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Introduction to Model Order Reduction II.2 The Projection Framework Methods. Luca Daniel Massachusetts Institute of Technology with contributions from: Alessandra Nardi, Joel Phillips, Jacob White. Projection Framework: Non invertible Change of Coordinates. Note: q << N. reduced state.
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Luca Daniel
Massachusetts Institute of Technology
with contributions from:
Alessandra Nardi, Joel Phillips, Jacob White
Reduction of number of equations: test by multiplying byVqT
Point Matching
II.2.b POD Principal Component Analysis
or SVD Singular Value Decomposition
or KLD KarhunenLo`eve Decomposition
or PCA Principal Component Analysis
Assuming A is nonsingular we can cast the dynamical linear system into a canonical form for moment matching model order reduction
Note: this step is not necessary, it just makes the notation simple for educational purposes
Taylor series expansion:
U
Aside on Krylov Subspaces  Definition
The order k Krylov subspace generated from matrix A and vector b is defined as
If
and
Then
Total of 2q moment of the transfer function will match
Combine point and moment matching: multipoint moment matching
Compare Pade’ Approximationsand Krylov Subspace Projection Framework
If U and V are such that:
Then the first q moments (derivatives) of the
reduced system match
apply k times lemma in next slide
Vectors{b,Eb,...,Ek1b}cannot be computed directly
Vectors will quickly line up with dominant eigenspace!
Generates new Krylov
subspace vector
For j = 1 to i
Orthogonalize new vector
Normalize new vector
Orthonormalization of U: The Arnoldi AlgorithmComputational Complexity
Normalize first vector
O(n)
sparse: O(n) dense:O(n2)
O(q2n)
O(n)
Orthonormalization of
the ith column ofUq
Orthonormalization of
all columns ofUq
So we don’t need to compute
the reduced matrix. We have it already:
If U and V are such that:
Then the first 2q moments of reduced system match
apply k times the lemma in next slide
Use Lanczos process to biorthonormalize the columns of U and V: gives very good numerical stability
Hence choosing Krylov subspace
s2
s1
matches first kj of transfer function around each expansion point sj
s1=0
s3
i.e. A is negative semidefinite
Example Finite Difference System from on Poisson Equation (heat problem)
We already know the Finite Difference matrices is positive semidefinite. Hence A or E=A1 are negative semidefinite.
i.e. E is negative semidefinite
same matrix
nxn
nxq
nxq
Congruence Transformation Preserves Negative Definiteness of E (hence passivity and stability)If we use
i.e. E is positive
semidefinite
i.e. A is negative
semidefinite
+


Example. hStateSpace Model from MNA of R, L, C circuitsLemma: A is negative semidefinite if and only if
When using MNA
For immittance systems
in MNA form
A is Negative
Semidefinite
E is Positive
Semidefinite
A different implementation of case #1:
V=U, UTU=I, Arnoldi Krylov Projection Framework:
Use Arnoldi: Numerically very stable
Used Lemma: If U is orthonormal (UTU=I) and b is a vector such that
Proof:
long coplanar Tline,
shorted on other side
dielectric layer
admittance [S]
100
__ with dielectrics
  w/o dielectrics
101
102
103
104
0
1
2
3
4
5
6
frequency [Hz]
x 108
current and charge conservation
Volume Integral Formulation including Dielectricsconductors
dielectrics
conductors
dielectrics
using Galerkin
congruence transformation
preserves positive definiteness
diagonal with
positive coef.
is block diagonal and the blocks are all positive,
hence is positive semidefinite and so is
congruence transformation
preserves positive definiteness
is block diagonal and the blocks are all positive semidefinite, hence is also positive semidefinite
admittance [S]
100
(order 16)
__ with dielectrics, reduced model
o with dielectrics, full system
(order 700)
101
102
103
104
0
1
2
3
4
5
6
frequency [Hz]
x 108
admittance [S]
100
101
102
103
(order 16)
__ with dielectrics, reduced model
o with dielectrics, full system
(order 700)
104
0
1
2
3
4
5
6
x 108
frequency [Hz]
Note: NOT TO SCALE!
reduced filament widths
for visualization purposes
admittance [S]
100
__ with dielectrics, reduced model (order 12)
o with dielectrics, full system (order 600)
  without dielectrics
101
102
103
104
0
1
2
3
4
5
6
frequency [Hz]
x 108
Energy of the output y(t) starting from state x with no input:
Observability Gramian:
Note: it is also the solution Lyapunov equation
Note: If x=xi the ith eigenvector of Wo :
Hence: eigenvectors of Wo corresponding to small
eigenvalues do NOT produce much energy at the output
(i.e. they are not very observable):
Idea: let’s get rid of them!
Minimum amount input energy required to drive the
system to a specific state x :
It is also the solution of
Inverse of Controllability Gramian:
Note: If x=xi the ith eigenvector of Wc:
Hence: eigenvectors of Wc corresponding to small
eigenvalues do require a lot of input energy in order
to be reached (i.e. they are not very controllable):
Idea: let’s get rid of them!
Fortunately the eigenvalues of the product (Hankel singular
values) do not change when changing coordinates:
Diagonal matrix with eigenvalues of the product
The eigenvectors change
But not the eigenvalues
And since Wc and Wo are symmetric, a change of coordinate matrix U can be found that diagonalizes both:
In Balanced coordinates the Gramians are equal and diagonal
Selection of vectors for the columns of the reduced order projection matrix.
In balanced coordinates it is easy to select the best vectors for the reduced model: we want the subspace of vectors that are at the same time most controllable and observable:
simply pick the eigenvectors corresponding to the largest entries on the diagonal (Hankel singular values).
In other words the ones corresponding to the largest
eigenvalues of the controllability and observability Grammians product.
Truncated Balance Realization Summary
TBR Model Not Positive Real
Accurate over a narrow band.
Matching function values and derivatives.
Cheap: O(qn).
Use it as a FIRST STAGE REDUCTION
Truncated Balanced Realization and Hankel Reduction
Optimal (best accuracy for given size q, and apriori error bound.
Expensive: O(n3)
USE IT AS A SECOND STAGE REDUCTION
Two Complementary ApproachesKrylov reduction (Wi , Vi):
Ai = WiTAVi
Bi = WiTB
Ci = CVi
Initial Model:
(A B C), n
Intermediate Model:
(Ai Bi Ci), ni
TBR reduction (Wt , Vt):
Ar = WtTAVt
Br = WtTB
Cr = CVt
Reduced Model:
(Ar Br Cr), q
Numerical Simulation
Quick intro to PDE Solvers
Quick intro to ODE Solvers
Model Order reduction
Linear systems
Common engineering practice
Optimal techniques in terms of model accuracy
Efficient techniques in terms of time and memory
NonLinear Systems
Parameterized Model Order Reduction
Linear Systems
NonLinear Systems
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