Orhan Karsligil's Ideas, Thoughts and Collection of Resources

Multiscale Methods to Speed up QR problems

My thesis was based on Multiscale Modelling. The building blocks of this methodology are a type of filters called Wavelets. Using wavelets it is possible to transform a first order system to time-frequency domain. In this domain the model representation is not anymore the simple linear time line but a tree structure where both time line (horizontal) and scales (frequency bands - vertical) are represented.

Once the model is transformed into this new domain it is also possible to create algorithms to solve a QR Programming problem based on this first order system. Instead of solving the problem globally it is possible to solve a subset which is a close approximation. This fast solution represents fine resolution at the beginning of the time line and a coarser resolution in the future where uncertainity in most physical systems make finer calculations obsolote anyhow.

I am going to go into more details in the comments section of this post in the next couple days. I will formulate a sample case and demonstrate the solution algorithm. I hope to have some links ready at some point too. Please send comments if you have any.

UPDATE: What is QR? (click here for the source)

The quadratic programming problem involves minimization of a quadratic function subject to linear constraints. Most codes use the formulation

where is symmetric, and the index sets and specify the inequality and equality constraints, respectively.

The difficulty of solving the quadratic programming problem depends largely on the nature of the matrix Q.