Wavelet Methods - Computational Methods for Quantitative Finance - page 173

N L ,we

=

( w 0 , w 1 ,..., w r ) , w j ∈ C This system is block-upper-triangular. With w

obtain its solution by solving

λ j + 1 M

2 A w j =

k

+ s j (12.14)

g j − l = j + 1 T j + 1 ,l + 1 M w l .By( 12.14 ), an hp -dG
time step of order r amounts to solving r

= =

for j r,..., 0, where s j

+ 1 linear systems with a matrix of the

form

k
2 A ,
where λ is an eigenvalue of C . For the preconditioning of the linear system, we
define the matrix S and the scaled matrix B

B = λ M +

N L N L by

∈ R × R

Re (λ) I

2 D

2

k

B

S − 1 BS − 1 , S = + , = (12.15)

where D is the diagonal preconditioner with entries 2 αl as in ( 12.9 ). The precon-
ditioned linear equations corresponding to ( 12.14 ) are solved approximately with
incomplete GMRES( m 0 ) iteration (restarted every m 0 ≥ 1 iterations). We then ob-

tain [123, Theorem 4]

Theorem 12.3.4 Let the assumptions of Theorem 12.3.2 hold . Then , choosing the
number and order of time steps such that M = r = O ( | log h | ) and in each time step

) 5 n G = O ( | log h | GMRES iterations implies that

u(T ) − U dG (T ) L 2 (G) ≤ Ch p , (12.16)

where U dG denotes the ( perturbed ) hp -dG approximation of the exact solution u to
( 12.4 ) obtained by the incomplete GMRES ( m 0 ) method .

Applying the matrix compression techniques, the judicious combination of ge-
ometric mesh refinement and linear increase of polynomial degrees in the hp -dG
time-stepping scheme, an appropriate number of GMRES iterations, results in lin-
ear (up to some logarithmic terms) overall complexity of the fully discrete scheme
( 12.10 ) for the solution of the parabolic problem ( 12.4 ).

Example 12.3.5 As in Example 10.6.3, we consider the variance gamma model [118]
with parameter σ = 0 . 3, ϑ = 0 . 25 and θ =− 0 . 3. For the compression scheme, we

1, a = use a 8, the absolute value of the entries in the

stiffness matrix A and the compressed matrix A are shown in Fig. 12.5 . Here, large
entries are colored red. For the stiffness matrix, blue entries are small but non-zero
whereas for the compressed matrix blue entries are zero either due to the first or
second compression. One clearly sees that the compression scheme neglects small
entries.

= 1, p = 2, p = 2. For L =