COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Adaptive Meshfree Methods Using Local Nodes and Radial Basis Functions
G. R. Liu 1*, Bernard B. T. Kee 1, Z. H. Zhong 2, G. Y. Li 2, X. Han 2
1
2
Centre for Advanced computations in Engineering Science (ACES), 10 Kent Ridge Crescent, National University of
Singapore, Department of Mechanical Engineering, Singapore 11920
State Key Laboratory of Advanced Technology for Vehicle & Manufacture, M.O.E., Hunan University, Changsha,
410082 China
Email: [email protected], [email protected]
Abstract In this paper, adaptive meshfree methods using local nodes and radial basis functions (RBFs) which based
on strong-form formulation is presented. In this present formulation, radial basis functions are used in the function
approximation for the discretization of the governing system equations. Regularization techniques are suggested and
examined to stabilize the solutions in order to obtain stable and accurate results. Different schemes for constructing
regularization matrix are compared and discussed. As stability is restored, meshfree strong-form method using local
nodes and RBFs can facilitate an easier implementation for adaptive analysis to achieve desired accuracy. Residual
based error indicator is devised in the adaptive scheme. A simple and practical node refinement procedure is presented
for node insertion at each adaptive step.
Key words: adaptive meshfree methods, radial basis functions, strong-form formulation, regularization technique,
error indicator
INTRODUCTION
Since Kansa’s work [12] in 1990, radial basis functions (RBFs) have been extensively used for solving partial
differential equations (PDEs) numerically [11,17,21]. In these methods, full and ill-conditioning coefficient matrix is
always obtained as RBFs is used globally with all the nodes in the problem domain. Recently, methods that use local
nodes and RBFs for solving PDEs has been proposed [4-9], which resulted in banded coefficient matrix. It is found that
the stability of the solution becomes, however, an important issue [2,6,7].
Radial point collocation method (RPCM) is a meshfree strong-form method using radial basis functions with
irregularly distributed local nodes [4-7]. It is regarded as a truly meshfree method as mesh is required neither in the
formulation procedure nor function approximation. Since no mesh is used, mesh-related problems can therefore be
avoided. The strong formulation is simple and straightforward [1,2], which can facilitate an easier implementation of
adaptive analysis if effective stabilization techniques can be developed.
In this paper, regularization techniques, which are often been used for stabilizing the solution of ill-posed inverse
problem [3], are suggested to restore the stability of the RPCM solution. Four different regularization schemes are
provided for constructing the regularization matrix that modifies the system coefficient matrix and hence provides
stability in the solution.
FUNCTION APPROXIMATION
A local radial point interpolation scheme is used to approximate unknown field function using radial basis functions
(RBFs) locally. Consider a smooth field function u can be approximated at interest point x in the problem domain as
u h (x ) = ∑ ai ri ( x − x i ) + ∑ b j p j
n
m
i =1
j =1
(1)
where n is the number of supporting nodes in the local support domain, m is the number of monomials in polynomial
— 95 —
function, ri
( ⋅ ) is radial basis function and
p j is the monomial of polynomial function for augmentation. ai and
b j are the coefficients of radial basis function and monomial of polynomial function.
By enforcing the interpolation passing through the nodal values, the following expression can be obtained,
⎡ u1 ⎤ ⎡ r1 ( x1 − x1 ) r1 ( x1 − x 2 )
⎢u ⎥ ⎢ r ( x − x ) r ( x − x )
1
2
2
2
⎢ 2⎥ = ⎢ 2 2
M
M
⎢M⎥ ⎢
⎢ ⎥ ⎢
⎣un ⎦ ⎣ rn ( x n − x1 ) rn ( x n − x 2 )
L r1 ( x1 − x n
L r2 ( x 2 − x n
O
M
L rn ( x n − x n
)
)
)
p1 (x1 ) L
p1 (x 2 ) L
M
O
p1 (x n ) L
⎡ a1 ⎤
pm (x1 )⎤ ⎢ M ⎥
⎥⎢ ⎥
pm (x 2 )⎥ ⎢ an ⎥
⎢ ⎥
M ⎥ ⎢ b1 ⎥
⎥
pm (x n )⎦ ⎢ M ⎥
⎢ ⎥
⎣bm ⎦
or
⎡a⎤
U = [R P] ⎢ ⎥,
⎣b ⎦
(2)
where U is the vector of unknown nodal values, a and b are the vector of coefficients of radial basis functions and
monomials of polynomial function respectively.
With orthogonal condition [13,14],
PT a = 0
(3)
The vectors of coefficients can be obtained as
⎡a⎤ ⎡ R
⎢b⎥ = ⎢PT
⎣ ⎦ ⎣
−1
P ⎤ ⎧U ⎫
−1 ⎧U ⎫
⎨ ⎬=G ⎨ ⎬
⎥
0⎦ ⎩ 0 ⎭
⎩0⎭
(4)
Therefore, the approximated field function u at interest point x can be expressed as
u h (x ) = [φ1 (x ) φ2 (x ) L φn (x )]U = Φ(x )U
(5)
where φ (x ) is the shape function called RPIM shape function.
The derivative of the field function can be easily obtained by differentiating the shape functions. For example, the first
derivative of field function with respect to k can expressed as
u,k (x ) = Φ,k (x )U
h
(6)
The details of constructing the RPIM shape functions using local nodes and general forms of RBFs with arbitrary real
parameters can be found in e.g., [1,2]. In this work, we use 16 nodes in the local domain and second order polynomials
for augmentation.
RADIAL POINT COLLOCATION METHOD (RPCM)
Consider an unknown field function u in a problem domain governed by PDEs in the forms of
L(u ) = f in Ω ,
(7)
with Neumann boundary conditions
B (u ) = g on Γt ,
(8)
and Dirichlet boundary conditions
u = u on Γ u ,
(9)
where L( ) and B( ) are differential operators.
Assume that the field function is smooth and can be approximated using RPIM shape functions which created by local
nodes, the above PDEs can be collocated at the field nodes in the problem domain and on the boundaries respectively
as shown in Fig. 1.
The discretized PDEs can be expressed as a set of algebraic equations as
L(ui ) = f i in Ω ,
(10)
— 96 —
Figure 1: Discretization of problem domain
with Neumann boundary conditions
B (ui ) = g i on Γt ,
(11)
and Dirichlet boundary conditions
ui = ui on Γ u ,
(12)
where subscript “i” denotes the equation at the ith collocation points.
The discretized governing system equations at all nodes in the problem domain can be then assembled and expressed in
the following matrix form:
KU = F ,
(13)
where K is the stiffness matrix, U is the vector of unknown nodal values and F denotes the force vector. Note that the
stiffness matrix is generally unsymmetric.
The vector of unknown nodal values can be obtained as
U = K −1F ,
(14)
if K is well-conditioned and invertible.
It is known that K is not always well-conditioned and its characteristics can be changed significantly while subjected
to small changes in the problem setting (such as change in node distribution) [2], therefore Eq. (13) may not be always
solvable, or the solution may not be stable. Attempts have been made in stabilizing the solutions as summarized in ref.
[2]. The following section discusses the regularization technique that used for stabilizing the solution.
REGULARIZATION PROCEDURE
The regularization techniques, which are commonly applied for stabilizing the inverse problem, was initially proposed
by Liu et al. [10] to stabilize the RPCM solution of forward problem. The basic idea is to modify the system coefficient
matrix using additional information about the problem, and the procedure is as follows.
A functional Π is first defined as follows [3,18],
Π = {KU − F}T {KU − F}+ α {K r U − Fr }T {K r U − Fr },
(15)
where α is the regularization factor, K and F are the stiffness matrix and force vector defined in Eq. (13), Kr and Fr
are regularization matrix and regularization vector which are to be constructed using additional information of the
problem. These additional information can be obtained by collocating the regularization equations at regularization
points. The first term of the functional Π is the L2 norm of the residual of the discretized governing system equations
and the second term is the L2 norm of the residual of the discretized regularization equations. The details of
constructing regularization matrix and regularization vector will be given later.
Seek for the minimal of the functional Π with respect to the U leads to the following equation set.
— 97 —
∂Π
T
= 2K T {KU − F}+ 2αK r {K r U − Fr } = 0 .
∂U
(16)
The above equation can be rearranged in the following form,
[K K + αK
T
T
r
]
T
ˆ U = Fˆ ,
K r U = K T F + αK r Fr or K
(17)
where K̂ is the regularized coefficient matrix and F̂ is the regularized force vector. One should note from Eq. (17)
that the regularized coefficient matrix is symmetric and positive definite (SPD). The vector of unknown nodal values U
can be obtained as
ˆ −1Fˆ ,
U=K
(18)
if K̂ is well-conditioned and invertible.
This method is known as the regularized least-squares radial point collocation method (RLS-RPCM) [10]. Proper
choice of Kr and α can make K̂ well-conditioned, which ensures stable solution.
REGULARIZATION SCHEMES
In this section, four different regularization schemes are provided for constructing the regularization matrix and
regularization vector.
1. Scheme I The governing equation Eq. (7) is used as a regularization equation:
L(ur ) = Fr ,
(19)
where subscript “r” denotes the regularization points.
The regularization points are all the nodes in the problem domain and on the boundaries as shown in the Fig. 2.
Figure 2: Regularization points of Scheme I
Hence, the regularization equation can be collocated at the regularization points and expressed in the following matrix
form
K r1U = Fr1 ,
(20)
where K r 1 is the regularization matrix and Fr 1 denotes as the regularization vector of scheme I.
2. Scheme II The PDEs that describe boundary conditions, Eq. (8) and Eq. (9), are used as the regularization equations
in scheme II. The regularization equation along the Neumann boundary is given as
B (ur ) = g r on Γ t ,
(21)
and the regularization equation along the Dirichlet boundary condition is given as
ur = ur on Γ u .
(22)
Regularization points are only allocated in between the boundary nodes as shown in Fig. 3.
— 98 —
Figure 3: Regularization points of Scheme II
By collocating Eq. (21) and (22) at the regularization points, the following matrix form can be obtained as
K r 2 U = Fr 2 ,
(23)
where K r 2 is the regularization matrix and Fr 2 denotes as the regularization vector of scheme II.
3. Scheme III In the third scheme, the problem domain is first discretized using Delaunay diagram. The regularization
points are allocated at the center of Delaunay cells as shown in Fig. 4.
Figure 4: Regularization points of Scheme III
Similar to the scheme I, the governing equation for interior nodes, Eq. (7), is used as the regularization equation. The
collocated regularization equation at regularization points can be expressed in the following matrix form,
K r 3U = Fr 3 .
(24)
where K r 3 is the regularization matrix and Fr 3 denotes as the regularization vector of scheme III.
4. Scheme IV In this scheme, the governing system equations, Eq. (7) to (9), are used as the regularization equations.
Regularization equations are given as
L(ur ) = Fr in Ω ,
(25)
with Neumann boundary conditions
B (ur ) = g r on Γt ,
(26)
and Dirichlet boundary conditions
ur = ur on Γu .
(27)
The regularization points in the problem domain are the same as the interior field nodes. The regularization points
lying along the boundary are allocated in between the field nodes along the boundary as shown in Fig. 5.
— 99 —
Figure 5: Regularization points of Scheme IV
The regularization equations can be collocated at regularization points respectively and assembled in the following
matrix form as
K r 4 U = Fr 4 ,
(28)
where K r 4 is the regularization matrix and Fr 4 denotes as the regularization vector of scheme IV.
DETERMINATION OF REGULARIZATION FACTOR
The degree of regularization depends on the regularization factor α . If α = 0 , the regularized least-squares radial
point collocation method (RLS-RPCM) will be reduced to the original radial point collocation method. Therefore, it is
very crucial to determine an appropriate regularization factor in the regularization procedure.
In this work, the L-curve approach [15] is adopted to determine the α . The idea of L-curve approach is to achieve an
optimal regularization factor such that the residual of the both terms of the functional Π in Eq. (15) are minimal. The
general guideline for determining an appropriate regularization factor is that α should be positive and just small
enough to prevent the ill-condition. From our observation, the optimal regularization factors for scheme I, scheme II
and scheme III are is remained unchanged throughout the adaptive analysis. Hence, regularization factor is only
required to determine at the initial step of the adaptive analysis.
Scheme IV is a very special case, the vector of unknown nodal values can be solved either by discretized governing
system equations, Eq. (13), or discretized regularization equations, Eq. (24). However, none of the solutions is stable.
By solving the same status of problem with two differences sets of discretized system equations, the regularization
procedure provides a compromise solution between two set of discretized system equations. Therefore, in this scheme,
α = 1 is a logical choice to be used.
ADAPTIVE STRATEGY
In this work, a residual based error indicator is used. The problem domain can be discretized using Delaunay diagram.
The governing equation is satisfied solely at the field nodes but elsewhere. The local error indicator can be evaluated at
the mid point of the Delaunay cell as
ηj =
1
A j Lu j − f j
3
L2
,
(29)
where A j is the area of the jth Delaunay cell and Lu j − f j
L2
is the L2 norm of the residual of the governing equation
at the mid point of the cell. The estimated global residual norm can therefore be obtained by integrating Eq. (29) over
the entire problem domain as
nc
1
j =1 3
η g = ∫ηl d Ω = ∑ Aj Lu j − f j
Ω
L2
(30)
,
where nc is the total number of Delaunay cells in the problem domain.
Additional node will be inserted in the middle of the jth Delaunay cell if the local error indicator for the jth cell meets the
refinement criteria,
— 100 —
η j ≥ κ lηm ,
(31)
where κ l is a predefined coefficient for local refinement and
ηm = max (η j ) .
(32)
j
Adaptive process will progress until the stopping criteria based on the global error indicator is met as
η g ≤ κ gη mg ,
(33)
where κ g is the global residual tolerant and ηmg is the maximum of the estimated global residual norm throughout the
adaptive process.
NUMERICAL EXAMPLE
In this numerical example, adaptive meshfree method using radial basis functions and local nodes is demonstrated.
Multi-quadratic (MQ) RBFs augmented with completed second order polynomial function is used for the function
approximation. The shape parameters of MQ-RBFs with arbitrary real parameters are adopted, and the recommended
parameters by Liu et al. [20] are used: q = 1.03, α c = 3.0 . The coefficient for local refinement and global residual
tolerant are predefined as κ l = 0.1 and κ g = 0.2 for all the cases studied in this work.
The following norm is defined to indicate the true error of the approximated solution:
∑ (s exact − s appr )
2
∑ (s exact )
2
e=
.
(34)
where s stands for the solution that can be displacements or stresses.
In this work, a benchmark plane stress problem of solid mechanics is studied. A cantilever beam subjected to a
parabolic shear stress at right end is considered (see Fig. 6). The geometrics and material properties are given as:
Young’s modulus E = 3 × 107 , Poisson’s ratio v = 0.3 , length of cantilever is L = 48.0m and the height is
P
H = 12.0m . The parabolic shear stress at the free end is given as τ xy = − ( H 2 / 4 − y 2 ) , where I is the moment of
2I
inertial of the cross section of cantilever beam and P = 1000N .
Figure 6: Cantilever beam subjected to a parabolic shear stress at the right end
The governing equations of the solid mechanics problem are well known as:
σ ij , j + bi = 0 in Ω ,
(35)
with Neumann boundary condition
σ ij n j = ti on Γ t ,
(36)
and Dirichlet boundary condition
ui = ui on Γ u .
(37)
In this example, the exact displacements are imposed along the left edge and Neumann boundary conditions are
imposed along the rest of the edges. The analytical solution of this problem can be found in [19].
The results of adaptive analysis using radial basis functions associated with different regularization schemes are shown
as follows.
— 101 —
1. Scheme I In this scheme, α = 0.01 is used as a regularization factor. The adaptive analysis started with 55 regular
distributed nodes and ended with 1258 nodes randomly distributed in the problem domain at 4th step as shown in
Fig. 7. The estimated global residual norms are plotted in Fig. 8, and the error norms of displacements are shown in
Fig. 9.
Figure 7: Nodal distribution of Scheme I in adaptive analysis
Figure 8: Estimated global residual norms of Scheme I in adaptive analysis
The error norms of the displacements have been drastically reduced from 9.38% and 10.39% to 0.69% and 0.58% in
the x, y directions respectively. The error norms of σ xx and τ xy are plotted in Fig. 10. The error norms of the stresses
reach 0.72% and 1.37% respectively at the final step of the adaptive analysis.
Figure 9: Error norms of displacements of Scheme I in adaptive analysis
— 102 —
Figure 10: Error norms of stresses of Scheme I in adaptive analysis
2. Scheme II In scheme II, α = 0.1 is used as a regularization factor Similar to scheme I, the adaptive analysis takes
four steps to complete in this case. The nodes has been increased from 55 regularly distributed nodes to 1185
irregularly distributed nodes in the problem domain at the final step as shown in Fig. 11. The estimated global residual
norms are plotted in Fig. 12, and the error norms of displacements are shown in Fig. 13.
Figure 11: Nodal distribution of Scheme II in adaptive analysis
Figure 12: Estimated global residual norms of Scheme II in adaptive analysis
From Fig. 13, the error norms of the displacements are shown tremendously reduced from 2.72% and 1.05% to 0.05%
and 0.06 % in the x and y direction respectively. The error norms of the σ xx and τ xy have been reduced from 4.55%
and 21.73% to 0.18% and 0.65% respectively, as shown in Fig. 14.
— 103 —
Figure 13: Error norms of displacements of Scheme II in adaptive analysis
Figure 14: Error norms of stresses of Scheme II in adaptive analysis
3. Scheme III In this scheme, α = 1 × 10 −5 is used as a regularization factor. The adaptive analysis takes five steps to
complete in this case. Nodal distributions at 1st, 3rd, 4th and 5th step are plotted in Fig. 15. The adaptive analysis ended
with 1533 nodes in the problem domain. The estimated residual norms are shown in Fig. 16 and the error norms of the
displacements are plotted in Fig. 17.
Figure 15: Nodal distribution of Scheme III in adaptive analysis
— 104 —
Figure 16: Estimated global residual norms of Scheme III in adaptive analysis
As shown in Fig. 17, the error norms of the displacements are reduced significantly from 135% and 134% to 5.10%
and 5.15% in the x and y direction. The error norms of the σ xx and τ xy have achieved 5.29% and 5.93% from 148%
and 244% respectively as shown in Fig. 18.
Figure 17: Error norms of displacements of Scheme III in adaptive analysis
Figure 18:. Error norms of stresses of Scheme III in adaptive analysis
4. Scheme IV The adaptive analysis of scheme IV began with 55 evenly distributed nodes and ended with 1277 nodes
at fourth step as shown in Fig. 19. The estimated global residual norms are plotted in Fig. 20, and the error norms of
displacements are plotted in Fig. 21.
— 105 —
Figure 19: Nodal distribution of Scheme IV in adaptive analysis
Figure 20: Estimated global residual norms of Scheme IV in adaptive analysis
Fig. 21 shows error norms of the displacements have reduced to 0.20% ad 0.10% from 3.16% and 3.97% in x, y
direction respectively. The error norms of the σ xx and τ xy are reduced gradually from 4.41 % and 19.62% to 0.26%
and 0.81% as shown in Fig. 22.
Figure 21: Error norms of displacements of Scheme IV in adaptive analysis
— 106 —
Figure 22: Error norms of stresses of Scheme IV in adaptive analysis
5. Results comparison To make a comparison among different schemes, the error norms of the displacements and
stresses at each adaptive step are listed in the Table 1 and Table 2.
Table 1 Error norm of displacements at each adaptive step
Displacements
error norm
Step 1
Step 2
Step 3
Step 4
Step 5
Scheme
u
v
u
v
u
v
u
v
u
v
I
9.38%
10.39%
2.69%
2.73%
0..66%
0.31%
0.69%
0.58%
-
-
II
2.72%
1.05%
1.85%
1.38%
2.05%
1.26%
0.05%
0.06%
-
-
III
135%
134%
7.42%
6.76%
19.08% 17.53%
5.06%
5.26%
5.10%
5.15%
IV
3.16%
3.97%
0.81%
0.76%
1.21%
0.20%
0.10%
-
-
RPCM
175%
179%
205%
201%
47.66% 51.15% 56.58% 53.79%
5.76%
4.63%
0.87%
*In the case of using RPCM without stabilization technique is used, twenty of the local nodes is used to construct the
shape functions.The number of nodes in the problem domain at each step is: 55,112,221,375,593.
Table 2 Error norm of stresses at each adaptive step
Stresses
error
norm
Step 1
Step 2
Step 3
Step 4
Step 5
Scheme
σ xx
τ xy
σ xx
τ xy
σ xx
τ xy
σ xx
τ xy
σ xx
τ xy
I
11.07%
25.48%
2.69%
6.00%
0.94%
2.81%
0.72%
1.37%
-
-
II
4.55%
21.73%
2.18%
6.77%
2.71%
3.76%
0.18%
0.65%
-
-
III
148%
244%
7.94%
14.40%
19.00%
22.62%
5.27%
8.10%
5.29%
5.93%
IV
4.41%
19.62%
1.39%
7.22%
1.34%
2.61%
0.26%
0.81%
-
-
RPCM
194%
320%
198%
234%
55.99%
100%
53.06%
78.01%
5.94%
6.06%
*In this problem, error norm of σ yy is not compared as the exact value of σ yy is equal to zero in the entire domain.
From the above examples, it is clearly seen that the global residual norm is reduced monotonically in adaptive process.
For schemes I, II and IV, the global residual norm converge almost linearly. This is a clear evidence that the schemes
are generally stable. The error norms of displacement or stresses are reduced as the adaptive steps forward, but not
always in a monotonic fashion. Note that our control of error is residual based, and hence one can have very good
— 107 —
understanding on the accuracy of the solution. This ensures the reliability of our solution obtained which is extremely
important in stress analysis of the practical structure designs.
The deflection, normal stress σ xx and shear stress τ xy of the cantilever beam computed by different regularization
schemes at final adaptive step are plotted in Fig. 23, Fig. 24 and Fig. 25. The deflections along y = 0 m are plotted
with analytical solution as shown in Fig. 23. Scheme I, II and IV are in very good agreement with the analytical
solution. Solutions of scheme III are obviously possesses larger error than other schemes, however the solution is still
closed to the analytical solution.
x 10
1
-3
Deflection of cantilever beam along y=0
Scheme I
Scheme II
Scheme III
Scheme IV
Analytical Solution
0
-1
Deflection uy
-2
-3
-4
-5
-6
-7
-8
-9
0
10
20
30
x-axis along y=0
40
50
Figure 23: The deflections along y = 0 m
The approximated normal stress σ xx along x = L / 2 at final step using different schemes are also compared with
analytical solution and shown in Fig. 24. Again, all the results are in very good agreement with analytical solution,
however the performance of the scheme I, II and IV are still better that scheme III.
σ xx along x=L/2
1500
1000
Scheme I
Scheme II
Scheme III
Scheme IV
Analytical Solution
σ xx
500
0
-500
-1000
-1500
-6
-4
-2
0
y-axis
2
4
6
Figure 24: The normal stress σ xx along x = L / 2
The comparison of the approximated shear stresses τ xy along x = L / 2 at final step using different schemes are given
in the Fig. 25. The solution of Scheme II and scheme IV are better than the scheme I and scheme III. Still, all the
solutions are very closed to the analytical solution.
— 108 —
τ
xy
along x=L/2
20
Scheme I
Scheme II
Scheme III
Scheme IV
Analytical Solution
0
-20
τ xy
-40
-60
-80
-100
-120
-140
-6
-4
-2
0
2
y-axis along x=L/2
4
6
Figure 25: The shear stresses τ xy along x = L / 2
Among all the schemes, the accuracy of the solutions of scheme III is the lowest and the number of nodes used are the
largest. Poor performance is observed in the Fig. 23 to Fig. 25 compared to other schemes. Scheme II and Scheme IV
are the best two schemes. Both schemes achieve higher accuracy in term of displacements and stresses at almost the
same amount of nodes used in the problem domain as shown in Table 1 and Table 2. The reason may be the fact that
these two scheme pay special attention to the the boundary of problem, which is important as discussed in [2]. Scheme
IV still stands an additional advantage against other schemes as regularization factor is not required to be determined.
In some cases, determination of the regularization factor can be tricky and not always easy. A regularization scheme,
which can get rid of the determination of regularization factor, is still of course preferred.
CONCLUSION
In this paper, four different regularization schemes for constructing the regularization matrix and force vector are
proposed. All the regularization schemes have shown regularization least-squares procedure is an effective measure to
stabilize the solutions of the radial point collocation method. As the stability is restored, adaptive meshfree
strong-form methods have been successfully implemented and demonstrated in the numerical example to obtain the
solution of desired accuracy. Comparison of among different schemes is made and discussed. Both scheme II and
scheme IV achieve higher accuracy in the adaptive analysis compared to scheme I and scheme III. Scheme IV is the
most preferred scheme as the determination of the regularization factor is not required.
Acknowledgements
The supports of Centre for ACES and NUS are gratefully acknowledged.
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