ExamplesLet us also revisit the shallow water equations

system of PDEs is defined as8>>>>>>><

>>>>>>>:

@h

@t+r · (hr�) = 0, in ⌦⇥ T ,

@�

@t+

1

2|r�|2 + h = 0, in ⌦⇥ T ,

h(0, x; ⌘h) = h0(x; ⌘h), in ⌦,

�(0, x; ⌘h) = �0(x; ⌘h), in ⌦,

(8.9)

with spatial coordinates x 2 ⌦, time t 2 T , state variables h,� : ⌦⇥ T 7! R, the first derivative operator@@t in time t and r· and r divergence and gradient differential operators in x, respectively. The variable� is the scalar potential of the fluid and h represents the height of the free-surface normalized by itsmean value. The system (8.9) is coupled with periodic boundary conditions for both the state variables.The evolution problem (8.9) admits a canonical symplectic Hamiltonian form (8.1) with Hamiltonian

H(h,�; ⌘) =1

2

Z

⌦

�h|r�|2 + h2

�dx. (8.10)

CP: We should add a numerical test where we monitor the projection error of the full model solutionat current time in the space spanned by the reduced basis at current time. As the rank is updated weshould see a decrease in the projection error. This would show that adding the error/residual as newcolumns (Section 6.3) is improving the approximation.NR: Done in 5; to be commented

8.2.1 SWE-1D

For this example, we set ⌦ = [�10, 10] and we consider the parametric domain � =⇥

110 ,

17

⇤⇥

⇥210 ,

1510

⇤.

The discrete set of parameters �h is obtained by uniformly sampling � with 10 samples per dimension,for a total of p = 100 different configurations. Problem (8.9) is completed by the initial condition

(h0(x; ⌘h) = 1 + ↵e��x2

,

�0(x; ⌘h) = 0,(8.11)

with ⌘h = (↵,�) and where ↵ controls the amplitude of the initial hump in the depth h and � its width(see Figure 1). The potential � is initially set to 0.

�10 �8 �6 �4 �2 0 2 4 6 8 10

1

1.05

1.1

1.15

spatial domain

h0(x

;⌘h)

Figure 1: SWE-1D: Initial value for the depth h for 1D test case. Different colors correspond to different parametervalues ⌘h = (↵,�).

We consider the second order accurate central finite difference scheme to discretize the differentialoperators in (8.9), represented by Dx. The domain ⌦ is partitioned into N � 1 equispaced intervals, as inthe linear wave equation case in Section 8.1, with N = 1000 and a resulting mesh width of �x = 2 · 10�2.

18

system of PDEs is defined as8>>>>>>><

>>>>>>>:

@h

@t+r · (hr�) = 0, in ⌦⇥ T ,

@�

@t+

1

2|r�|2 + h = 0, in ⌦⇥ T ,

h(0, x; ⌘h) = h0(x; ⌘h), in ⌦,

�(0, x; ⌘h) = �0(x; ⌘h), in ⌦,

(8.9)

with spatial coordinates x 2 ⌦, time t 2 T , state variables h,� : ⌦⇥ T 7! R, the first derivative operator@@t in time t and r· and r divergence and gradient differential operators in x, respectively. The variable� is the scalar potential of the fluid and h represents the height of the free-surface normalized by itsmean value. The system (8.9) is coupled with periodic boundary conditions for both the state variables.The evolution problem (8.9) admits a canonical symplectic Hamiltonian form (8.1) with Hamiltonian

H(h,�; ⌘) =1

2

Z

⌦

�h|r�|2 + h2

�dx. (8.10)

CP: We should add a numerical test where we monitor the projection error of the full model solutionat current time in the space spanned by the reduced basis at current time. As the rank is updated weshould see a decrease in the projection error. This would show that adding the error/residual as newcolumns (Section 6.3) is improving the approximation.NR: Done in 5; to be commented

8.2.1 SWE-1D

For this example, we set ⌦ = [�10, 10] and we consider the parametric domain � =⇥

110 ,

17

⇤⇥

⇥210 ,

1510

⇤.

The discrete set of parameters �h is obtained by uniformly sampling � with 10 samples per dimension,for a total of p = 100 different configurations. Problem (8.9) is completed by the initial condition

(h0(x; ⌘h) = 1 + ↵e��x2

,

�0(x; ⌘h) = 0,(8.11)

with ⌘h = (↵,�) and where ↵ controls the amplitude of the initial hump in the depth h and � its width(see Figure 1). The potential � is initially set to 0.

�10 �8 �6 �4 �2 0 2 4 6 8 10

1

1.05

1.1

1.15

spatial domain

h0(x

;⌘h)

Figure 1: SWE-1D: Initial value for the depth h for 1D test case. Different colors correspond to different parametervalues ⌘h = (↵,�).

We consider the second order accurate central finite difference scheme to discretize the differentialoperators in (8.9), represented by Dx. The domain ⌦ is partitioned into N � 1 equispaced intervals, as inthe linear wave equation case in Section 8.1, with N = 1000 and a resulting mesh width of �x = 2 · 10�2.

18

system of PDEs is defined as8>>>>>>><

>>>>>>>:

@h

@t+r · (hr�) = 0, in ⌦⇥ T ,

@�

@t+

1

2|r�|2 + h = 0, in ⌦⇥ T ,

h(0, x; ⌘h) = h0(x; ⌘h), in ⌦,

�(0, x; ⌘h) = �0(x; ⌘h), in ⌦,

(8.9)

with spatial coordinates x 2 ⌦, time t 2 T , state variables h,� : ⌦⇥ T 7! R, the first derivative operator@@t in time t and r· and r divergence and gradient differential operators in x, respectively. The variable� is the scalar potential of the fluid and h represents the height of the free-surface normalized by itsmean value. The system (8.9) is coupled with periodic boundary conditions for both the state variables.The evolution problem (8.9) admits a canonical symplectic Hamiltonian form (8.1) with Hamiltonian

H(h,�; ⌘) =1

2

Z

⌦

�h|r�|2 + h2

�dx. (8.10)

CP: We should add a numerical test where we monitor the projection error of the full model solutionat current time in the space spanned by the reduced basis at current time. As the rank is updated weshould see a decrease in the projection error. This would show that adding the error/residual as newcolumns (Section 6.3) is improving the approximation.NR: Done in 5; to be commented

8.2.1 SWE-1D

For this example, we set ⌦ = [�10, 10] and we consider the parametric domain � =⇥

110 ,

17

⇤⇥

⇥210 ,

1510

⇤.

The discrete set of parameters �h is obtained by uniformly sampling � with 10 samples per dimension,for a total of p = 100 different configurations. Problem (8.9) is completed by the initial condition

(h0(x; ⌘h) = 1 + ↵e��x2

,

�0(x; ⌘h) = 0,(8.11)

with ⌘h = (↵,�) and where ↵ controls the amplitude of the initial hump in the depth h and � its width(see Figure 1). The potential � is initially set to 0.

�10 �8 �6 �4 �2 0 2 4 6 8 10

1

1.05

1.1

1.15

spatial domain

h0(x

;⌘h)

Figure 1: SWE-1D: Initial value for the depth h for 1D test case. Different colors correspond to different parametervalues ⌘h = (↵,�).

We consider the second order accurate central finite difference scheme to discretize the differentialoperators in (8.9), represented by Dx. The domain ⌦ is partitioned into N � 1 equispaced intervals, as inthe linear wave equation case in Section 8.1, with N = 1000 and a resulting mesh width of �x = 2 · 10�2.

18

Let us define uh(t; ⌘h) := (hh,�h) = (h1, . . . , hN ,�1, . . . ,�N ) for all t 2 T and ⌘h 2 �h as the set ofdegrees of freedom. The semi-discrete formulation of (8.9) represents a canonical Hamiltonian systemwith the gradient of the Hamiltonian function with respect to uh given by

rHh(uh; ⌘h) =

0

@1

2|Dx�h|2 + hh

�Dx (h�Dx�h)

1

A .

The related discrete Hamiltonian is

Hh(uh; ⌘h) =1

2

NX

i=1

✓hi

✓�i+1 � �i�1

2�x

◆2

+ h2i

◆.

The solution uh(t; ⌘h) is computed using a uniform computational step size �t = 10�3 in the time intervalT = [0, 7]. We use the implicit midpoint rule as time integrator because it is a symplectic integrator.

50 100 150 200 25010�14

10�11

10�8

10�5

10�2(a)

index

sing

ular

valu

es

global

local

0 2 4 60

20

40

60(b)

time [s]

✏-ra

nk

10�1

10�3

10�5

10�7

10�9

0 5 10 15 20

10�3

10�1

Figure 2: SWE-1D: 2a) Singular values decay of the global snapshots matrix S and time average of the singularvalues decay of the local trajectories matrix St. The values are normalized using the largest singular value foreach case. 2b) ✏-rank of the local trajectories matrix St for different values of ✏.

19

Let us define uh(t; ⌘h) := (hh,�h) = (h1, . . . , hN ,�1, . . . ,�N ) for all t 2 T and ⌘h 2 �h as the set ofdegrees of freedom. The semi-discrete formulation of (8.9) represents a canonical Hamiltonian systemwith the gradient of the Hamiltonian function with respect to uh given by

rHh(uh; ⌘h) =

0

@1

2|Dx�h|2 + hh

�Dx (h�Dx�h)

1

A .

The related discrete Hamiltonian is

Hh(uh; ⌘h) =1

2

NX

i=1

✓hi

✓�i+1 � �i�1

2�x

◆2

+ h2i

◆.

The solution uh(t; ⌘h) is computed using a uniform computational step size �t = 10�3 in the time intervalT = [0, 7]. We use the implicit midpoint rule as time integrator because it is a symplectic integrator.

50 100 150 200 25010�14

10�11

10�8

10�5

10�2(a)

index

sing

ular

valu

es

global

local

0 2 4 60

20

40

60(b)

time [s]

✏-ra

nk

10�1

10�3

10�5

10�7

10�9

0 5 10 15 20

10�3

10�1

Figure 2: SWE-1D: 2a) Singular values decay of the global snapshots matrix S and time average of the singularvalues decay of the local trajectories matrix St. The values are normalized using the largest singular value foreach case. 2b) ✏-rank of the local trajectories matrix St for different values of ✏.

19

Examples

102 103 104

10�4

10�3

10�2

2n1 = 6

2n1 = 8

2n1 = 10

2n1 = 12

2n = 10

2n = 20

2n = 30

2n = 40

2n = 60

2n = 80

runtime [s]

E(T

)

Not adapted

r = 1.02, c = 1.2

r = 1.2, c = 1.2

Global Method

Full Model

102 103 104

10�4

10�3

10�2

2n1 = 6

2n1 = 8

2n1 = 10

2n1 = 12

2n = 10

2n = 20

2n = 30

2n = 40

2n = 60

2n = 80

runtime [s]

E(T

)

Not adapted

r = 1.02, c = 1.2

r = 1.2, c = 1.2

Global Method

Full Model

Figure 3: SWE-1D: Error as a function of the runtime for the Complex SVD method ( ), the dynamical RBmethod ( ) and the adaptive dynamical RB method for different values of the hyperparameters r and c ( , )for the simulations of all the sampled parameters in �h. The considered error metric is the error defined in(8.3) at t = T . The runtime is defined as the sum of the durations of the offline and online stages in case of theComplex SVD (global method), as the time required to evolve basis and coefficients ((5.3)) for the dynamicalRB. For the adaptive dynamical RB case, we add the time required to compute the error indicator and updatethe dimension of the approximating manifold. For the sake of comparison, we report the timing required by thehigh-fidelity solver ( ) to compute the numerical solution 8⌘h 2 �h.

20

0 1 2 3 4 5 6 7

10�3.5

10�3

(a)

E(t)

0 1 2 3 4 5 6 7

8

10

12

14

16 (b)

n⌧

0 1 2 3 4 5 6 7

10�4

10�3.5

(c)

E(t)

0 1 2 3 4 5 6 7

10

15

20(d)

n⌧

0 1 2 3 4 5 6 7

10�5

10�4

10�3(e)

time [s]

E(t)

Not adapted Target

r = 1.02, c = 1.2 r = 1.05, c = 1.2

r = 1.1, c = 1.2 r = 1.2, c = 1.2

0 1 2 3 4 5 6 7

15

20

25 (f)

time [s]

n⌧

Figure 4: SWE-1D: On the left, we report the evolution over time of the error E(t) ((8.3)) for the adaptive andnot adaptive proposed dynamical RB method for different values of the hyperparameters r and c and differentdimensions 2n1 of the approximating manifold of the initial condition. In particular, we consider the cases2n1 = 8(4a), 2n1 = 10 (4c) and 2n1 = 12 (4e) and we compare the errors to the error obtained by solving thesemidiscrete formulation (8.9) using the high fidelity solver with the initial condition (8.11) projected onto asymplectic manifold of dimension 2n1. On the right, the dimension of the dynamical reduced basis over time,driven by the error indicator (6.5) is shown for 2n1 = 8 (4b), 2n1 = 10 (4d) and 2n1 = 12 (4f). In the not adaptivesetting, the dimension does not change with time.

0

2

4

6

·10�4

(a)

E?(t)

Not adapted

r = 1.1, c = 1.2, 2n1 = 8

0

1

2

3

4·10�4

(b)

E?(t)

Not adapted

r = 1.2, c = 1.2, 2n1 = 10

0 1 2 3 4 5 6 7

10

15

20(c)

time [s]

n⌧

0 1 2 3 4 5 6 7

10

15

20 (d)

time [s]

n⌧

Figure 5: SWE-1D: In 5a) and 5b) we show the evolution of the error E?(t)(8.4) for different values of thehyperparameters r and c and initial dimension 2n1 of the approximating manifold. This error represents theinaccuracy introduced by projecting the solution obtained via the high-fidelity solver R(t), 8⌘h 2 �h, on the spacespanned by the dynamical basis V (t) computed in (5.3). In 5c) and 5d), we report the corresponding evolutionsof the sizes of the approximating manifolds.

21

2n=10

2n=12

ExamplesSimilar example for 2d shallow water equation (2n=6)

�3

�1

1

3

u

✓t;

✓1 3,17 10

◆◆

1

1.1

1.2

1.3

�3 �1 1 3

�3

�1

1

3

P2n⌧

i=1Ui(t)Zi

✓t;

✓1 3,17 10

◆◆

�3 �1 1 3 �3 �1 1 3 �3 �1 1 3

t = 0s t = 5s t = 15s t = 20s

Figure 7: SWE-2D: High-fidelity solution (first row) and adaptive dynamical reduced solution (second row) for

the parameter value (↵,�) =

✓13,1710

◆. In the reduced approach, we consider r = 1.1 and c = 1.3 and 2n1 = 6.

103 104 10510�5

10�4

10�3

10�2

10�1

100

2n1 = 4

2n1 = 6

2n1 = 8

2n1 = 10

runtime [s]

E(t)

Not adapted

r = 1.1 c = 1.2

r = 1.2 c = 1.3

r = 1.3 c = 1.1

r = 1.1 c = 1.3

Full Model

103 104 10510�5

10�4

10�3

10�2

10�1

100

2n1 = 4

2n1 = 6

2n1 = 8

2n1 = 10

runtime [s]

E(t)

Not adapted

r = 1.1 c = 1.2

r = 1.2 c = 1.3

r = 1.3 c = 1.1

r = 1.1 c = 1.3

Full Model

Figure 8: SWE-2D: Error as a function of the runtime for the dynamical RB method ( ) and the adaptivedynamical RB method for different values of the hyperparameters r and c ( , , , ) for the simulation of allthe sampled parameters in �h. The considered error metric is the error defined in (8.3) at t = T . The runtimeis defined as the time required to evolve basis and coefficients ((5.3)) for the dynamical RB. For the adaptivedynamical RB case, we add the time required to compute the error indicator and update the dimension of theapproximating manifold. For the sake of comparison, we report the timing required by the high-fidelity solver () to compute the numerical solution 8⌘h 2 �h.

23

�3

�1

1

3

u

✓t;

✓1 3,17 10

◆◆

1

1.1

1.2

1.3

�3 �1 1 3

�3

�1

1

3

P2n⌧

i=1Ui(t)Zi

✓t;

✓1 3,17 10

◆◆

�3 �1 1 3 �3 �1 1 3 �3 �1 1 3

t = 0s t = 5s t = 15s t = 20s

Figure 7: SWE-2D: High-fidelity solution (first row) and adaptive dynamical reduced solution (second row) for

the parameter value (↵,�) =

✓13,1710

◆. In the reduced approach, we consider r = 1.1 and c = 1.3 and 2n1 = 6.

103 104 10510�5

10�4

10�3

10�2

10�1

100

2n1 = 4

2n1 = 6

2n1 = 8

2n1 = 10

runtime [s]

E(t)

Not adapted

r = 1.1 c = 1.2

r = 1.2 c = 1.3

r = 1.3 c = 1.1

r = 1.1 c = 1.3

Full Model

103 104 10510�5

10�4

10�3

10�2

10�1

100

2n1 = 4

2n1 = 6

2n1 = 8

2n1 = 10

runtime [s]E(t)

Not adapted

r = 1.1 c = 1.2

r = 1.2 c = 1.3

r = 1.3 c = 1.1

r = 1.1 c = 1.3

Full Model

Figure 8: SWE-2D: Error as a function of the runtime for the dynamical RB method ( ) and the adaptivedynamical RB method for different values of the hyperparameters r and c ( , , , ) for the simulation of allthe sampled parameters in �h. The considered error metric is the error defined in (8.3) at t = T . The runtimeis defined as the time required to evolve basis and coefficients ((5.3)) for the dynamical RB. For the adaptivedynamical RB case, we add the time required to compute the error indicator and update the dimension of theapproximating manifold. For the sake of comparison, we report the timing required by the high-fidelity solver () to compute the numerical solution 8⌘h 2 �h.

23

Examples

0 2 4 6 8 10 12 14 16 18 20

10�2

10�1(a)

E(t)

0 1 2 3 4 5 6 7

4

6

8

10 (b)

2n⌧

0 2 4 6 8 10 12 14 16 18 20

10�3

10�2

10�1

100 (c)

E(t)

Not adapted Target

r = 1.1, c = 1.2 r = 1.2, c = 1.3

r = 1.3, c = 1.1 r = 1.1, c = 1.3

0 2 4 6 8 10 12 14 16 18 205

10

15 (d)

2n⌧

0 2 4 6 8 10 12 14 16 18 20

10�4

10�3

10�2 (e)

time [s]

E(t)

0 2 4 6 8 10 12 14 16 18 20

10

15

20 (f)

time [s]

2n⌧

Figure 9: SWE-2D: On the left, we report the evolution over time of the error E(t) ((8.3)) for the adaptiveand not adaptive proposed dynamical RB method for different values of the hyperparameters r and c anddifferent dimensions n0 of the approximating manifold of the initial condition. In particular, we consider thecases 2n1 = 4(9a), 2n1 = 6 (9c) and 2n1 = 8 (9e) and we compare the errors to the error obtained by solvingthe semidiscrete formulation (8.9) using the high fidelity solver with the initial condition (8.12) projected onto asymplectic manifold of dimension 2n1. On the right, the dimension of the dynamical reduced basis over time,driven by the error indicator (6.5) is shown for 2n1 = 4 (9b), 2n1 = 6 (9d) and 2n1 = 8 (9f). In the not adaptivesetting, the dimension does not change with time.

8.3 Schrödinger equationTest with example from optic fibers (?).We can re-write this section directly for the 2D case

We consider the cubic Schrödinger equation in the spatial domain ⌦ := [0, L],8<

:i@u

@t+

@2u

@x2+ ⌘|u|2u = 0, in ⌦⇥ T ,

u(0, x; ⌘) = u0(x; ⌘), in ⌦,(8.13)

with periodic boundary conditions. We consider as variables the real and imaginary part of the solutionu so that u = q + ip. Then, the Schrödinger equation (8.13) can be written as a Hamiltonian system incanonical symplectic form with Hamiltonian

H(q, p; ⌘) =1

2

Z L

0

✓✓@q

@x

◆2

+

✓@p

@x

◆2

� ⌘

2(q2 + p2)2

◆dx. (8.14)

Let us consider a partition of the spatial domain ⌦ into N � 1 equispaced intervals (xi, xi+1) withxi = (i � 1)�x for i = 1, . . . , N � 1, and �x = L/N . We consider a finite difference discretizationwhere the second order spatial derivative is approximated using a centered finite difference scheme. Letuh(t; ⌘h) := (q1, . . . , qN , p1, . . . , pN ), for all t 2 T and ⌘h 2 �h, where {qi}Ni=1 and {pi}Ni=1 are the degrees

24

Error control for 2d shallow water equation

ExamplesAs a last example, we consider the Poisson-Vlasov equation

0 0.5 1 1.5 2 2.5 3

10�3

10�2

10�1 (a)

E(t)

0 0.5 1 1.5 2 2.5 3

10

20

30 (b)

2n⌧

0 0.5 1 1.5 2 2.5 3

10�4

10�3

10�2

10�1

100101

(c)

E(t)

Not adapted r = 1.2, c = 1.05

r = 1.2, c = 1.1 r = 1.1, c = 1.05

0 0.5 1 1.5 2 2.5 3

10

20

30 (d)

2n⌧

Figure 16: NLS-2D: On the left, we report the evolution over time of the error E(t) ((8.3)) for the adaptive andnot adaptive proposed dynamical RB method for different values of the hyperparameters r and c and differentdimensions 2n1 of the approximating manifold of the initial condition. In particular, we consider the cases2n1 = 6(16a) and 2n1 = 8 (16c) and we compare the errors to the error obtained by solving the semidiscreteformulation (8.9) using the high fidelity solver with the initial condition (8.17) projected onto a symplecticmanifold of dimension 2n1. On the right, the dimension of the dynamical reduced basis over time, driven by theerror indicator (6.5) is shown for 2n1 = 6 (16b) and 2n1 = 8 (16d). In the not adaptive setting, the dimensiondoes not change with time.

8.4 Vlasov–Poisson plasma modelThe Vlasov–Poisson system describes the dynamics of a collisionless magnetized plasma under the actionof a self-consistent electric field. The evolution of the plasma at any time t 2 T ⇢ R is described interms of the distribution function fs(t, x, v) (s denotes the particle species) in the Cartesian phasespace domain (x, v) 2 ⌦ := ⌦x ⇥ ⌦v ⇢ R2. Assume that ⌦x := Td = Rd/(2⇡Zd) is the d-dimensionaltorus and ⌦v := Rd. We consider the 1D-1V Vlasov–Poisson problem (d = 1): For fs

0 2 V|t=0, find

fs 2 C1(T ;L2(⌦)) \ C0(T ;V ), and the electric field E 2 C0(T ;L2(⌦x)) such that

@tfs + v @xf

s +qs

msE @vf

s = 0, in ⌦⇥ T , 8s,

@xE =X

s

qsZ

⌦v

fs dv, in ⌦x ⇥ T ,

fs(0, x, v) = fs0 , in ⌦.

Here qs the charge and ms the particle mass, and the space V is defined as

V := {f(t, ·, ·) 2 L2(⌦) : f > 0, f(t, ·, v) ⇠ e�v2

as |v| ! 1},

The electric field can be written as the spatial derivative of the electric potential �, namely E(t, x) =�@x�(t, x). The Vlasov–Poisson problem can then be recast as

@tfs + v @xf

s � qs

ms@x�@vf

s = 0, in ⌦⇥ T , 8s,

��x� = ⇢(x) :=X

s

qsZ

⌦v

fs dv, in ⌦x ⇥ T .(8.18)

As shown in [7, Sections 1 and 2], the Vlasov–Poisson problem admits a Hamiltonian formulation with aLie–Poisson bracket and Hamiltonian given by the sum of the kinetic and electric energy as

H(f) =X

s

ms

2

Z

⌦v2fs(t, x, v) dx dv +

1

2

Z

⌦x

|E(t, x)|2 dx. (8.19)

29

It is solved using a PIC method, ieFor the semi-discrete approximation of problem (8.18) we consider a particle method couple with aH1-conforming discretization of the Poisson problem for the electric potential. The distribution functionfs is approximated by the superposition of P 2 N computational macro-particles each having a weight!s` , so that

fs(t, x, v) ⇡ fsh(t, x, v) =

PX

`=1

!s` S(x�Xs

` (t))S(v � V s` (t)),

where Xs(t) and V s(t) are the vector of the position and velocity of the macro-particles, respectively,and S is a compactly supported shape function, here chosen to be the Dirac delta. The idea of particlemethods is to derive the time evolution of the approximate distribution function fs

h by advancing themacro-particles along the characteristics of the Vlasov equation. As we will show in the following, asuitable discretization of the electromagnetic field allows to recast the characteristic equations as aHamiltonian system, to which we can apply the proposed dynamical model order reduction. Particlemethods, like PIC, are widely use in the numerical simulation of plasma models. However, the slowconvergence decay requires the use of many particles to achieve sufficient accuracy and therefore PICmethods are considered cumbersome. Model order reduction, in the number of macro-particles, of thesesemi-discrete schemes can be crucial and extremely beneficial.

On a partition of the spatial domain ⌦x, we consider the finite element discretization of the Poissonequation (8.18) in the space Pk⇤0(⌦x) ⇢ H1(⌦x) of continuous piecewise polynomial functions of degreeat most k � 1. The semi-discrete variation problem reads: find �h 2 C1(T ;Pk⇤0(⌦x)) such that

ah(�h, ) = l( ), 8 2 Pk⇤0(⌦x). (8.20)

where the bilinear form ah corresponds to the Laplace operator, while the linear function l is associatedwith the density ⇢; thereby

ah(', ) :=

Z

⌦x

rx'(x) ·rx (x) dx, l( ) :=X

s

qsZ

⌦fs(t, x, v) (x) dv dx, 8 ,' 2 Pk⇤

0(⌦x).

Let {�0i }Nki=1 be a basis of the space Pk⇤0(⌦x), then the time-dependent algebraic system ensuing from

(8.20) readsL� =

X

s

⇤0(Xs)>Msq

where � 2 RNk is the vector of degrees of freedom of �h, Msq 2 RP⇥1 is the row vector of entries

(Msq )` = qs`!

s` , while the matrices L 2 RNk⇥Nk and ⇤0(Xs) 2 RP⇥Nk are defined as

Li,j := (rx�0j ,rx�

0i )L2(⌦x), (⇤0(Xs))`,i := �0i (X

s` ). (8.21)

The Hamiltonian (8.19) is approximated as

Hh(fh) =X

s

PX

`=1

ms

2!s`V

s` (t)

2 +1

2

Z

⌦x

|@x�h(t, x)|2 dx

=1

2

X

s

V s(t)>MpVs(t) +

1

2�(t)>L�(t)

=1

2

X

s

V s(t)>MpVs(t) +

1

2�(t)>

X

s

⇤0(Xs)>Msq

=1

2

X

s

V s(t)>MpVs(t) +

1

2

X

s

(Msq )

>⇤0(Xs)L�1⇤0(Xs)>Msq ,

(8.22)

where Mp = diag(ms1!

s1, . . . ,m

sP!

sP ), and diag(d) denotes the diagonal matrix with diagonal elements

given by the vector d. By differentiating the discrete Hamiltonian Hh in (8.22) with respect to the vector

30

Classic problem in plasma physics/fusion/astrophysics etc

ExamplesPerformance the same as for simpler problems

100 200 300 400 500 600 70010�14

10�11

10�8

10�5

10�2(a)

index

sing

ular

valu

es

global

local

0 2 4 60

20

40

60

80(b)

time [s]

✏-ra

nk10�1

10�3

10�5

10�7

10�9

0 5 10 15 2010�4

10�3

10�2

10�1

100

Figure 18: V1D1V: 17a) Singular values decay of the global snapshots matrix S and time average of the singularvalues decay of the local trajectories matrix St. The values are normalized using the largest singular value foreach case. 17b) ✏-rank of the local trajectories matrix St for different values of ✏.

103 10410�5

10�4

10�3

10�2

2n1 = 42n1 = 6

2n1 = 82n1 = 10

2n1 = 12

2n = 10

2n = 20

2n = 40

2n = 60

runtime [s]

E(t)

Not adapted

r = 1.2 c = 1.1

r = 1.3 c = 1.1

r = 1.5 c = 1.1

Global

Full Model

Figure 19: V1D1V: Error as a function of the runtime for the Complex SVD method ( ), the dynamicalRB method ( ) and the adaptive dynamical RB method for different values of the hyperparameters r and c( , , , ) for the simulation of all the sampled parameters in �h. The considered error metric is the errordefined in (8.3) at t = T . The runtime is defined as the sum of the durations of the offline and online stages incase of the Complex SVD (global method), as the time required to evolve basis and coefficients ((5.3)) for thedynamical RB. For the adaptive dynamical RB case, we add the time required to compute the error indicator andupdate the dimension of the approximating manifold. For the sake of comparison, we report the timing requiredby the high-fidelity solver ( ) to compute the numerical solution 8⌘h 2 �h.

32

100 200 300 400 500 600 70010�14

10�11

10�8

10�5

10�2(a)

index

sing

ular

valu

es

global

local

0 2 4 60

20

40

60

80(b)

time [s]

✏-ra

nk

10�1

10�3

10�5

10�7

10�9

0 5 10 15 2010�4

10�3

10�2

10�1

100

Figure 18: V1D1V: 17a) Singular values decay of the global snapshots matrix S and time average of the singularvalues decay of the local trajectories matrix St. The values are normalized using the largest singular value foreach case. 17b) ✏-rank of the local trajectories matrix St for different values of ✏.

103 10410�5

10�4

10�3

10�2

2n1 = 42n1 = 6

2n1 = 82n1 = 10

2n1 = 12

2n = 10

2n = 20

2n = 40

2n = 60

runtime [s]

E(t)

Not adapted

r = 1.2 c = 1.1

r = 1.3 c = 1.1

r = 1.5 c = 1.1

Global

Full Model

Figure 19: V1D1V: Error as a function of the runtime for the Complex SVD method ( ), the dynamicalRB method ( ) and the adaptive dynamical RB method for different values of the hyperparameters r and c( , , , ) for the simulation of all the sampled parameters in �h. The considered error metric is the errordefined in (8.3) at t = T . The runtime is defined as the sum of the durations of the offline and online stages incase of the Complex SVD (global method), as the time required to evolve basis and coefficients ((5.3)) for thedynamical RB. For the adaptive dynamical RB case, we add the time required to compute the error indicator andupdate the dimension of the approximating manifold. For the sake of comparison, we report the timing requiredby the high-fidelity solver ( ) to compute the numerical solution 8⌘h 2 �h.

32

To summarize

‣The development of reduced order stable methods for time-dependent nonlinear problems is more complex that for traditional reduced models.

‣The Hamiltonian interpretation of problems yields access to a number of powerful tools

‣Local and adaptive techniques helps with cost

‣Structure preserving reduction methods are unknown

‣Errors — a priori and a posteriori — are poorly understood