Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
An Introduction to Jacobian-Free Newton-Krylov
with An Application to Non-Equilibrium
Radiation Diffusion
BOBBY PHILIP
Computer Science and Mathematics Division
Oak Ridge National Laboratory, U.S.A.
[email protected]
CIMPA Research School
Indian Institute of Science
July 17, 2013
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Preconditioned Krylov Methods
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Non-Equilibrium Radiation-Diffusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Conclusion
1
This research was conducted in part under the auspices of the Office of
Advanced Scientific Computing Research, Office of Science, U.S. Department
of Energy under Contract No. DE-AC05- 00OR22725 with UT-Battelle, LLC.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Krylov Methods
Original linear system:
Ax = b,
A ∈ Rn×n , b, x ∈ Rn .
BOBBY PHILIP
Nonlinear Methods
(1)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Krylov Methods
Original linear system:
Ax = b,
A ∈ Rn×n , b, x ∈ Rn .
Assume A is symmetric positive-definite (SPD), and we wish to
find a solution efficiently and robustly.
BOBBY PHILIP
Nonlinear Methods
(1)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Krylov Methods
Original linear system:
Ax = b,
A ∈ Rn×n , b, x ∈ Rn .
(1)
Assume A is symmetric positive-definite (SPD), and we wish to
find a solution efficiently and robustly.
Equivalent to solving the minimization problem: Find x ∈ Rn
minimizing:
1
f (y ) = minn ( y T Ay − b T y )
(2)
y ∈R 2
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Steepest Descent Method
General Idea:
◮
Pick initial guess: x0
◮
Compute direction of steepest descent:
−∇f (xk ) = b − Axk = rk
◮
New approximation: xk+1 = xk + αk rk
◮
αk comes from line search to minimize f (xk+1 ):
dxk+1
< rk , rk >
df (xk+1 )
= ∇f (xk+1 )T ·
= 0 =⇒ αk =
dαk
dαk
< Ark , rk >
◮
Note:
d 2 f (xk +αk rk )
dα2k
>0
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Steepest Descent Method
◮
◮
◮
f (xi ) = f (x + ei ) = f (x) + 12 eiT Aei
ei+1 = ei − αk ri
Convergence rate:
||ei ||A ≤
κ−1
κ+1
i
||e0 ||A ,
κ=
where ||ei ||A =< Aei , ei >
◮
λmax (A)
,
λmin (A)
Same search directions often visited multiple times leading to
slow convergence
BOBBY PHILIP
Nonlinear Methods
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Conjugate Directions Method
◮
Choose n A−orthogonal search directions d0 , d1 , · · · , dn−1
(Adi , dj ) = 0 for i 6= j
◮
Pick initial guess: x0
◮
Compute search vector: xk+1 = xk + αk dk
◮
Perform line search to choose αk by minimizing f (xk+1 ):
df (xk+1 )
dxk+1
= ∇f (xk+1 )T ·
= 0 =⇒< Aek+1 , dk >= 0
dαk
dαk
◮
Leads to: αk =
<Aek ,dk >
<Adk ,dk >
=
BOBBY PHILIP
<rk ,dk >
<Adk ,dk >
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Conjugate Directions Method
Eliminates error in search directions progressively
Pk
◮ xk+1 = xk + αk dk =⇒ ek+1 = ek − αk dk = e0 −
i=0 αi di
Pn−1
◮ Let e0 =
i=0 δi di .
◮ Then
δi
=
=
◮
This gives: ek+1
< Ae0 , di >
< Adi , di >
P
< A(e0 − i−1
j=0 αj dj ), di >
< Adi , di >
< Aei , di >
=
< Adi , di >
= αi
P
P
= e0 − ki=0 αi di = n−1
i=k+1 αi di
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Conjugate Directions Method
◮
Error minimization approach
◮
Converges in n iterations if search directions available
◮
Where do the n A-orthogonal search directions come from?
◮
Given n linearly independent vectors u0 , u1 , · · · , un−1 use the
Gram-Schmidt (GS) process (or modified GS) to
A-orthogonalize.
dk = uk −
with βik =
k−1
X
βik di
i=0
<Adi ,uk >
<Adi ,di >
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Conjugate Directions Method
◮
Properties:
◮
◮
◮
◮
Pros:
◮
◮
◮
(rk , dj ) = 0 for j < k since by design (Aek , dj ) = 0
Pj−1
(rk , uj ) = 0 for j < k since uj = dj + i=0 βij di
(rk , dk ) = (rk , uk ) for k = 0, 1, · · · , n − 1
Optimal for error minimization
Converges in n iterations if search directions available
Cons:
◮
◮
All search vectors have to be stored for orthogonalization
Full method is O(n3 ) for general bases
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Conjugate Gradients Method
◮
What if uk are chosen to be the rk like in Steepest Descent?
◮
Then, xk+1 = xk + αk dk gives αk Adk = rk − rk+1
◮
Or, αk (Adk , rj ) = (rk , rj ) − (rk+1 , rj ) = 0 if j < k
◮
Therefore at every iteration the current search direction dk is
A-orthogonal to all previous search directions except dk−1
◮
The Gram-Schmidt orthogonalization now reduces to
orthogonalizing against the previous search direction:
dk = rk − βk,k−1 dk−1
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Krylov Subspace
Since xk+1 = (xk + αk dk ), we have:
xk+1
∈ x0 + span{d0 , d1 , . . . , dk }
= x0 + span{d0 , Ad0 , . . . , Ak d0 }
= x0 + span{r0 , Ar0 , . . . , Ak r0 }
= x0 + Kk+1 (A; r0 )
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Conjugate Gradients Algorithm
◮
r0 ← b − Ax0 , d0 ← r0 .
◮
For j = 0,1,... until convergence do:
◮
α ← (rj , rj )/(Adj , dj )
◮
◮
◮
◮
◮
xj+1 ← xj + αdj
rj+1 ← rj − αAdj
β ← (rj+1 , rj+1 )/(rj , rj )
dj+1 ← rj+1 + βdj
enddo
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG Convergence
CG will converge to the solution in n iterations, and the
convergence rate is given by:
kx − xk kA
≤2
kx − x0 kA
√
κ−1 k
√
,
κ+1
κ=
but ..... n can be very large.
BOBBY PHILIP
Nonlinear Methods
λmax (A)
,
λmin (A)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG Convergence
CG will converge to the solution in n iterations, and the
convergence rate is given by:
kx − xk kA
≤2
kx − x0 kA
√
κ−1 k
√
,
κ+1
κ=
but ..... n can be very large.
CG converges quickly when:
BOBBY PHILIP
Nonlinear Methods
λmax (A)
,
λmin (A)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG Convergence
CG will converge to the solution in n iterations, and the
convergence rate is given by:
kx − xk kA
≤2
kx − x0 kA
√
κ−1 k
√
,
κ+1
κ=
but ..... n can be very large.
CG converges quickly when:
◮
κ(A) is small;
BOBBY PHILIP
Nonlinear Methods
λmax (A)
,
λmin (A)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG Convergence
CG will converge to the solution in n iterations, and the
convergence rate is given by:
kx − xk kA
≤2
kx − x0 kA
√
κ−1 k
√
,
κ+1
κ=
but ..... n can be very large.
CG converges quickly when:
◮
κ(A) is small;
◮
eigenvalues of A are clustered;
BOBBY PHILIP
Nonlinear Methods
λmax (A)
,
λmin (A)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG Convergence
CG will converge to the solution in n iterations, and the
convergence rate is given by:
kx − xk kA
≤2
kx − x0 kA
√
κ−1 k
√
,
κ+1
κ=
but ..... n can be very large.
CG converges quickly when:
◮
κ(A) is small;
◮
eigenvalues of A are clustered;
◮
distinct eigenvalues are few.
BOBBY PHILIP
Nonlinear Methods
λmax (A)
,
λmin (A)
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Preconditioning
Most systems do not have these nice properties.
The aim of preconditioning would be to transform the system
Ax = b,
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Preconditioning
Most systems do not have these nice properties.
The aim of preconditioning would be to transform the system
Ax = b,
into a system:
M −1 Ax = M −1 b,
such that both systems have the same solution, but the new system
now possesses one or more of the attributes mentioned, i.e.,
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Preconditioning
Most systems do not have these nice properties.
The aim of preconditioning would be to transform the system
Ax = b,
into a system:
M −1 Ax = M −1 b,
such that both systems have the same solution, but the new system
now possesses one or more of the attributes mentioned, i.e.,
◮
κ(M −1 A) is small;
◮
eigenvalues of M −1 A are clustered;
◮
distinct eigenvalues are few.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
What M would do the job?
Requirements on a preconditioner:
◮
M −1 A must be well conditioned
◮
M must be cheap to apply
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
What M would do the job?
Requirements on a preconditioner:
◮
M −1 A must be well conditioned
◮
M must be cheap to apply
These conflicting requirements must be balanced depending on the
application.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
What M would do the job?
Requirements on a preconditioner:
◮
M −1 A must be well conditioned
◮
M must be cheap to apply
These conflicting requirements must be balanced depending on the
application.
◮
Extreme 1: M = A =⇒ κ(M −1 A) = 1;
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
What M would do the job?
Requirements on a preconditioner:
◮
M −1 A must be well conditioned
◮
M must be cheap to apply
These conflicting requirements must be balanced depending on the
application.
◮
Extreme 1: M = A =⇒ κ(M −1 A) = 1;
◮
Extreme 2: M = I .
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
What M would do the job?
Requirements on a preconditioner:
◮
M −1 A must be well conditioned
◮
M must be cheap to apply
These conflicting requirements must be balanced depending on the
application.
◮
Extreme 1: M = A =⇒ κ(M −1 A) = 1;
◮
Extreme 2: M = I .
.... and somewhere in between is what we want.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
”Finding a good preconditioner to solve a given
sparse linear system is often viewed as a
combination of art and science....” Saad.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Splitting based preconditioners
Consider a splitting
A = M − N.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Splitting based preconditioners
Consider a splitting
A = M − N.
The associated stationary iteration is
x k+1 = M −1 Nx k + M −1 b.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Splitting based preconditioners
Consider a splitting
A = M − N.
The associated stationary iteration is
x k+1 = M −1 Nx k + M −1 b.
If this iteration converges, kI − M −1 Ak < 1 (can be relaxed), it
will converge to the fixed point of
(I − M −1 N)x = M −1 b,
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Splitting based preconditioners
But, I − M −1 N = I − M −1 (M − A) = M −1 A, and so we converge
to the solution of
M −1 Ax = M −1 b,
- the preconditioned system.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Splitting based preconditioners
But, I − M −1 N = I − M −1 (M − A) = M −1 A, and so we converge
to the solution of
M −1 Ax = M −1 b,
- the preconditioned system.
Thus, we can use preconditioners based on splittings. Examples
are:
◮
Jacobi
◮
(Symmetric) Gauss-Seidel
◮
(Symmetric) SOR
◮
Block versions of the above stationary splittings
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Splitting based preconditioners
But, I − M −1 N = I − M −1 (M − A) = M −1 A, and so we converge
to the solution of
M −1 Ax = M −1 b,
- the preconditioned system.
Thus, we can use preconditioners based on splittings. Examples
are:
◮
Jacobi
◮
(Symmetric) Gauss-Seidel
◮
(Symmetric) SOR
◮
Block versions of the above stationary splittings
These preconditioners are often cheap to implement and apply, but
may not provide significant improvements.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Factorization based preconditioners
◮
◮
◮
For SPD matrices we can factor A = LLt - a Cholesky
factorization. For sparse A this can introduce significant fill-in
and is expensive.
An incomplete Cholesky factorization A ≈ LLt is an
approximate Cholesky factorization where the sparsity pattern
of L is limited, based in some fashion on the sparsity pattern
of A. A = LLt + R
M = LLt can then be used as a preconditioner.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Factorization based preconditioners
◮
For SPD matrices we can factor A = LLt - a Cholesky
factorization. For sparse A this can introduce significant fill-in
and is expensive.
An incomplete Cholesky factorization A ≈ LLt is an
approximate Cholesky factorization where the sparsity pattern
of L is limited, based in some fashion on the sparsity pattern
of A. A = LLt + R
M = LLt can then be used as a preconditioner.
◮
Advantages:
◮
◮
◮
◮
Useful where black-box preconditioners required
Disadvantages;
◮
◮
Not easily parallelized
The factorization LLt can be unstable
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Multigrid preconditioners
◮
Multigrid methods optimal for elliptic problems.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Multigrid preconditioners
◮
◮
Multigrid methods optimal for elliptic problems.
Provide grid independent convergence rates.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Multigrid preconditioners
◮
◮
◮
Multigrid methods optimal for elliptic problems.
Provide grid independent convergence rates.
Example from Wathen:
− ▽2 u = f in Ω, u = g on ∂Ω.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Multigrid preconditioners
◮
◮
◮
Multigrid methods optimal for elliptic problems.
Provide grid independent convergence rates.
Example from Wathen:
− ▽2 u = f in Ω, u = g on ∂Ω.
A 2D FD discretization yields a block tridiagonal A, with
◮
◮
◮
λmin ≈ 2π 2 ,
λmax ≈ 8h−2 ,
and κ ≈ O(h−2 ).
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Multigrid preconditioners
◮
◮
◮
Multigrid methods optimal for elliptic problems.
Provide grid independent convergence rates.
Example from Wathen:
− ▽2 u = f in Ω, u = g on ∂Ω.
A 2D FD discretization yields a block tridiagonal A, with
◮
◮
◮
◮
◮
λmin ≈ 2π 2 ,
λmax ≈ 8h−2 ,
and κ ≈ O(h−2 ).
Unpreconditioned CG: h−dependent convergence rate.
CG preconditioned with multigrid: h−independent rate.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Multigrid preconditioners
◮
◮
◮
Multigrid methods optimal for elliptic problems.
Provide grid independent convergence rates.
Example from Wathen:
− ▽2 u = f in Ω, u = g on ∂Ω.
A 2D FD discretization yields a block tridiagonal A, with
◮
◮
◮
◮
◮
◮
λmin ≈ 2π 2 ,
λmax ≈ 8h−2 ,
and κ ≈ O(h−2 ).
Unpreconditioned CG: h−dependent convergence rate.
CG preconditioned with multigrid: h−independent rate.
Requires more effort to implement for average user (but more
packages are available now).
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG: Incorporating Preconditioning
◮
The preconditioned system is:
M −1 Ax = M −1 b
with M, A SPD. However, M −1 A is not SPD in general and it
would appear we no longer can apply CG.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG: Incorporating Preconditioning
◮
The preconditioned system is:
M −1 Ax = M −1 b
with M, A SPD. However, M −1 A is not SPD in general and it
would appear we no longer can apply CG.
◮
But M can be factored as M = LLt . Moreover, M −1 A and
L−1 AL−t have the same eigenvalues.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
CG: Incorporating Preconditioning
◮
The preconditioned system is:
M −1 Ax = M −1 b
with M, A SPD. However, M −1 A is not SPD in general and it
would appear we no longer can apply CG.
◮
◮
But M can be factored as M = LLt . Moreover, M −1 A and
L−1 AL−t have the same eigenvalues.
We can now apply CG to the preconditioned system
L−1 AL−t z = L−1 b, Lt x = z.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Split-Preconditioned (Transformed PCG)
◮
r0 ← b − L−1 AL−t x0 , d0 ← r0 .
◮
For j = 0,1,... until convergence do:
◮
α ← (rj , rj )/(L−1 AL−t dj , dj )
◮
◮
◮
◮
◮
xj+1 ← xj + αdj
rj+1 ← rj − αL−1 AL−t dj
β ← (rj+1 , rj+1 )/(rj , rj )
dj+1 ← rj+1 + βdj
enddo
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Untransformed PCG
Transformed CG would require computing the factorization
M = LLt . This can be avoided in practice and leads to the
following algorithm:
◮
r0 ← b − Ax0 , d0 ← M −1 r0 .
◮
For j = 0,1,... until convergence do:
◮
α ← (M −1 rj , rj )/(Adj , dj )
◮
◮
◮
◮
◮
xj+1 ← xj + αdj
rj+1 ← rj − αAdj
β ← (M −1 rj+1 , rj+1 )/(M −1 rj , rj )
dj+1 ← M −1 rj+1 + βdj
enddo
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
GMRES
◮
◮
◮
The Krylov space: Ki (A; r0 ) = span{r0 , Ar0 , · · · , Ai−1 r0 }
The Generalized Minimum Residual (GMRES) algorithm is a
minimum residual approach
i-th step of GMRES: Find xi ∈ Ki (A; r0 ) such that
xi = argmin (||b − Ax||2 )
x∈Ki (A;r0 )
◮
The Krylov vectors Aj r0 form an ill-conditioned basis for
Ki (A; r0 ) and cannot be directly used
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Arnoldi iteration
◮
◮
Given A ∈ Rn×n there exists a unitary matrix Q ∈ Rn×n and
an upper Hessenberg matrix H ∈ Rn×n such that Q T AQ = H
or equivalently AQ = QH.
Let Q = [q1 q2 . . . qn ]. By comparing columns of AQ and QH
we can show
AQk = Qk+1 H̃
where Qk = [q1 q2 . . . qk ] and H̃ ∈ R(k+1)×k is the upper left
matrix block of H which is upper Hessenberg also.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Arnoldi iteration
The Arnoldi iteration is used to generate the vectors qi from a
starting 2 norm unit vector q0
◮
◮
r0 ← q0 , h10 = 1, k = 0.
While (hk+1,k 6= 0)
◮
◮
◮
◮
qk+1 ← rk /hk+1,k
k ←k +1
rk ← Aqk
for j ← 1 · · · k
◮
◮
◮
hjk ← (rk , qj )
rk ← rk − hjk qj
hk+1,k ← krk k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
◮
◮
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
q0 , q1 , . . . , qi form an orthonormal basis for the vector space
Ki+1 (A; q0 ) = span{q0 , Aq0 , . . . , Ai q0 }
Let xi = Qi yi . Then the i-th iteration of GMRES becomes
yi
= argmin(||AQi y − b||2 )
y ∈Ri
= argmin(||Qi+1 H̃i y − b||2 )
y ∈Ri
∗
= argmin(||Qi+1
(Qi+1 H̃i y − b)||2 )
y ∈Ri
∗
= argmin(||H̃i y − Qi+1
b||2 )
y ∈Ri
= argmin(||H̃i y − ||b||e1 ||2 )
y ∈Ri
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
GMRES Algorithm
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
GMRES Algorithm
◮
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
k ←k +1
vk+1 ← Aqk
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
GMRES Algorithm
◮
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
◮
k ←k +1
vk+1 ← Aqk
for j ← 1 · · · k
◮
◮
◮
◮
hjk ← (vk+1 , qj )
vk+1 ← vk+1 − hjk qj
hk+1,k ← kvk+1 k
qk+1 ← vk+1 /hk+1,k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
GMRES Algorithm
◮
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
◮
k ←k +1
vk+1 ← Aqk
for j ← 1 · · · k
◮
◮
◮
◮
◮
◮
◮
hjk ← (vk+1 , qj )
vk+1 ← vk+1 − hjk qj
hk+1,k ← kvk+1 k
qk+1 ← vk+1 /hk+1,k
min kβe1 − Hk y k k2 over Rk
ρ ← kβe1 − Hk y k2
xk ← x 0 + Q k y k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
GMRES: Observations
◮
GMRES converges monotonically ||rk+1 ||2 < ||rk ||2
◮
GMRES converges in atmost n steps
◮
The storage and computational cost of each step increases
due to the increasing size of the Krylov space
◮
GMRES(m) restarted versions of GMRES
◮
Strong preconditioners can limit the size of the required
Krylov space resulting in an efficient and robust solver
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Preconditioning GMRES
◮
Left preconditioning: L−1 Ax = L−1 b
◮
Right preconditioning: AR −1 z = b, Rx = z
◮
Split preconditioning: L−1 AR −1 z = L−1 b, Rx = z
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Left Preconditioned GMRES
◮
r0 ← L−1 (b − Ax0 ), ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Left Preconditioned GMRES
◮
◮
r0 ← L−1 (b − Ax0 ), ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
k ←k +1
vk+1 ← L−1 Aqk
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Left Preconditioned GMRES
◮
◮
r0 ← L−1 (b − Ax0 ), ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
◮
k ←k +1
vk+1 ← L−1 Aqk
for j ← 1 · · · k
◮
◮
◮
◮
hjk ← (vk+1 , qj )
vk+1 ← vk+1 − hjk qj
hk+1,k ← kvk+1 k
qk+1 ← vk+1 /hk+1,k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Left Preconditioned GMRES
◮
◮
r0 ← L−1 (b − Ax0 ), ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
◮
k ←k +1
vk+1 ← L−1 Aqk
for j ← 1 · · · k
◮
◮
◮
◮
◮
◮
◮
hjk ← (vk+1 , qj )
vk+1 ← vk+1 − hjk qj
hk+1,k ← kvk+1 k
qk+1 ← vk+1 /hk+1,k
min kβe1 − Hk y k k2 over Rk
ρ ← kβe1 − Hk y k2
xk ← x0 + Qk y k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Left Preconditioned GMRES
Points to note:
◮
◮
The Arnoldi process now constructs an orthogonal basis for
(r0 , (L−1 A)r0 , · · · , (L−1 A)k r0 ).
All residuals and norms calculated in the algorithm now
correspond to the preconditioned residuals, L1− (b − Axk ).
◮
The original residuals and norms for the unpreconditioned
system cannot be accessed easily, they have to be explicitly
calculated.
◮
Special care must be taken when specifying stopping criteria
due to the previous point.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Right Preconditioned GMRES
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Right Preconditioned GMRES
◮
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
k ←k +1
vk+1 ← AM −1 qk
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Right Preconditioned GMRES
◮
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
◮
k ←k +1
vk+1 ← AM −1 qk
for j ← 1 · · · k
◮
◮
◮
◮
hjk ← (vk+1 , qj )
vk+1 ← vk+1 − hjk qj
hk+1,k ← kvk+1 k
qk+1 ← vk+1 /hk+1,k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Right Preconditioned GMRES
◮
◮
r0 ← b − Ax0 , ρ ← kr0 k, β ← ρ, q0 ← r0 /ρ, k = 0.
while ρ > ǫkbk andk < kmax do
◮
◮
◮
k ←k +1
vk+1 ← AM −1 qk
for j ← 1 · · · k
◮
◮
◮
◮
◮
◮
◮
hjk ← (vk+1 , qj )
vk+1 ← vk+1 − hjk qj
hk+1,k ← kvk+1 k
qk+1 ← vk+1 /hk+1,k
min kβe1 − Hk y k k2 over Rk
ρ ← kβe1 − Hk y k2
xk ← x0 + M −1 Qk y k
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Right Preconditioned GMRES
Points to note:
◮
◮
The Arnoldi process now constructs an orthogonal basis for
(r0 , (AM −1 )r0 , · · · , (AM −1 )k r0 ).
All residuals and norms calculated in the algorithm now
correspond to the original residuals, (b − Axk ).
◮
No special care needs to be taken when specifying stopping
criteria due to the previous point.
◮
Flexible preconditioned versions of GMRES are possible with
right preconditioning.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Steepest Descent
Conjugate Directions Method
Conjugate Gradients Method
Preconditioning
Krylov Methods for Non-Symmetric Systems
Split Preconditioned GMRES
Points to note:
◮
Suitable when A is nearly symmetric and P ≈ LLt .
◮
Combines left and right preconditioning aspects.
◮
Residual norms correspond to left preconditioning.
◮
Stopping criteria suffer same problems as left preconditioning
alone.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
Model equations:
∂E
− ∇ · (Dr ∇E ) = σa (T 4 − E )
∂t
∂T
− ∇ · (Dt ∇T ) = −σa (T 4 − E )
∂t
BOBBY PHILIP
Nonlinear Methods
in Ω = [0, 1]3
in Ω = [0, 1]3
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
Model equations:
∂E
− ∇ · (Dr ∇E ) = σa (T 4 − E )
∂t
∂T
− ∇ · (Dt ∇T ) = −σa (T 4 − E )
∂t
Constitutive law: σa =
z3
T3
BOBBY PHILIP
Nonlinear Methods
in Ω = [0, 1]3
in Ω = [0, 1]3
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
Model equations:
∂E
− ∇ · (Dr ∇E ) = σa (T 4 − E )
∂t
∂T
− ∇ · (Dt ∇T ) = −σa (T 4 − E )
∂t
Constitutive law: σa =
Diffusion coefficients:
in Ω = [0, 1]3
in Ω = [0, 1]3
z3
T3
1
Dr
=
Dt
= kT 5/2
BOBBY PHILIP
3σa +
k∇E k
E
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
Model equations:
∂E
− ∇ · (Dr ∇E ) = σa (T 4 − E )
∂t
∂T
− ∇ · (Dt ∇T ) = −σa (T 4 − E )
∂t
BOBBY PHILIP
Nonlinear Methods
in Ω = [0, 1]3
in Ω = [0, 1]3
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
Model equations:
∂E
− ∇ · (Dr ∇E ) = σa (T 4 − E )
in Ω = [0, 1]3
∂t
∂T
− ∇ · (Dt ∇T ) = −σa (T 4 − E )
in Ω = [0, 1]3
∂t
Initial conditions:
E = E0 , T = (E0 )1/4
at t = 0
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Non-Equilibrium Radiation-Diffusion
Model equations:
∂E
− ∇ · (Dr ∇E ) = σa (T 4 − E )
in Ω = [0, 1]3
∂t
∂T
− ∇ · (Dt ∇T ) = −σa (T 4 − E )
in Ω = [0, 1]3
∂t
Initial conditions:
E = E0 , T = (E0 )1/4
at t = 0
Boundary conditions:
1
E
n · Dr ∇E + = R
2
4
n · Dr ∇E = 0
n · ∇T = 0
BOBBY PHILIP
on ∂ΩR , t ≥ 0
on ∂ΩN , t ≥ 0
on ∂Ω, t ≥ 0
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Previous Work
◮
Rider, Knoll and Olson (JQSRT, 63, 1999; JCP, 152, 1999)
introduced the idea of physics based preconditioning in 1D
◮
Mousseau, Knoll, Rider (JCP, 2000) and Mousseau, Knoll
(JCP, 2003) demonstrated effectiveness for 2D problems
◮
Mavriplis (JCP, 175, 2002) compared Newton-Multigrid and
FAS using agglomeration ideas on unstructured grids.
◮
Stals (ETNA, 15, 2003), Newton-Multigrid and FAS, local
refinement on unstructured grids for equilibrium radiation
diffusion.
◮
Lowrie (JCP, 2004) compares different time integration
methods for non-equilibrium radiation diffusion
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Previous Work
◮
Brown, Shumaker, Woodward (JCP, 2005) consider fully
implicit methods and high order time integration.
◮
Shestakov, Greenough, and Howell (JQSRT, 2005) consider
pseudo-transient continuation on AMR grids using an
alternative formulation.
◮
Glowinski, Toivanen (JCP, 2005) consider using automatic
differentiation and system multigrid.
◮
Pernice, Philip (SISC, 2006), use JFNK with FAC
preconditioners on AMR grids for equilibrium
radiation-diffusion on SAMR grids.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Structured Adaptive Mesh Refinement
Structured adaptive mesh refinement (SAMR) represents a locally
refined mesh as a union of logically rectangular meshes.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Structured Adaptive Mesh Refinement
Structured adaptive mesh refinement (SAMR) represents a locally
refined mesh as a union of logically rectangular meshes.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Structured Adaptive Mesh Refinement
Structured adaptive mesh refinement (SAMR) represents a locally
refined mesh as a union of logically rectangular meshes.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Structured Adaptive Mesh Refinement
Structured adaptive mesh refinement (SAMR) represents a locally
refined mesh as a union of logically rectangular meshes.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Structured Adaptive Mesh Refinement
Structured adaptive mesh refinement (SAMR) represents a locally
refined mesh as a union of logically rectangular meshes.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Spatial Discretization:
◮
Method of Lines (MOL) approach
◮
Cell Centered Finite Volume Discretization
◮
Face centered diffusion coefficients
◮
Fluxes computed at cell faces
◮
Material discontinuities aligned with cell faces for simplicity
◮
Linear interpolation at coarse-fine interfaces to provide
centered ghost cell data
◮
Coarse-fine interpolation is programming intensive to account
for all special cases
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Space Discretization: Coarse-Fine Boundaries
(a) Convex face
(b) Convex edge
(d) Concave edge
(f) Sibling edge
BOBBY PHILIP
(c) Convex point
(e) Concave point
(g) Sibling point
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Space Discretization: Coarse-Fine Boundaries
(a) 3 patches.
(c) Coarse fine boundary fragments of the top face of
the patch at the bottom.
BOBBY PHILIP
(b) The coarse fine boundary
fragments of the right face of the
patch on the left.
(d) Coarse fine boundary fragments of the front face of the
patch at the back.
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Space Discretization: Coarse-Fine Interpolation
c1
c3
c1
i
f
c2
g
i
c3
i
c4
ci (i = 1, 2, 3, 4)
coarse ghost cell value
coarse ghost cells
i
interpolated value
fine ghost cells
f
fine cell value
fine cell
g
fine ghost cell value
Methods
Figure Nonlinear
:
BOBBY PHILIP
c2
c4
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Explicit, Semi-Implicit, or Implicit?
◮
Stiff IBVP that we want to evolve at dynamical timescale
◮
Explicit methods: do not meet criteria here
Semi-implicit methods:
◮
◮
Pros:
◮
◮
◮
potential overall cost savings
reuse off the shelf solver components
Cons:
◮
◮
◮
timesteps limited by stability restrictions, typically stronger
than accuracy restrictions
problem specific analysis
AMR implementations can be complex
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Explicit, Semi-Implicit, or Implicit?
Implicit methods:
◮ Pros:
◮
◮
◮
◮
Cons:
◮
◮
Step at dynamic timescale of problem instead of finest scales
Timestep limited primarily by accuracy considerations, stability
for nonlinear systems less of a problem
AMR implementations less complex than for semi-implciit
Efficient monolithic nonlinear solver required
We pick BDF2: 3 step method with quick damping of error
modes, 2nd order in time, potentially expand to BDF1-5
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: BDF2
αn2
1 + 2αn n+1
u
− (1 + αn )un +
un−1 = ∆tn f (un+1 )
1 + αn
1 + αn
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: BDF2
αn2
1 + 2αn n+1
u
− (1 + αn )un +
un−1 = ∆tn f (un+1 )
1 + αn
1 + αn
with
∆tn
∆tn−1
E
u =
T
∇ · (Dr ∇E ) + σa (T 4 − E )
f (u) =
∇ · (Dt ∇T ) − σa (T 4 − E )
αn =
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Timestep Control
◮
First step: Backward Euler with user selected initial guess for
timestep
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Timestep Control
◮
◮
First step: Backward Euler with user selected initial guess for
timestep
Subsequent steps:
◮
◮
◮
◮
constant fixed timestep
ramp to constant final timestep
limit relative change in energy
predictor-corrector with adaptive timestep selection
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Predictor-Corrector1
1
‘Incompressible Flow and the Finite Element Method, Volume 2, Isothermal Lamin
Flow‘, Gresho and Sani
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Predictor-Corrector1
◮
Predict using generalized leapfrog:
un+1
= un + (1 + αn ) ∆tn u̇n − αn2 un − un−1
P
1
‘Incompressible Flow and the Finite Element Method, Volume 2, Isothermal Lamin
Flow‘, Gresho and Sani
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Predictor-Corrector1
◮
Predict using generalized leapfrog:
un+1
= un + (1 + αn ) ∆tn u̇n − αn2 un − un−1
P
◮
Solve for un+1 using un+1
as an initial guess for BDF2 step
P
1 + 2αn n+1
αn2
u
− (1 + αn )un +
un−1 − ∆tn f (un+1 ) = 0
1 + αn
1 + αn
1
‘Incompressible Flow and the Finite Element Method, Volume 2, Isothermal Lamin
Flow‘, Gresho and Sani
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Time Discretization: Timestep Control
◮
Estimate local error
e n ≡ un − u(tn ) ≈
◮
◮
◮
αn−1 + 1 n
(u − unP )
3αn−1 + 2
Choose stepsize to satisfy : ||e n+1 || ≤ ǫ
Control theoretic controller (PI.4.7):
0.4/3 n−1 0.7/3
||e
||
∆tn+1 = ∆tn ||eǫn ||
||e n ||
Choice of ǫ and || · || is important
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Components
◮
Time Integrator (BDF2)
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Components
◮
Time Integrator (BDF2)
◮
Nonlinear Solver
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Components
◮
Time Integrator (BDF2)
◮
Nonlinear Solver
◮
Linear Solver
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Components
◮
Time Integrator (BDF2)
◮
Nonlinear Solver
◮
Linear Solver
◮
Preconditioner
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Components
◮
Time Integrator (BDF2)
◮
Nonlinear Solver
◮
Linear Solver
◮
Preconditioner
.... on AMR grids.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Nonlinear systems
Implicit time discretizations lead to a nonlinear system of equations
that needs to be solved at each timestep
F (un+1 ) = 0
where
F (un+1 ) ≡
1 + 2αn n+1
αn2
u
− (1 + αn )un +
un−1 − ∆tn f (un+1 )
1 + αn
1 + αn
with un+1 a cell centered vector over an AMR mesh.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Inexact Newton Methods
◮
Let F : Rn → Rn and consider solving F (u) = 0.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Inexact Newton Methods
◮
◮
Let F : Rn → Rn and consider solving F (u) = 0.
The k th step of classical Newton’s method requires solution of
the Newton equations:
F ′ (uk )sk = −F (uk ).
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Inexact Newton Methods
◮
◮
Let F : Rn → Rn and consider solving F (u) = 0.
The k th step of classical Newton’s method requires solution of
the Newton equations:
F ′ (uk )sk = −F (uk ).
◮
With inexact Newton methods, we only require
kF (uk ) + F ′ (uk )sk k ≤ ηk kF (uk )k,
This can be done with any iterative method.
BOBBY PHILIP
Nonlinear Methods
ηk > 0.
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Jacobian-free Newton-Krylov (JFNK)
◮
System multigrid could be used directly, or
◮
Krylov subspace methods - need Jacobian-vector products,
which can be approximated by
F ′ (uk )v ≈
F (uk + εv) − F (uk )
,
ε
√
ε ≈ O( ǫmach ).
◮
The resulting Jacobian-free Newton-Krylov (JFNK) method is
easier to implement because only function evaluation and
preconditioning setup/apply is required.
◮
ε must take into account accuracy, efficiency, and
non-negativity considerations
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioning
In order for a Newton-Krylov method to be efficient, effective
preconditioners are required.
The aim of preconditioning would be to transform the system
Ax = b,
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioning
In order for a Newton-Krylov method to be efficient, effective
preconditioners are required.
The aim of preconditioning would be to transform the system
Ax = b,
into a system:
P −1 Ax = P −1 b,
such that both systems have the same solution, but the new
system now possesses one or more of the attributes below:
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioning
In order for a Newton-Krylov method to be efficient, effective
preconditioners are required.
The aim of preconditioning would be to transform the system
Ax = b,
into a system:
P −1 Ax = P −1 b,
such that both systems have the same solution, but the new
system now possesses one or more of the attributes below:
◮ κ(P −1 A) is small;
◮ eigenvalues of P −1 A are clustered;
◮ distinct eigenvalues are few.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioned Krylov Methods
◮
Right-preconditioning of the Newton equations is used, i.e.,
we solve
(F ′ (uk )P −1 )Psk = −F (uk ).
where P is the preconditioner.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioned Krylov Methods
◮
◮
Right-preconditioning of the Newton equations is used, i.e.,
we solve
(F ′ (uk )P −1 )Psk = −F (uk ).
where P is the preconditioner.
For JFNK this requires the Jacobian-vector products:
(F ′ (uk )P −1 )v ≈
BOBBY PHILIP
F (uk + εP −1 v) − F (uk )
.
ε
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioned Krylov Methods
◮
◮
Right-preconditioning of the Newton equations is used, i.e.,
we solve
(F ′ (uk )P −1 )Psk = −F (uk ).
where P is the preconditioner.
For JFNK this requires the Jacobian-vector products:
(F ′ (uk )P −1 )v ≈
◮
F (uk + εP −1 v) − F (uk )
.
ε
The approximate Jacobian-vector is computed in two steps:
◮
◮
Solve y = P −1 v approximately
(uk )
.
Compute F (uk +εy)−F
ε
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Linear Systems
The Jacobian systems at each Newton step are of the form:
δE
−rE
L δT
= −r
T
where
L≈
I
∆t
− ∇ · Drk ∇ + σa I
−σa I
BOBBY PHILIP
I
∆t
−σa (T k )3
− ∇ · Dtk ∇ + σa (T k )3
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioner: Operator Split
We use a splitting of the form shown in our preconditioner
L ≈ P1 P 2
where
P1 =
I
∆t
− ∇ · Drk ∇
0
I
∆t
0
− ∇ · Dtk ∇
and
P2 =
(1 + ∆tσa )I
−∆tσa I
−∆tσa (T k )3
I + ∆tσa (T k )3
Systems involving P1 are solved using the FAC/AFACx methods
and systems involving P2 are solved using cell-wise inversion.
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Regridding: Time Integration
◮
Linear interpolation causes jump in time derivative
◮
Warm restart to minimize timestep changes
◮
Cold restart results in time step cuts
◮
Resolve at existing step to minimize perturbation to solution
◮
Regrid can lead to non-positive values
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Numerics: Simulation Software
◮
SAMRAI package for AMR
◮
PETSc SNES package for inexact Newton
◮
PETSc Krylov solver - GMRES
◮
SAMRSolvers package for multilevel preconditioners and
operators - FAC, AFACx, MDS
◮
NRDF application code with implicit time integration
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Numerics: Solver parameters
◮
◮
◮
◮
timestepper tolerance: 1.0e − 5
nonlinear solver absolute tolerance: 1.0e − 10
nonlinear solver relative tolerance: 1.0e − 12
step tolerance: 1.0e − 10
◮
variable forcing term: ηk = 0.01 initially
◮
max. gmres subspace dimension: 50
◮
max. linear iterations: 100
◮
FAC V-cycle, R-B Gauss Seidel (1 pre and 0 post smooths)
◮
final time: 0.5
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Numerics: Material Properties (Atomic Number)
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Nonlinear Iterations
Levels
1
2
3
4
5
163
-
2.99
2.96
2.64
2.71
323
2.99
2.96
2.63
2.71
-
643
2.96
2.64
2.71
-
-
1283
2.61
2.72
-
-
-
2563
3.04
-
-
-
-
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Nonlinear Iterations
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BOBBY PHILIP
Nonlinear Methods
�����
�����
�����
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Linear Iterations
Levels
1
2
3
4
5
163
-
7.71
7.21
6.63
6.92
323
7.68
7.22
6.61
6.91
-
643
7.15
6.63
6.92
-
-
1283
6.43
6.86
-
-
-
2563
6.72
-
-
-
-
BOBBY PHILIP
Nonlinear Methods
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Performance: Linear Iterations
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BOBBY PHILIP
Nonlinear Methods
�����
�����
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Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Relative Degrees of Freedom
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BOBBY PHILIP
Nonlinear Methods
����
����
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Number of Timesteps
Levels
1
2
3
4
5
163
-
1333
3251
8204
8697
323
1334
3251
8204
8704
-
643
3253
8204
8704
-
-
1283
8207
8734
-
-
-
2563
8698
-
-
-
-
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Timestep Variation
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BOBBY PHILIP
Nonlinear Methods
�����
�����
�����
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Performance: Timestep Variation
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BOBBY PHILIP
Nonlinear Methods
�����
�����
Model Problem
Discretization
Solution Methodology
Numerical Results
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
1,805.92
1,804.22
3,842.55
2,584.29
3,351.01
3,297.35
7,247.99
0.5
4,257.58
7,131.46
7,075.36
1
·104
8,360.88
Time in seconds
1.5
13,446.3
Performance: Wallclock Time
0
Total
Preconditioner
128b1l
64b2l
32b3l
Nonlinear function
16b4l
Figure : Timings for 128b1l equivalent AMR grids on 128 cores
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Accuracy: Temporal
t
0.05
∆t
0.15
0.25
0.35
0.45
L2 Error: Energy Density
2.0e-04
4.53e-05
3.93e-05
2.87e-05
2.47e-05
2.36e-05
1.0e-04
1.13e-05
9.80e-06
7.10e-06
6.10e-06
5.80e-06
5.0e-05
2.30e-06
2.00e-06
1.50e-06
1.30e-06
1.20e-06
L2 Error: Temperature
2.0e-04
2.43e-05
2.06e-05
1.89e-05
1.87e-05
1.82e-05
1.0e-04
6.00e-06
5.10e-06
4.70e-06
4.60e-06
4.50e-06
5.0e-05
1.20e-06
1.00e-06
1.00e-06
9.00e-07
9.00e-07
Table : Temporal L2 norm errors on a 16b4l grid
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Accuracy: Spatial, Single Material
t
0.05
Grid
0.11
0.27
0.36
0.45
L2 Error: Energy Density
16b1l
1.22e-01
3.46e-01
4.65e-01
4.22e-01
3.77e-01
16b2l
2.28e-01
1.99e-01
1.47e-01
1.35e-01
1.13e-01
16b3l
9.27e-02
5.64e-02
4.92e-02
3.20e-02
4.13e-02
16b4l
1.94e-02
1.30e-02
1.07e-02
1.06e-02
9.15e-03
128b1l
1.94e-02
1.30e-02
1.07e-02
1.06e-02
9.12e-03
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Accuracy: Spatial, Single Material
t
0.05
Grid
0.11
0.27
0.36
0.45
L2 Error: Temperature
16b1l
5.57e-02
1.61e-01
2.30e-01
2.34e-01
2.29e-01
16b2l
8.03e-02
8.05e-02
7.98e-02
7.04e-02
6.12e-02
16b3l
2.55e-02
2.28e-02
2.17e-02
1.69e-02
1.99e-02
16b4l
5.15e-03
4.37e-03
3.81e-03
4.14e-03
4.20e-03
128b1l
5.15e-03
4.37e-03
3.81e-03
4.14e-03
4.19e-03
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Accuracy: Spatial, Multimaterial
t
0.04
Grid
0.10
0.25
0.35
0.46
L2 Error: Energy Density
16b1l
6.01e-02
3.32e-01
4.99e-01
4.49e-01
3.77e-01
16b2l
2.21e-01
2.12e-01
1.55e-01
1.28e-01
1.15e-01
16b3l
9.92e-02
6.82e-02
4.12e-02
3.84e-02
4.28e-02
16b4l
2.38e-02
1.39e-02
1.11e-02
1.44e-02
2.29e-02
128b1l
2.38e-02
1.39e-02
1.11e-02
1.44e-02
2.29e-02
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Model Problem
Discretization
Solution Methodology
Numerical Results
Accuracy: Spatial, Multimaterial
t
0.04
Grid
0.10
0.25
0.35
0.46
L2 Error: Temperature
16b1l
3.15e-02
1.52e-01
2.32e-01
2.23e-01
2.06e-01
16b2l
7.78e-02
8.97e-02
7.17e-02
7.08e-02
7.86e-02
16b3l
2.76e-02
2.26e-02
1.90e-02
2.65e-02
3.39e-02
16b4l
5.53e-03
4.43e-03
7.63e-03
1.11e-02
1.64e-02
128b1l
5.53e-03
4.43e-03
7.63e-03
1.11e-02
1.64e-02
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
Software
◮
BoxMG, parallel open source geometric black box multigrid
solver (LANL)
◮
SMG, PFMG, BoomerAMG, parallel open source multigrid
solvers (LLNL)
◮
PETSc parallel multigrid solver (ANL)
◮
LAMG, parallel algebraic multigrid solver (LANL)
◮
SAMRSolvers, Multilevel FAC, AFAC, AFACx solvers,
ORNL/Philip
BOBBY PHILIP
Nonlinear Methods
Preconditioned Krylov Methods
Non-Equilibrium Radiation-Diffusion
Conclusion
References
◮
Iterative Methods for Large Linear Systems, Hank A. van der
Vorst
◮
Matrix Computations, Golub and Van Loan
◮
Iterative Methods for Sparse Linear Systems, Youssef Saad
◮
Iterative Methods for Linear and Nonlinear Systems, C. T.
Kelley
BOBBY PHILIP
Nonlinear Methods