 15:31 by potthast