John Reid’s influence in informatics and mathematical modelling Kaj Madsen Technical University of...
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Transcript of John Reid’s influence in informatics and mathematical modelling Kaj Madsen Technical University of...
John Reid’s influence in
informatics and mathematical modelling
Kaj MadsenTechnical University of Denmark
Numerical Analysis GroupComputer Science and Systems DivisionA.E.R.E. Harwell
Allan CurtisRoger FletcherMike PowellJohn Reid
August 1973
Space Mapping
Physical problem
Rf fine model
Rc
coarse model
Connect similar residuals
. xf*. xc*
P
( ) ( ( ))f f c fR x R P xJohn Bandler, 1993
Fortran programming
Polynomial zeros
Spring 1974:
John was Visiting Professor at
Institute for Numerical Analysis
Technical University of Denmark
Owe Axelson:Solution of linear systems of equations: iterative methods
J. Alan George: Solution of linear systems of equations: direct methods for finite element problems
John K. Reid:Solution of linear systems of equations: direct methods (general)
Axel Ruhe:Computation of eigenvalues and eigenvectors
Numerical Analysis
Per Christian Hansen
T. K. Jensen and P. C. Hansen, Iterative regularization with minimum-residual methods, BIT, 47 (2007), pp. 103–120.
Scientific Computing
Image Deblurring
deblurring
Io (moon of Jupiter)
blurring
John Reid & Conjugate Gradients
It took a few years for researchers to realize that it was more fruitful to consider the conjugate gradient method truly iterative. In 1972, John Reid was one of the first to point in this direction.
Henk A. van der VorstKrylov Subspace Iterations
Computing in Science and Engineering, IEEE, 2000
J. K. Reid, The Use of Conjugate Gradients for Systems of Equations Possessing ’Property A’, SIAM J. Numerical Analysis, 9 (1972), pp. 325–332.
T. K. Jensen and P. C. Hansen, Iterative regularization with minimum-residual methods, BIT, 47 (2007), pp. 103–120.
Example (2953903 = 345150 unknowns)
Scientific Computing
Computer Science
Wireless Networks for Smart Energy
Devices join and leave a secure wireless network as described by a Markov Chain.
When devices leave there is a risk that the security is compromised.
ZigBee devices contain tiny microprocessors have limited memory, and
are deployed in home and industrial settings.
What is the trade-off between installing
new security keys and the risk of security flaws?
Wireless Networks for Smart Energy
The question:
The model:
The matrix:ZigBee devices
contain tiny microprocessors have limited memory, and
are deployed in home and industrial settings.
Scientific ComputingOperations ResearchStatisticsImage Processing
Computer Science
Principal Component Analysis (PCA)[Karl Persson (1901)]
Brain Morphometry
Image from temagami.carleton.ca
The corpus callosum is the nerve fiber bundle that connects the two hemispheres of the brain.
Local atrophy correlates to loss of particular ability, e.g walking speed, verbal fluency (age-related degeneracy)
F
M
S A
P/T
V
In a study of 600 elderly the CC outline was extracted using automated image analysis on MRI brain images
Each outline is represented by a list of corresponding ”landmark” coordinates sampled along the outline
),...,,,,...,,( 2121 pp yyyxxxx
We want to find local (sparse) variations from the mean CC shape to be used in predicition of cognitive and clinical parameters such as max. walking speed and verbal fluency
The shape coordinates are projected onto the first few (sparse) principal components before regression
Reconstruction error
Transformation to PCA space and back
Sparse principal components
kkT
k
jjj
k
jj
n
ii
Ti
IAA
llxALxLALA
subject to
minarg)ˆ,ˆ(1
11
2
1
2
,
Transformation to k-D PCA-space
Elastic net type regularization
Keep loading matrix L near orthogonal
For k = 0, A = L is the ordinary principal component loadings. For positive ’s L is sparse
mean
slowerWalking Speed
Regression of walking speed on the sparse eigen modes identifies two significant modes representing atrophy in the nerve fibers connect the motor control centres and cognitive centres of the brain, respectively
Sparse Decomposition and Modeling of Anatomical Shape Variation Sjöstrand, Karl ; Rostrup, Egill ; Ryberg, Charlotte ; Larsen, Rasmus ; Studholme, Colin ; Baezner, Hansjoerg ; Ferro, Jose ; Fazekas, Franz ;
Pantoni, Leonardo ; Inzitari, Domenico ; Waldemar, Gunhildin journal: IEEE Transactions on Medical Imaging (ISSN: 0278-0062) , vol: 26, issue: 12, pages: 1625-1635, 2007
Scientific ComputingOperations ResearchStatisticsImage ProcessingSignal AnalysisComputer Science
…
A1,1A2,1A1,2
The CocKtail Party Problem
References
[1] P. Comon, Independent component analysis, A new concept?, Signal Processing (36)287-314,1994
[2] T. Bell, T. Sejnowski, An information maximisation approach to blind separation and blind deconvolution, Neural Computation (7) 1129-1159, 1995
[3] L. K. Hansen, J. Larsen and T. Kolenda, On Independent Component Analysis for Multimedia Signals, Multimedia Image and Video Processing, 175-199, 2000
IC1
IC2IC3
The CocKtail Party Problem
…
A1,1A2,1A1,2
Solution: As the distribution of unmixed speech signals are sparse optimizing for such that becomes sparse solves the above ambiguity up to scale and permutation of the sources. This solution can be obtained through a method named Independent Component Analysis (ICA) [1-3], i.e:
ICA solution for SMixture XTrue sources S
Problem: From mixture recover mixing matrix and underlying sources . There are infinitely many potential solutions, i.e.
where is an invertible matrix.
Mixed signals X are not in general as sparse as true underlying sources S.
Happy Birthday, John !