Techniques to zero out the cost function in optimal control theory for NMR/MRI applications
- Type:Master Thesis
- Field of Study:
Mechanical engineering, Electrical engineering, Physics, Mathematics
Optimal control theory can be employed in a variety of engineering problems including Nuclear Magnetic Resonance, electronics, physics, and dynamical analysis in Mechanical engineering. The main idea has been widely used and is very beneficial in cases where no analytical solutions exist, which is normally the case in complicated systems.
As a brief explanation, a dynamical system starts its propagation from its initial state at time t0 and goes to its final state at time t1, following its governing equations. For a certain dynamical system, we wish to achieve a specific state at t1, which is called “Target-State” in Optimal Control.
Basically, the process of Optimal Control consist of five steps.
- Guess certain values for all input variables and check the final state.
- Calculate “Cost Function” which is the difference between the final state and the Target-State.
- Based on the results, optimize the variables to reduce the Cost Function.
- Calculate the Cost Function again.
- Repeat steps two to four so that the Cost Function is zero.
In fact, when the Cost Function is zero, it means that we have reached our Target-State. So, all we need is to zero out the Cost Function to reach our wish!
Fig. 1- Cost Function
There are many techniques to optimize the variables in order to reach the Target-State. First of all, we need to obtain the derivative of the variables to update them in the next iterations. Many techniques are available for the derivative calculations. The simplest one is to increase the value by epsilon, calculate the new cost function, and obtain the derivative based on the difference between cost functions (which is the basic definition of the derivative). Another technique is to calculate the derivatives based on the results of Backward Propagation. Irrespective of the derivatives, it is also possible to alter the governing condition, which makes it easier or even sometimes possible to reach zero for the Cost Function. An example for changing the governing conditions can be any change in the values, which are considered as a constant value during the propagation.
Here at IMT, we use optimal control theory to optimize the excitation pulses in NMR (Nuclear Magnetic Resonance) and MRI (Magnetic Resonance Imaging). Our Target State is normally a certain phase map, which is expected to be achieved at the end of the excitation pulse. Despite the fact that any knowledge in NMR/MRI will be beneficial, it is not necessary to have such a knowledge to work on this project. Students who have experience in optimization and coding (MATLAB, Python, Mathematica,…) are privileged irrespective of the their field (Mechanical engineering, Electrical engineering, Physics or … ).
Starting date: Any time
Contract duration: 6 months
- Familiar with numerical solutions and coding (MATLAB, Python, …)
- Familiar with optimal control theory
- Enthusiast to work in an interdisciplinary environment
Dr. Mazin Jouda
Karlsruhe Institute of Technology
Institute of Microstructure Technology
P.O. Box 3640
phone: +49 721 608-22746
e-mail: mazin jouda∂kit edu