5.5 Computation of the left null singular vector
5.5.1 Doubly-periodic domain
The starting point for describing the particular LNSV computation procedure is a quali- tative observation of the structure of the LNSV of the discrete Poisson matrix associated with a doubly-periodic domain. Fig. 5.5 shows an example of the LNSV structure for a domain with 3× 3 subdomains, and N = 4. The observed LNSV structure and associ- ated computation procedure outlined below, can be directly extended to any number of subdomains.
25 50 75 100 125 150 175 200 225
−0.1 −0.05 0
Figure 5.5: Left null singular vector u0 for an example of a doubly periodic do-
main with 3× 3 subdomains, and N = 4. The vertical dashed lines separate the contributions of individual subdomains, i.e. the subvector
sv shown in 5.6.
Inspection of Fig. 5.5 shows a repetitive pattern/subvector of total size of (N + 1)2 el-
ements, shown in detail in Fig. 5.6. This subvector is denoted as sv, and it is repeated as many times as the number of subdomains (nsub) in the global domain (e.g. nsub = 9
5 10 15 20 25 −0.1
−0.05 0
Figure 5.6: Exploded view of the subvector sv for an example of a doubly periodic domain with 3× 3 subdomains, and N = 4.
for the case of 3× 3 subdomains). A general definition of sv is
sv = u0(p : p + (N + 1)2− 1)
with p = 1 + ( j− 1)(N + 1)2, where j = 1, . . . , nsub represents the j-th subdomain.
Based on this definition and our visual observations, we have found that, for the case of uniform-sized subdomains, we can construct the LNSV u0as
u0=
1nsub⊗ sv
k1nsub⊗ svk2
(5.38) where 1nsub is a vector of ones with nsub elements. For the more general case of sub-
domains with different dimensions, observation indicates that the magnitude of the el- ements of sv scales with the area of the particular subdomain it originates from. For a doubly-periodic domain, with any number of arbitrarily-sized subdomains, the global LNSV is then generally computed as
u0 =
a⊗ sv
ka ⊗ svk2
(5.39) where a is a vector of nsub elements, which contains the area of each subdomain.
Further analysis applied to the vector sv (Fig.5.6) reveals an additional level of Kro- necker product structure within it. As in Fig. 5.5, Fig. 5.6 also shows a repetitive pattern, denoted as uI(see Fig. 5.7), which repeats itself every N + 1 entries of sv with
varying magnitude. Specifically, the vector sv can be constructed as
1 2 3 4 5 −0.4 −0.3 −0.2 −0.1 0
Figure 5.7: Structure of the vector uIfor the case of N = 4.
where uI is a vector, regarded as a “core vector”, representing the contribution of an
appropriately defined subdomain one-dimensional subdomain to u0. Both a detailed
definition and computation procedure of uIare offered in section 5.5.3. Once the vector
sv is computed, the global LNSV u0 is calculated using on Eq. (5.39) and (5.40) as
u0= −
a⊗ uI⊗ uI
ka ⊗ uI⊗ uIk2
(5.41)
5.5.2
2D non-periodic domain
The same exercise can be performed for the more general non-periodic case. The struc- ture of the LNSV associated with the discrete Poisson matrix, for an example of 3× 3 subdomains and N = 4, is presented in Fig. 5.8. Effectively, the example subdomain consists of a central domain surrounded by eight subdomains, each of which has at least one physical boundary that is non-periodic. As in the previous case, the observed LNSV structure and associated computation procedure outlined below, can be directly extended to any number of subdomains.
In this non-periodic example, there are as many different types of patterns as there are subdomains with different combinations of boundary conditions along each interface (e.g. 9 for the case of 3× 3 subdomains in the example set-up of Fig. 5.8). Nonetheless, there is not a clear repetitive pattern as in the doubly-periodic case, which means that no
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−0.05 0
Figure 5.8: Structure of the of the left null singular vector u0for an example non-
periodic domain with 3× 3 subdomains, and N = 4
longer a simple Kronecker product, as the one used in Eq. (5.39), can be used to com- pute u0. As in the periodic case, the magnitude of the entries in each subvector is related
to the area of each subdomain. If we denote as sv(i) the sub-vector of size (N + 1)2that contains the contribution of the corresponding subdomain i, the global LNSV u0can be
computed as u0 = h sv(1)| . . . |sv(nsub)iT ksv(1)| . . . |sv(nsub)k 2 (5.42) where the concatenation (|) operator is applied in a row-wise sense such that the numer- ator of Eq. (5.42) is a vector with the following structure (see Eq. (5.43) )
h sv(1)| . . . |sv(nsub)iT = sv (1) 1 . . . sv (1) ns | {z } sv(1) sv(i)1 . . . sv(i)ns | {z } sv(i) sv(nsub)1 . . . sv(nsub)ns | {z } sv(nsub) T . (5.43)
In Eq. (5.42) and (5.43), nsub is the total number of subdomains, ns = (N + 1)2 is the
total number of points per subdomain, and sv(i)j is the j-th element of the vector sv(i), the
non-periodic analog of vector sv defined in the previous section. Similarly to Eq. (5.40), the subvector sv(i)corresponding to a subdomain i is computed as
sv(i) =−ai(uz⊗ ux) (5.44)
where ai is the area of the subdomain i, and uz and ux are core vectors, non-periodic
analogs of uIused in the doubly-periodic case, which, however, are determined by the
type of boundary the subdomain i has in the vertical and horizontal direction, respec- tively. The possible choices of uzand uxfor the 9 different type of subdomains (in terms
of their patching/boundary conditions) of Fig. 5.9 are given in Table I. Specifically, these nine different subdomain locations where a subdomain i can be located are four corners (BL, BR, TL, and TR), four sides (B, T, L, and R), and the interior (I). Once the vectors uB, uT, uL, uRand uIare available, the null singular vector u0 can be computed
for any 2D non-periodic domain discretized with rectangular subdomains. In the next section, a procedure to identify and compute these vectors is presented.
BL B BR
L I R
TL T TR
Figure 5.9: Schematic of a general non-periodic domain. It shows the possible locations of subdomain i subject to different patching/boundary con- ditions. Location ux uz BL uL uB B uI uB BT uR uB L uL uI I uI uI R uR uI TL uL uT T uI uT TR uR uT
Table I: Possible choices of the core vectors uxand uz, used in the computation of
u0, depending on the location of the subdomain under consideration, as