4.3 Derivation of the DSFEM framework
4.3.3 Strategy for constructing random blocks
Off-diagonal partitions
In light of the prior derivations, a desirable condition for positive (semi-)definiteness of the block matrix G has been acquired. In fact, Theorem 4.3.4 proves very convenient to enable the creation of a method that encompasses all the properties outlined in Section 4.3.1. To achieve that, we firstly claim:
Proposition 4.3.7. Any positive semi-definite matrix with the same diagonal blocks A, D as the original G can robustly be generated as a realisation of eG, by applying a suitable SVD perturbation to block B (implicitly, via L+ABL
+∗
D ), provided the conditions
of Theorem 4.3.4 are met.
Proof. To simplify the notation, G (with no subscript) has been taken as the original matrix, on which eG is based. Let:
A = QAΣAQ
∗
A, D = QDΣDQ
∗
D (4.31)
be singular value decompositions of A, D, and Z and its SVD be: Z := L+ABL
+∗
D = UZΣZV
∗
Z (4.32)
which is simply W∗ from the proof of Theorem 4.3.4. Then B can be expressed in terms of Z, pre and postmultiplied by the factors LA and LD:
LAZL ∗ D= LAL + ABL +∗ D L ∗ D= B (4.33)
The second part of the equality is only possible to ascertain since Theorem 4.3.4(b) and (c) are fulfilled. Temporarily, let us assume A, D are either fixed, or their perturbation is of no interest. Now, the earlier representation of B will be used for ’implicitly’ defining eB.
Construction 4.3.8 (Indirect generation of eB). Let eRA, eRD be stochastic unitary
matrices of the same size as A and D, respectively. Additionally, let eΣZ ∈ R be a
random diagonal matrix of the size and rank of ΣZ, with elements not exceeding unity.
Then we can write e Z0 := UZΣeZV ∗ Z (4.34a) e Z := eRAZe0Re∗D = ( eRAUZ) eΣZ( eRDVZ) ∗ (4.34b) e B := LAZLe ∗D = LAReAZe0Re∗DL∗D (4.34c)
Using Construction 4.3.8, make the observation that (4.34b) explicitly provides an SVD of eZ in terms of a singular value decomposition of Z. By virtue of the definitions in Construction 4.3.8, the validity of condition (a) of Theorem 4.3.4 is naturally preserved upon substituting B with its random counterpart. Remember that A and D were intentionally kept fixed, thus the transition from Z to eZ does not affect their factors LA and LD, and is contained entirely in eB. The validation of the remaining two
requirements of Theorem 4.3.4 is rather trivial:
range( eB) = range(LAZLe ∗D) ⊆ range(LA) = range(A)
range( eB∗) = range(LDZe∗L∗A) ⊆ range(LD) = range(D)
A couple of essential properties of defining eB as per the above way are immediately evident. Firstly, if no further restrictions are imposed on eRA, eRD and eΣZ, the domain
of eB is precisely {B : G 0}, i.e. the ensemble of all matrices B for which eG with constant diagonal blocks is positive semi-definite.
More formally, assume there exists some B1 that is not a possible realisation of eB, and
yet satisfies the prescriptions of Theorem 4.3.4. Then there must exist an SVD of the corresponding Z1 that is not in the domain of eZ, which implies either ΣZ1 is not in
the domain of eΣZ, or that @RA1 : RA1UZ = UZ1. All of the preceding statements are
obvious contradictions, the last one because we can always select RA1 = UZ1UZ−1, as
the RHS forms a unitary matrix. Identical argumentation can also be used for the case of RD1, VZ, VZ1. With this, the proposition is proven.
The direct implication of Proposition 4.3.7 is that the elusive requirement (3) of Sec- tion 4.3.1 can be fully satisfied. In addition, the fact that any possible PSD matrix, having the same diagonal blocks as the original deterministic one, can be sampled from
e
G, means there exist no restrictions on the outcomes of the physical quantity or process described by eG. This translates into Construction 4.3.8 being ’non-restrictive’, in the sense that there are no physical uncertainties that cannot be reproduced. In contrast, the CBSM cannot replicate variability in the interface between subsystems. Overall, existing techniques are unable to individually control the relation between two DOF sets, i.e. the B block, linking DOFs corresponding to A, to those of D.
Diagonal blocks
What remains to be demonstrated is that the diagonal partitions are also amenable to such treatment. Since A and D are already PSD matrices, it is possible to use existing techniques to model eA, eD, like the nonparametric probabilistic approach, described in Section 2.3.4. One of its benefits is that it promises to preserve the rank and nullspace of the random square blocks. Another basic idea that achieves the same is a direct eigenvalue modification of the form
e A = QAΣeAQ ∗ A, D = Qe DΣeDQ ∗ D (4.35)
following the notation adopted in (4.31) for the factorisations of A and D. Restricting the subsequent comments to eA for brevity, we can require
from the same physical constraints as the ones imposed on the whole matrix eG. Fur- thermore, preserving the rank of A also keeps its range and nullspace unchanged, as depicted in (4.6). Then the two image inclusion conditions of Theorem 4.3.4 automat- ically remain satisfied upon substitution of A with a realisation of eA in G. However, this does not guarantee that part (a) is still valid. Indeed, retracing the steps of the proof of the theorem, we can write:
e G/ eA = eD − B∗Ae+B = eLD I − ( eL+DB ∗ e L+∗A )( eL + DB ∗ e L+∗A ) ∗ e L∗D
Now, we can expand the term eL+DB
∗
e
L+∗A using the factorisations defined in (4.30), and
accounting for the fact the eigenvalue matrices of the diagonal blocks are random, due to (4.35): e L+DB ∗ e L+∗A = XDΣe 1/2+ D Q ∗ DB ∗Q AΣe 1/2+∗ A | {z } e J∗ XA∗ (4.36)
and as XA, XD are unitary
σi( eL+DB
∗
e
L+∗A ) = σi( eJ ) (4.37)
but from Theorem 4.3.4(a), for eG to be positive semi-definite, σi( eJ ) ≤ 1 ∀i. This does
not hold in general:
Example 4.3.9. Consider the matrix
G =1 1 1 1
with blocks A = B = D = 1, eigenvalues (2, 0) and QA = ΣA = · · · = J = 1. We can
freely select a realisation A1 = ΣA1 such that 0 < A1 < 1, which conforms to (4.35).
However, eG then becomes indefinite. Explicitly, keeping D fixed and calling J1 the
corresponding realisation of eJ , we obtain J1 = A−11 > 1 for the above choice of A1.
Irrespectively of that, we can still use the idea of Construction 4.3.8 in order to remedy the issue that arises. Indeed, since the crucial range inclusion conditions of Theo- rem 4.3.4 are not broken, we can make the following assertion:
Construction 4.3.10 (Generation of eB from sampled eA, eD). Let As and Ds be
realisations of eA and eD, as specified by (4.35). We can then compute the respective Zs= L+AsBL
+∗
Ds = UZsΣZsV
∗
Zs
where, as displayed, ΣZs may have diagonal values exceeding unity. Define
e Zs0 := UZsΣeZsVZs∗ (4.38a) e Zs:= eRAZes0Re∗D = ( eRAUZs) eΣZs( eRDVZs) ∗ (4.38b) e B := LAsZesL∗Ds = LAsReAZes0Re∗DL∗Ds (4.38c) in the same way as in (4.34).
The only principal difference with the case when the diagonal submatrices were kept fixed is that now eΣZsmight be impossible to remain static, and could incur a potentially
larger change of its singular values, in order to guarantee all of the latter still have an upper bound of 1. Observe that Construction 4.3.10 is invariant with respect to how
e
A and eD have been defined in order to retrieve the realisations As and Ds. Without
loss of generality, we could have assumed e
A = eQAΣeAQe∗A, D = ee QDΣeDQe∗D (4.39) instead of (4.35), as long as eQA, eQD are such that the image and kernel of the non-
stochastic original matrices are preserved. As explained, this should be the case if they are extracted by the the nonparametric probabilistic approach. Otherwise, barring further restrictions, (4.39) describes all possible pairs of matrices whose dimensions are equal to those of A and D. Not only would the preceding analyses be invalidated, since range(B) ⊆ range(A) and range(B∗) ⊆ range(D) are no longer certain, but the physical significance of A and D would be violated.
Sampling eG
At this stage, it is feasible to construct a holistic process for the computation of reali- sations of the full random Hermitian matrix eG, with all blocks being stochastic. Algorithm 1 Sampling eG based on G
1: procedure SampleMatrix(A,B,D,Ns) . Input: blocks of G, Ns 2: As← SampleBlock(A) 3: Ds← SampleBlock(D) 4: LAs ← Decompose(As) . Any LAs : LAsL∗As = As 5: LDs ← Decompose(Ds) . Any LDs : LDsL∗Ds = Ds 6: Zs← L+AsBL +∗ Ds 7: UZs, ΣZs, VZs← SVD(Zs) 8: k = 1 9: while k ≤ Ns do
10: RAk ← SampleUnitary(n) . Same size as A
11: RDk ← SampleUnitary(m) . Same size as D
12: ΣZs,k ← SampleDiag(ΣZs) . σi(ΣZs) ≤ 1 13: Zs,k ← RAsUZsΣZs,kVZs∗R ∗ Ds 14: Bk← LAsZs,kL∗Ds 15: k ← k + 1 16: end while 17: return As, Ds, B1, . . . , BNs 18: end procedure
In Algorithm 1, the sampling routines represent the functions that would be used in practice to obtain realisations of the corresponding stochastic matrices. As already discussed, the ones operating on the blocks A and D may be specified based on either (4.35) or (4.39), depending on the chosen underlying method.
The loop from 1 to Ns is intentionally defined to demonstrate the idea of extracting
multiple realisations of eB per single one of the diagonal partitions. This corresponds to creating Ns instances of eG, acquired via Construction 4.3.8, with eA and eD fixed to
As and Ds, respectively. The resultant sample of eG is the set
Gs,i= As Bi Bi∗ Ds : i = 1, . . . , Ns
The purpose of such an approach is to avoid the unknown and expectedly large com- putational cost of sampling the diagonal blocks with, say, the nonparametric model of uncertainty. Even in the case of the eigenvalue modification of (4.35) being used, calcu- lation of the factors LA, LD and their pseudoinverses is still a costly operation that is
preferably averted. Moreover, lines 6 and 7 of Algorithm 1 represent computationally heavy operations which need not be repeated until Asand Dschange (e.g. to the next
realisation As+1 and Ds+1).