(Sparse)LinearSolvers
ABAB
A
x =
BA
x =
B
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(Sparse)
Linear
Solvers
A
B
A
B
A
x = B
Ax = B
Why?Why?
Many
geometry
processing
applications
boil
down
to:
solve one or more linear systemssolve
one
or
more
linear
systems
ParameterizationEditing Reconstruction Fairing Morphing
Don
’t you just invert
A
Don t
you
just
invert
A
-^
Even
if
invertible
V
i^
O(
Very expensive
O(
n
Usually
n
number
of
vertices/faces
Problem definition
-^
Input
Problem
definition
Matrix
A
mxn
Vector
B
mx
-^
Output^ –
Vector
x
nx
Such
that
Ax = B
-^
Small time and memory complexity
-^
Small
time
and
memory
complexity
-^
Use
additional
information
on
the
p
roblem p
Properties of
linear
systems for DGP
Properties
of
linear
systems
for
DGP
-^
Symmetric
p
ositive
definite
(p
ositive
y
p
(p
eigenvalues)^ –
Many
times
A
is
the
Laplacian matrix
Laplacian systems
are
usually
SPD
-^
A
remains,
b
changes
many
right
hand
sides
Interactive
applications
Mesh
geometry
same
matrix
for
X, Y, Z
Linear solvers zooLinear
solvers
zoo
-^
A
is
square
and
regular
-^
Indirect
solvers
iterative
-^
JacobiJacob
-^
Gauss
‐Seidel
-^
Conjugate
gradient
-^
Direct
solvers
factorization
-^
LU
-^
QRQR
-^
Cholesky
-^
Multigrid solvers
-^
Multigrid
solvers
8
Jacobi iterative
solver
Jacobi
iterative
solver
-^
Let
x
*^
be the exact solution
of
Ax
*^
=
b
Let
x
be
the
exact
solution
of
Ax
b
-^
Jacobi
method
Set
While
not
converged
Values
from
previous iteration
-^
For
i^
= 1
to
n
previous
iteration
10
Jacobi
iterative
solver
Pros
Si
l^
t^
i^
l^
t
-^
Si
mple
t
o
implement
No
need
for
sparse
data
structure
-^
Low
memory
consumption
O(
n
-^
Takes
advantage of sparse
structure
Takes
advantage
of
sparse
structure
b
ll li
d
-^
Can
b
e
parallelized
11
Direct solvers
factorization
Direct
solvers
factorization
•^
If A is diagonal/triangular the system is easy to solveIf
A
is
diagonal/triangular
the
system
is
easy
to
solve
•^
Idea:
-^
Idea:^ –
Factor
A
using
simple
”^
matrices x^1
Solve
using
k
easy
systems
13
Direct solvers
factorization
Direct
solvers
factorization
-^
Factoring
harder than solving
-^
Factoring
harder
than
solving
-^
Added
benefit
multiple
right
hand
sides
Factor only once!Factor
only
once!
14
Solving easy matricesSolving
easy
matrices
Lower triangularLower
triangular
-^
Forward
substitution
-^
Start from
x
-^
Start
from
x
1
16
Solving easy matricesSolving
easy
matrices
Upper triangularUpper
triangular
-^
Backward
substitution
-^
Start from
x
-^
Start
from
x
n
17
QR factorizationQR
factorization
-^
A
= QR
A
QR
Q
orthogonal
Æ
Q
T^
=
Q
R
upper
triangular
-^
Exists
for
any
matrix
-^
Solve
using
Rx
Q
Tb
19
Cholesky factorizationCholesky
factorization
•^
A = LL
T
L
lower
triangular
•^
Exists for square symmetric positive definiteExists
for
square
symmetric
positive
definite
matrices
-^
Solve
using
Lx
b
and
L
T^ x
x
1
20