Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 238

yielding an updated global analysis over the entire model grid. In the case of weather
models it has been shown that dynamics in local regions can be regarded as relatively
low-dimensional in comparison to the global weather dynamics [ 24 ].
The LETKF has several advantages. First it is computationally efficient because
the analysis in each local region can be done in parallel where the dimensionality
of the matrices in the Kalman filter Eqs. ( 40 )-( 42 ) is reduced. Second the LETKF
uses a convenient basis to perform the optimization contributing significantly to
the computational performance of the algorithm. Third the analysis is computed
without use of the model equations; thus, the LETKF is model independent ,akey
feature with regard to GBM where the biology is poorly understood and modeling
efforts are in their infancy. Studies have proven the LETKF to be an accurate and
efficient data assimilation approach for weather and ocean models when the analysis
is updated over sufficiently small intervals [ 11 , 13 , 30 ].

Notation

We now derive the LETKF data assimilation procedure. At time t n − 1 we start with
an analysis ensemble consisting of m -dimensional model vectors

u a ( i ) { : i = 1 ... k }.

t n − 1

For the case of the Eikenberry model each u a ( i )

t n − 1 is comprised of density of the
proliferating and migrating cells, chemorepellent, and ECM at every grid point of
the model geometry. The mean is regarded as the best estimate of the most likely
state of the system. Each ensemble member is advanced under the model until
time t n . This yields the background ensemble

u b ( i )

{ = ... },

: i 1 k

t n

where

u b ( i ) u a ( i )

= (

t n − 1 ,

) .

F t n − 1

t n

For the remainder of this chapter we will simplify the notation by omitting all
time subscripts, which are assumed to be t n . The sample background mean and
analysis mean are defined by

k
i = 1 u b ( i ) ,

k − 1

u b =

k
i = 1 ( u b ( i ) − u b )( u b ( i ) − u b )

) − 1

T

P b =( k − 1 (43)

) − 1 U b U b

T

=( k − 1 ( ) , (44)