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Juan Ignacio Polanco
BasisSplines.jl
Commits
89256cf1
Commit
89256cf1
authored
Apr 17, 2020
by
Juan Ignacio Polanco
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Allow nonsymmetric Galerkin matrices
parent
12889843
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src/galerkin.jl
src/galerkin.jl
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src/galerkin.jl
View file @
89256cf1
"""
galerkin_matrix(
B::BSplineBasis, [MatrixType = BandedMatrix{Float64}];
Ndiff::Val =
Val(0
),
Ndiff::Val =
(Val(0), Val(0)
),
)
Compute Galerkin mass matrix.
Compute Galerkin mass
or stiffness
matrix.
Definition:
Definition
of mass matrix
:
M[i, j] = ⟨ bᵢ, bⱼ ⟩ for i = 1:N and j = 1:N,
...
...
@@ 18,13 +18,16 @@ between functions.
To obtain a matrix associated to the Bspline derivatives, set the `Ndiff`
argument to the order of the derivative.
For instance, if `Ndiff = Val(1)`, this returns the matrix `⟨ bᵢ', bⱼ' ⟩`.
For instance, if `Ndiff = (Val(0), Val(2))`, this returns the matrix
`⟨ bᵢ, bⱼ'' ⟩`.
Note that the Galerkin matrix is
symmetric, positive definite and
banded,
Note that the Galerkin matrix is banded,
with `k + 1` and `k + 2` for `k` even and odd, respectively.
This function always returns a
Moreover, if both derivative orders are the same, the matrix is
symmetric and positive definite.
In those cases, a
[`Symmetric`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/index.html#LinearAlgebra.Symmetric)
view of an underlying matrix.
view of an underlying matrix
is returned
.
By default, the underlying matrix holding the data is a `BandedMatrix` that
defines the upper part of the symmetric matrix.
...
...
@@ 34,43 +37,52 @@ Other types of container are also supported, including regular sparse matrices
function
galerkin_matrix
(
B
::
BSplineBasis
,
::
Type
{
M
}
=
BandedMatrix
{
Float64
};
Ndiff
::
Val
=
Val
(
0
),
Ndiff
=
(
Val
(
0
),
Val
(
0
)
),
)
where
{
M
<:
AbstractMatrix
}
N
=
length
(
B
)
A
=
allocate_galerkin_matrix
(
M
,
N
,
order
(
B
))
S
=
Symmetric
(
A
)
galerkin_matrix!
(
S
,
B
,
Ndiff
=
Ndiff
)
deriv
=
_galerkin_make_Ndiff
(
Ndiff
)
symmetry
=
deriv
[
1
]
===
deriv
[
2
]
A
=
allocate_galerkin_matrix
(
M
,
N
,
order
(
B
),
symmetry
)
# Make the matrix symmetric if possible.
S
=
symmetry
?
Symmetric
(
A
)
:
A
galerkin_matrix!
(
S
,
B
,
Ndiff
=
deriv
)
end
allocate_galerkin_matrix
(
::
Type
{
M
},
N
,
k
)
where
{
M
<:
AbstractMatrix
}
=
allocate_galerkin_matrix
(
::
Type
{
M
},
N
,
etc
...
)
where
{
M
<:
AbstractMatrix
}
=
M
(
undef
,
N
,
N
)
allocate_galerkin_matrix
(
::
Type
{
SparseMatrixCSC
{
T
}},
N
,
k
)
where
{
T
}
=
allocate_galerkin_matrix
(
::
Type
{
SparseMatrixCSC
{
T
}},
N
,
etc
...
)
where
{
T
}
=
spzeros
(
T
,
N
,
N
)
function
allocate_galerkin_matrix
(
::
Type
{
M
},
N
,
k
)
where
{
M
<:
BandedMatrix
}
function
allocate_galerkin_matrix
(
::
Type
{
M
},
N
,
k
,
symmetry
)
where
{
M
<:
BandedMatrix
}
# The upper/lower bandwidths are:
#  for even k: Nb = k / 2 (total = k + 1 bands)
#  for odd k: Nb = (k + 1) / 2 (total = k + 2 bands)
# Note that the matrix is also symmetric, so we only need the upper band.
# Note that if the matrix is also symmetric, then we only need the upper
# band.
Nb
=
(
k
+
1
)
>>
1
M
(
undef
,
(
N
,
N
),
(
0
,
Nb
))
bands
=
symmetry
?
(
0
,
Nb
)
:
(
Nb
,
Nb
)
M
(
undef
,
(
N
,
N
),
bands
)
end
"""
galerkin_matrix!(S::Symmetric, B::BSplineBasis; Ndiff::Val = Val(0))
galerkin_matrix!(A::AbstractMatrix, B::BSplineBasis;
Ndiff::Val = (Val(0), Val(0)))
Fill preallocated Galerkin ma
ss ma
trix.
Fill preallocated Galerkin matrix.
It is assumed that the `Symmetric` view looks at the data in the upper part of
its parent. In other words, it was constructed with the `uplo = :U` option,
which is the default in Julia. See the [Julia
docs](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/index.html#LinearAlgebra.Symmetric)
for details.
The matrix may be a `Symmetric` view, in which case only one half of the matrix
will be filled. Note that, for the matrix to be symmetric, both derivative orders
in `Ndiff` must be the same.
See also [`galerkin_matrix`](@ref).
"""
function
galerkin_matrix
!
(
S
::
Symmetric
,
B
::
BSplineBasis
;
Ndiff
::
Val
=
Val
(
0
))
function
galerkin_matrix
!
(
S
::
AbstractMatrix
,
B
::
BSplineBasis
;
Ndiff
=
(
Val
(
0
),
Val
(
0
)))
N
=
size
(
S
,
1
)
if
N
!=
length
(
B
)
...
...
@@ 79,11 +91,6 @@ function galerkin_matrix!(S::Symmetric, B::BSplineBasis; Ndiff::Val = Val(0))
fill!
(
S
,
0
)
# The matrix is symmetric, so we fill only the upper part.
# For now we assume that S uses the upper part of its parent.
@assert
S
.
uplo
===
'U'
A
=
parent
(
S
)
k
=
order
(
B
)
t
=
knots
(
B
)
h
=
(
k
+
1
)
÷
2
# k/2 if k is even
...
...
@@ 92,21 +99,34 @@ function galerkin_matrix!(S::Symmetric, B::BSplineBasis; Ndiff::Val = Val(0))
# Quadrature information (weights, nodes).
quad
=
_quadrature_prod
(
k
)
# Upper part: j >= i
deriv
=
_galerkin_make_Ndiff
(
Ndiff
)
if
S
isa
Symmetric
deriv
[
1
]
===
deriv
[
2
]

error
(
"matrix will not be symmetric with Ndiff =
$
Ndiff"
)
fill_upper
=
S
.
uplo
===
'U'
fill_lower
=
S
.
uplo
===
'L'
A
=
parent
(
S
)
else
fill_upper
=
true
fill_lower
=
true
A
=
S
end
for
j
=
1
:
N
# We're only visiting the elements that have nonzero values.
# In other words, we know that S[i, j] = 0 outside the chosen interval.
istart
=
clamp
(
j

h
,
1
,
N
)
istart
=
fill_upper
?
clamp
(
j

h
,
1
,
N
)
:
j
iend
=
fill_lower
?
clamp
(
j
+
h
,
1
,
N
)
:
j
bj
=
BSpline
(
B
,
j
)
tj
=
support
(
bj
)
fj
=
x
>
bj
(
x
,
Ndiff
)
for
i
=
istart
:
j
fj
=
x
>
bj
(
x
,
deriv
[
2
]
)
for
i
=
istart
:
iend
bi
=
BSpline
(
B
,
i
)
ti
=
support
(
bi
)
fi
=
x
>
bi
(
x
,
Ndiff
)
fi
=
x
>
bi
(
x
,
deriv
[
1
]
)
t_inds
=
intersect
(
ti
,
tj
)
# common support of b_i and b_j
@assert
!
isempty
(
t_inds
)
# there is a common support (the Bsplines see each other)
@assert
length
(
t_inds
)
==
k
+
1

(
j

i
)
@assert
length
(
t_inds
)
==
k
+
1

abs
(
j

i
)
A
[
i
,
j
]
=
_integrate_prod
(
fi
,
fj
,
t
,
t_inds
,
quad
)
end
end
...
...
@@ 114,6 +134,9 @@ function galerkin_matrix!(S::Symmetric, B::BSplineBasis; Ndiff::Val = Val(0))
S
end
_galerkin_make_Ndiff
(
v
::
Val
)
=
(
v
,
v
)
_galerkin_make_Ndiff
(
v
::
Tuple
{
Vararg
{
<:
Val
,
2
}})
=
v
# Generate quadrature information for Bspline product.
# Returns weights and nodes for integration in [1, 1].
#
...
...
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