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Krylov Subspace Methods
for Linear Systems and Matrix Equations
V. Simoncini
Dipartimento di Matematica
Universita di [email protected]
http://www.dm.unibo.it/˜simoncin
Krylov subspace methods – p. 1
Projection methods for large-scale problems
Given the system of n equations
F(x) = 0 x ∈ Rn
- Construct approximation space Km (m = dim(Km) )
- Find x ∈ Km such that x ≈ x
? Projection onto a much smaller space m ¿ n
Approximation process:
residual: r := F(x)
Construct (left) space Lm of dimension m and impose
r ⊥ Lm
Krylov subspace methods – p. 2
Projection methods for large-scale problems
Given the system of n equations
F(x) = 0 x ∈ Rn
- Construct approximation space Km (m = dim(Km) )
- Find x ∈ Km such that x ≈ x
? Projection onto a much smaller space m ¿ n
Approximation process:
residual: r := F(x)
Construct (left) space Lm of dimension m and impose
r ⊥ Lm
Krylov subspace methods – p. 2
Challenges
How to select Km
How to derive good Lm so that it also carries other good properties
General idea:
Construct sequence of approximation spaces Km ⊂ Km+1 such that
xm ∈ Km and xm → x as m → ∞
(in some sense)
Analogously, Lm ⊂ Lm+1
Krylov subspace methods – p. 3
Challenges
How to select Km
How to derive good Lm so that it also carries other good properties
General idea:
Construct sequence of approximation spaces Km ⊂ Km+1 such that
xm ∈ Km and xm → x as m → ∞
(in some sense)
Analogously, Lm ⊂ Lm+1
Krylov subspace methods – p. 3
Challenges
How to select Km
How to derive good Lm so that it also carries other good properties
General idea:
Construct sequence of approximation spaces Km ⊂ Km+1 such that
xm ∈ Km and xm → x as m → ∞
(in some sense)
Analogously, Lm ⊂ Lm+1
Krylov subspace methods – p. 3
More specific problems
F(x) = 0
A ∈ Rn×n
Ax− b = 0 (ϕk(A)x− b = 0) where ϕk polynomial of degree k
Ax − λMx = 0 (ϕk(λ, A0, A1, . . . , Ak)x = 0)
AX + XAT + Q = 0
Evaluation of Transfer and other matrix functions
Krylov subspace methods – p. 4
More specific problems
F(x) = 0
A ∈ Rn×n
Ax− b = 0 (ϕk(A)x− b = 0) where ϕk polynomial of degree k
Ax − λMx = 0 (ϕk(λ, A0, A1, . . . , Ak)x = 0)
AX + XAT + Q = 0
Evaluation of Transfer and other matrix functions
Krylov subspace methods – p. 4
More specific problems
F(x) = 0
A ∈ Rn×n
Ax− b = 0 (ϕk(A)x− b = 0) where ϕk polynomial of degree k
Ax − λMx = 0 (ϕk(λ, A0, A1, . . . , Ak)x = 0)
AX + XAT + Q = 0
Evaluation of Transfer and other matrix functions
Krylov subspace methods – p. 4
More specific problems
F(x) = 0
A ∈ Rn×n
Ax− b = 0 (ϕk(A)x− b = 0) where ϕk polynomial of degree k
Ax − λMx = 0 (ϕk(λ, A0, A1, . . . , Ak)x = 0)
AX + XAT + Q = 0
Evaluation of Transfer and other matrix functions
Krylov subspace methods – p. 4
Choosing Lm
Focus on linear system:
Ax = b, A nonsing.
xm ∈ Km:
? Lm = Km If A symmetric positive definite,
rm = b − Axm ⊥ Lm ⇔ ‖x − xm‖A = minex∈Km
!
? Lm = AKm
rm = b − Axm ⊥ Lm ⇔ ‖rm‖2 = minex∈Km
!
“Optimal” properties hold for any choice of Km Eiermann & Ernst A.N. ’01
Typically: Lm = Km ⇒ FOM, CG Lm = AKm ⇒ GMRES, MINRES
Krylov subspace methods – p. 5
Choosing Lm
Focus on linear system:
Ax = b, A nonsing.
xm ∈ Km:
? Lm = Km If A symmetric positive definite,
rm = b − Axm ⊥ Lm ⇔ ‖x − xm‖A = minex∈Km
!
? Lm = AKm
rm = b − Axm ⊥ Lm ⇔ ‖rm‖2 = minex∈Km
!
“Optimal” properties hold for any choice of Km Eiermann & Ernst A.N. ’01
Typically: Lm = Km ⇒ FOM, CG Lm = AKm ⇒ GMRES, MINRES
Krylov subspace methods – p. 5
Choosing Lm
Focus on linear system:
Ax = b, A nonsing.
xm ∈ Km:
? Lm = Km If A symmetric positive definite,
rm = b − Axm ⊥ Lm ⇔ ‖x − xm‖A = minex∈Km
!
? Lm = AKm
rm = b − Axm ⊥ Lm ⇔ ‖rm‖2 = minex∈Km
!
“Optimal” properties hold for any choice of Km Eiermann & Ernst A.N. ’01
Typically: Lm = Km ⇒ FOM, CG Lm = AKm ⇒ GMRES, MINRES
Krylov subspace methods – p. 5
Choosing Lm
Focus on linear system:
Ax = b, A nonsing.
xm ∈ Km:
? Lm = Km If A symmetric positive definite,
rm = b − Axm ⊥ Lm ⇔ ‖x − xm‖A = minex∈Km
!
? Lm = AKm
rm = b − Axm ⊥ Lm ⇔ ‖rm‖2 = minex∈Km
!
“Optimal” properties hold for any choice of Km Eiermann & Ernst A.N. ’01
Typically: Lm = Km ⇒ FOM, CG Lm = AKm ⇒ GMRES, MINRES
Krylov subspace methods – p. 5
Relaxing optimality in Lm. Truncation
Example: A is non-normal, but “nice” Lm = AKm
0 10 20 30 40 50 60 70 8010−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
space dimension
norm
of
resid
ual
k = ∞
k = 5
(-): r ⊥ Lm (- -): r ⊥loc Lm k gives “locality”
Krylov subspace methods – p. 6
Relaxing optimality in Lm. Truncation
Example: A is non-normal, but “nice” Lm = AKm
0 10 20 30 40 50 60 70 8010−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
space dimension
norm
of re
sid
ual
k = ∞
k = 5
(-): r ⊥ Lm (- -): r ⊥loc Lm k gives “locality”
Krylov subspace methods – p. 6
Relaxing optimality in Lm. Truncation
Example: A is non-normal, more “nasty” Lm = AKm
0 10 20 30 40 50 60 70 8010−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
space dimension
norm
of re
sid
ual
k = ∞
k = 30
k = 20
k = 10
(-): r ⊥ Lm (- -): r ⊥loc Lm Simoncini &Szyld, Num.Math.’05
Krylov subspace methods – p. 7
Choosing Km
The “classical” approach
Km := Km(A, v) = span{v, Av, . . . , Am−1v}
(e.g., v = b in linear systems)
x ∈ Km, x = pm−1(A)v
Nice Properties:
For m sufficiently large, Km(A, v) invariant for A
Convergence (analysis) in terms of spectral properties of A
Variants of “basic” methods by acting on polynomial pm−1
This choice of Km is good, but no longer sufficiently good
Krylov subspace methods – p. 8
Choosing Km
The “classical” approach
Km := Km(A, v) = span{v, Av, . . . , Am−1v}
(e.g., v = b in linear systems)
x ∈ Km, x = pm−1(A)v
Nice Properties:
For m sufficiently large, Km(A, v) invariant for A
Convergence (analysis) in terms of spectral properties of A
Variants of “basic” methods by acting on polynomial pm−1
This choice of Km is good, but no longer sufficiently good
Krylov subspace methods – p. 8
Choosing Km
The “classical” approach
Km := Km(A, v) = span{v, Av, . . . , Am−1v}
(e.g., v = b in linear systems)
x ∈ Km, x = pm−1(A)v
Nice Properties:
For m sufficiently large, Km(A, v) invariant for A
Convergence (analysis) in terms of spectral properties of A
Variants of “basic” methods by acting on polynomial pm−1
This choice of Km is good, but no longer sufficiently good
Krylov subspace methods – p. 8
Getting greedy
Km = Km−k(A, v) ∪ Sk New question: Sk?
? Faster approximation when spectral a-priori knowledge is available(even for A Hermitian)
? Main motivating problem:
Large m may be required for accurate approximation xm
⇓
Computational/Memory costs increase nonlinearly with m
(A non-normal)
Krylov subspace methods – p. 9
Getting greedy
Km = Km−k(A, v) ∪ Sk New question: Sk?
? Faster approximation when spectral a-priori knowledge is available(even for A Hermitian)
? Main motivating problem:
Large m may be required for accurate approximation xm
⇓
Computational/Memory costs increase nonlinearly with m
(A non-normal)
Krylov subspace methods – p. 9
Getting greedy
Km = Km−k(A, v) ∪ Sk New question: Sk?
? Faster approximation when spectral a-priori knowledge is available(even for A Hermitian)
? Main motivating problem:
Large m may be required for accurate approximation xm
⇓
Computational/Memory costs increase nonlinearly with m
(A non-normal)
Krylov subspace methods – p. 9
Intermezzo. Restarting procedure
Computational/Memory costs increase nonlinearly with m
(A non-normal)
Restarting procedure: x0 initial approx, r0 = b − Ax0
Km(A, r0) → x(1)m , r(1)
m
Km(A, r(1)m ) → x(2)
m , r(2)m
... →...
Warning: Larger m not always implies faster convergence(Embree, Ernst, ...)
Krylov subspace methods – p. 10
Restarted Methods
Convergence strongly depends on choice of m ...
0 100 200 300 400 500 600 700 80010−10
10−8
10−6
10−4
10−2
100
102
GMRES(15)
Number of Matrix−vector multiplies
norm
of re
lative r
esid
ual
GMRES(20)
Krylov subspace methods – p. 11
Restarted Methods
Convergence strongly depends on choice of m ... true?
0 100 200 300 400 500 600 700 80010−10
10−8
10−6
10−4
10−2
100
102
GMRES(15)
FOM(15)
Global subspace generated
norm
of re
lative r
esid
ual
GMRES(20)
Krylov subspace methods – p. 12
Restarted Methods
Switch to FOM residual vector only at the very first restart
0 50 100 150 200 250 300 350 400 45010−10
10−8
10−6
10−4
10−2
100
102
Number of Matrix−vector multiplies
no
rm o
f re
lative
re
sid
ua
l
GMRES(15)
GMRES(15)+FOM
Pictures from Simoncini, SIMAX 2000. Intermezzo ends.
Krylov subspace methods – p. 13
Augmented projection spaces
Km = Km−k(A, v) ∪ Sk
Sk from spectral information of A
Km−k(A, v) ∪ Sk = span{v, Av, . . . , Am−k−1v, y1, y2, . . . , yk}
y1, . . . , yk approximate eigenvectors of A associated to cluster
(Baglama, Calvetti, Erhel, Morgan, Nabben, Reichel, Saad, Sorensen, Vuik ...)
Sk important space from previous restartsRanking based on “error” or “effectiveness” of the past spaces
(De Sturler, Baker, Jessup, Manteuffel, ...)
Krylov subspace methods – p. 14
Augmented projection spaces
Km = Km−k(A, v) ∪ Sk
Sk from spectral information of A
Km−k(A, v) ∪ Sk = span{v, Av, . . . , Am−k−1v, y1, y2, . . . , yk}
y1, . . . , yk approximate eigenvectors of A associated to cluster
(Baglama, Calvetti, Erhel, Morgan, Nabben, Reichel, Saad, Sorensen, Vuik ...)
Sk important space from previous restartsRanking based on “error” or “effectiveness” of the past spaces
(De Sturler, Baker, Jessup, Manteuffel, ...)
Krylov subspace methods – p. 14
Augmented method. Spectral Information available.Structural Dynamics problem: (AB−1 + σI)x = b
A,B complex sym. n = 86, 000. Solve for σ in a wide interval
Note: at each iteration solve system with matrix B
B = B(κ) κ related to artificial stiff springs at ground boundaries
B numerically singular as κ → 0
Krylov subspace methods – p. 15
Augmented method. Spectral Information available.Structural Dynamics problem: (AB−1 + σI)x = b
A,B complex sym. n = 86, 000. Solve for σ in a wide interval
Note: at each iteration solve system with matrix B
B = B(κ) κ related to artificial spring stiffness at ground boundaries
B numerically singular as κ → 0
Fill-in p=5 Fill-in p=15
E. Time # its E. Time # its
[s] (outer/ avg. inner) [s] (outer/ avg. inner)
κ = 0 ACG 14066 296/38 12790 281/33
κ = 100 14072 117/120 13739 121/102
κ = 1000 8694 88/96 8724 89/83
κ = 1000 unrealistic Perotti & Simoncini, ’02
ACG: Augmented CG, Saad & Yeung & Ehrel & Guyomarc’h, 2000
Krylov subspace methods – p. 16
Example of Augmented method. Error SpaceTricky way to enhance approximation space
A from L(u) = −1000∆u + 2e4(x2+y2)ux − 2e4(x2+y2)uy n = 40 000
0 200 400 600 800 1000 120010−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Iteration index
Rela
tive r
esid
ual norm
GMRES(50)
GMRES(150)
Krylov subspace methods – p. 17
Example of Augmented method. Error SpaceTricky way to enhance approximation space (E. De Sturler)
A from L(u) = −1000∆u + 2e4(x2+y2)ux − 2e4(x2+y2)uy n = 40 000
0 200 400 600 800 1000 120010−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Iteration index
Rela
tive r
esid
ual norm
GMRES(50)
GMRES(150)
GCROT(7,23,23,4,1,0)
Code: courtesy of Oliver Ernst see also Baker, Jessup, Manteuffel ’05
Krylov subspace methods – p. 18
Changing Km
Modify Km instead of augmenting it!
Increase flexibility
Cope with “involved” operators
...
Krylov subspace methods – p. 19
Changing Km. Flexible methods
Original problem
AP−1x = b P preconditioner
Km(AP−1, b) = span{b, AP−1b, . . . , (AP−1)m−1b}
at each iteration i: zi = P−1vi
Flexible variant: Saad, ’93
Iteration i: zi = P−1vi ⇒ zi = P−1i vi
xm ∈ span{b, z1, z2, . . . , zm−1} 6= Km(AP−1, b)
Krylov subspace methods – p. 20
Changing Km. Flexible methods
Original problem
AP−1x = b P preconditioner
Km(AP−1, b) = span{b, AP−1b, . . . , (AP−1)m−1b}
at each iteration i: zi = P−1vi
Flexible variant: Saad, ’93
Iteration i: zi = P−1vi ⇒ zi = P−1i vi
xm ∈ span{b, z1, z2, . . . , zm−1} 6= Km(AP−1, b)
Krylov subspace methods – p. 20
Flexible methods. An exampleFlexible method may be used as a Truncated/Augmented method.
z = P−1v ⇔ z ≈ A−1v span{b, z1, z2, . . . , zm−1}
A from L(u) = −∆u + 1000xux n = 900
0 200 400 600 800 1000 120010−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
F/G(20) FT/G(16−2) FT/G(12−4)
GM(20)
number of matrix−vector multiplies
norm
of
rela
tive r
esid
ual
Simoncini & Szyld, SINUM ’03
Krylov subspace methods – p. 21
Flexible methods. An exampleFlexible method may be used as a Truncated/Augmented method.
z = P−1v ⇔ z ≈ A−1v span{b, z1, z2, . . . , zm−1}
A from L(u) = −∆u + 1000xux n = 900
0 200 400 600 800 1000 120010−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
F/G(20) FT/G(16−2) FT/G(12−4)
GM(20)
number of matrix−vector multiplies
norm
of re
lative r
esid
ual
Simoncini & Szyld, SINUM ’03
Krylov subspace methods – p. 21
Flexible methods. A second exampleFlexible method may be used as a Truncated/Augmented method.z = P−1v ⇔ z ≈ A−1v span{b, z1, z2, . . . , zm−1}
A from L(u) = −1000∆u + 2e4(x2+y2)ux − 2e4(x2+y2)uy n = 40 000
0 500 1000 1500 2000 250010−7
10−6
10−5
10−4
10−3
10−2
10−1
100
number of matrix−vector multiplies
norm
of re
lative r
esid
ual
GM(30)
F/G(30)FT/G(10−10)
FT/G(20−5)
Simoncini & Szyld, SINUM ’03
Krylov subspace methods – p. 22
Changing Km. Inexact methods.
Original problem
Ax = b A Possibly not available exactly
Km(A, b) = span{b, Ab, . . . , Am−1b}
Inexact (relaxed) variant:
Iteration i: zi = Avi ⇒ zi = Avi + fi
xm ∈ span{b, z1, z2, . . . , zm−1} 6= Km(A, b)
♣ “Worse” than for Flexible methods:
rm = b − Axm not available!
Available: rm computable residual
Krylov subspace methods – p. 23
Changing Km. Inexact methods.
Original problem
Ax = b A Possibly not available exactly
Km(A, b) = span{b, Ab, . . . , Am−1b}
Inexact (relaxed) variant:
Iteration i: zi = Avi ⇒ zi = Avi + fi
xm ∈ span{b, z1, z2, . . . , zm−1} 6= Km(A, b)
♣ “Worse” than for Flexible methods:
rm = b − Axm not available!
Available: rm computable residual
Krylov subspace methods – p. 23
Changing Km. Inexact methods.
Original problem
Ax = b A Possibly not available exactly
Km(A, b) = span{b, Ab, . . . , Am−1b}
Inexact (relaxed) variant:
Iteration i: zi = Avi ⇒ zi = Avi + fi
xm ∈ span{b, z1, z2, . . . , zm−1} 6= Km(A, b)
♣ “Worse” than for Flexible methods:
rm = b − Axm not available!
Available: rm computable residual
Krylov subspace methods – p. 23
Inexact Methods. An example
if ‖fi‖ = O
(ε
‖ri−1‖
)∀i ⇒ ‖b − Axm‖ ≤ ε for m large enough
BT S−1B︸ ︷︷ ︸A
x = b
At each it. i solve:
Swi = Bvi
‖Swi − Bvi‖ ≤ εinner(‖fi‖)
δm = ‖rm − erm‖
0 20 40 60 80 100 12010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
number of iterations
magnitude
δm
||rm
||
||rm
||
εinner
~
Simoncini & Szyld, SISC ’03
Krylov subspace methods – p. 24
Inexact Methods. An example
if ‖fi‖ = O
(ε
‖ri−1‖
)∀i ⇒ ‖b − Axm‖ ≤ ε for m large enough
BT S−1B︸ ︷︷ ︸A
x = b
At each it. i solve:
Swi = Bvi
‖Swi − Bvi‖ ≤ εinner(‖fi‖)
δm = ‖rm − erm‖
0 20 40 60 80 100 12010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
number of iterations
magnitude
δm
||rm
||
||rm
||
εinner
~
Simoncini & Szyld, SISC ’03
Krylov subspace methods – p. 24
A different application. The Lyapunov equation
AX + XAT + Q = 0
with A dissipative, Q = BBT of low rank X ≈ X low rank
Standard Krylov approach: X ∈ Km(A, B) and
R = AX + XAT + Q ⊥ Lm = Km(A, B)
X = VmYmV Tm for some Ym Range(Vm) = Km(A, B) Saad, ’90
New “Enhanced” approach: X ∈ Kk(A, B) ∪ Kk(A−1, B) and
R = AX + XAT + Q ⊥ Lm = Kk(A, B) ∪ Kk(A−1, B)
? Very competitive w.r.to Cyclic ADI methodSimoncini, tr. 2006
Krylov subspace methods – p. 25
A different application. The Lyapunov equation
AX + XAT + Q = 0
with A dissipative, Q = BBT of low rank X ≈ X low rank
Standard Krylov approach: X ∈ Km(A, B) and
R = AX + XAT + Q ⊥ Lm = Km(A, B)
X = VmYmV Tm for some Ym Range(Vm) = Km(A, B) Saad, ’90
New “Enhanced” approach: X ∈ Kk(A, B) ∪ Kk(A−1, B) and
R = AX + XAT + Q ⊥ Lm = Kk(A, B) ∪ Kk(A−1, B)
? Very competitive w.r.to Cyclic ADI methodSimoncini, tr. 2006
Krylov subspace methods – p. 25
A different application. The Lyapunov equation
AX + XAT + Q = 0
with A dissipative, Q = BBT of low rank X ≈ X low rank
Standard Krylov approach: X ∈ Km(A, B) and
R = AX + XAT + Q ⊥ Lm = Km(A, B)
X = VmYmV Tm for some Ym Range(Vm) = Km(A, B) Saad, ’90
New “Enhanced” approach: X ∈ Kk(A, B) ∪ Kk(A−1, B) and
R = AX + XAT + Q ⊥ Lm = Kk(A, B) ∪ Kk(A−1, B)
? Very competitive w.r.to Cyclic ADI methodSimoncini, tr. 2006
Krylov subspace methods – p. 25
A different application. The Lyapunov equation
AX + XAT + Q = 0
with A dissipative, Q = BBT of low rank X ≈ X low rank
Standard Krylov approach: X ∈ Km(A, B) and
R = AX + XAT + Q ⊥ Lm = Km(A, B)
X = VmYmV Tm for some Ym Range(Vm) = Km(A, B) Saad, ’90
New “Enhanced” approach: X ∈ Kk(A, B) ∪ Kk(A−1, B) and
R = AX + XAT + Q ⊥ Lm = Kk(A, B) ∪ Kk(A−1, B)
? Very competitive w.r.to Cyclic ADI methodSimoncini, tr. 2006
Krylov subspace methods – p. 25
An example. Time-invariant linear system
x′ = xxx + xyy + xzz − 10xxx − 1000yxy − 10zxz + b(x, y)u(t)
A matrix 183 × 183
0 5 10 15 20 2510−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
CPU Time
no
rm o
f re
latv
e r
esid
ua
l
Enhanced Krylov
Standard Krylov
approximation space dim.: 146 (Standard Krylov) 112 (Enhanced Krylov)
Krylov subspace methods – p. 26
An example. Time-invariant linear system
x′ = xxx + xyy + xzz − 10xxx − 1000yxy − 10zxz + b(x, y)u(t)
A matrix 183 × 183
0 5 10 15 20 2510−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
CPU Time
no
rm o
f re
latv
e r
esid
ua
l
Enhanced Krylov
Standard Krylov
approximation space dim.: 146 (Standard Krylov) 112 (Enhanced Krylov)
Krylov subspace methods – p. 26
An example. Time-invariant linear system
x′ = xxx + xyy + xzz − 10xxx − 1000yxy − 10xz + b(x, y)u(t)
A matrix 183 × 183
0 5 10 15 20 2510−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
CPU Time
no
rm o
f re
latv
e r
esid
ua
l
Enhanced Krylov
Enhanced Krylov(reord)
Standard Krylov
approximation space dim.: 146 (Standard Krylov) 112 (Enhanced Krylov)
Krylov subspace methods – p. 27
Conclusions and Pointers
Projection is a versatile tool
Lots of room for improvements on hard problems
Survey
“ Recent computational developments in Krylov Subspace Methodsfor linear systems”
with Daniel Szyld, Temple University
To appear in J. Numerical Linear Algebra w/Appl. (352 refs)
This and other papers at
http://www.dm.unibo.it/˜simoncin
Krylov subspace methods – p. 28
Conclusions and Pointers
Projection is a versatile tool
Lots of room for improvements on hard problems
Survey
“ Recent computational developments in Krylov Subspace Methodsfor linear systems”
with Daniel Szyld, Temple University
To appear in J. Numerical Linear Algebra w/Appl. (352 refs)
This and other papers at
http://www.dm.unibo.it/˜simoncin
Krylov subspace methods – p. 28
Structural Dynamics Problem
Mq + Cq + Kq = f(t) n d.o.f.
Linearization under small deformations
Direct frequency analysis for damping modeling(as opposed to modal or time analysis)? ideal for mechanical properties depending on frequency- influence of deformation velocity on materials- presence of hysteretic damping, ...
In the frequency domain: (a = −(2πf)2q(f)):
(1
−(2πf)2K∗ +
1
i(2πf)CV + M
)a = b
C = CV + 12πf
CH , ⇒ K∗ := K + iCH
Krylov subspace methods – p. 29
Concrete Slab ElementsGround Elements
x y
z
SIDE VIEW
PLAN VIEW
Concrete Slab
Elements
Ground
y
xz
Elements
Krylov subspace methods – p. 30
Model Problem: 3D Soil-Structure interaction
? viscous damping at the boundary ⇒ K∗ singular
⇒ FE discretization, K∗ almost singular
Algebraic Linearization:(σ2K∗ + σiCV − M
)x = b, x = x(σ)
equivalent to
[iCV −M
−M 0
]
︸ ︷︷ ︸A
+σ
[K∗ 0
0 M
]
︸ ︷︷ ︸B
[y
x
]=
[b
0
]
with y = σx
Krylov subspace methods – p. 31
+
Krylov subspace methods – p. 32