POD/EIM nonlinear model reduction and POD/4-D VAR with TR...

0.

POD/EIM nonlinear model reduction andPOD/4-D VAR with TR for models of the SWE

R. Stefanescu1 and I. M. Navon2

September 3, 2012

Florida State University, Tallahassee, Florida, USA

[email protected],[email protected]


POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 1/101

Part1 - POD/DEIM nonlinear model reduction

1 POD/DEIM POD/EIM justification and methodology

2 Description of EIM algorithm

3 Limited - Area SWE

4 Generation of POD using FEM

5 POD FEM SWE Model

6 Numerical Results

7 Reduced order POD 4-D VAR

8 Trust Region POD

9 T-R POD algorithm

10 Illustration

11 Conclusions

1. POD/DEIM POD/EIM justification and methodology

POD/DEIM POD/EIM justification and methodology

Model order reduction : Reduce the computationalcomplexity/time of large scale dynamical systems byapproximations of much lower dimension with nearly the sameinput/output response characteristics.

Goal : Construct reduced-order model for different types ofdiscretization method (finite difference (FD), finite element(FEM), finite volume (FV)) of unsteady and/or parametrizednonlinear PDEs. E.g., PDE:

∂y

∂t(x , t) = L(y(x , t)) + F(y(x , t)), t ∈ [0,T ]

where L is a linear function and F a nonlinear one.



POD/DEIM methodology applied to FD SCHEMES

The corresponding FD scheme is a n dimensional ordinarydifferential system

d

dty(t) = Ay(t) + F(y(t)), A ∈ Rn×n,

where y(t) = [y1(t), y2(t), .., yn(t)] ∈ Rn and yi (t) ∈ R arethe spatial components y(xi , t), i = 1, .., n. F is a nonlinearfunction evaluated at y(t) componentwise, i.e.F = [F(y1(t)), ..,F(yn(t))]T , F : I ⊂ R→ R.

A common model order reduction method involves theGalerkin projection with basis Vk ∈ Rn×k obtained fromProper Orthogonal Decomposition (POD), for k n, i.e.y ≈ Vk y(t), y(t) ∈ Rk . Applying an inner product to theODE discrete system we get

d

dty(t) = V T

k AVk︸︷︷︸k×k

y(t) + V Tk F(Vk y(t))︸︷︷︸

N(y)

(1)


POD/DEIM methodology applied to FD SCHEMES

The efficiency of POD - Galerkin technique is limited to thelinear or bilinear terms. The projected nonlinear term stilldepends on the dimension of the original system

N(y) = V Tk︸︷︷︸

k×n

F(Vk y(t))︸︷︷︸n×1

.

To mitigate this inefficiency we introduce ”Discrete EmpiricalInterpolation Method (DEIM) ” for nonlinear approximation.For m n

N(y) ≈ V Tk U(PTU)−1︸︷︷︸

precomputed k×m

F(PTVk y(t))︸︷︷︸m×1

.




POD/EIM methodology applied to FE SCHEMES

The corresponding Finite Element (FE) scheme is a ndimensional ordinary differential system

Mhd

dty(t) = Khy(t) + Nh(y(t)), Mh,Kh ∈ Rn×n, (2)

y(t) = [y1(t), y2(t), .., yn(t)] ∈ Rn, yi (t) ∈ R.

y(t, x) 'n∑

j=1

ψj(x)yj(t) = Ψ(x)y(t), Ψ(x) ∈ R1×n.





Nh(y(t)) ∈ Rn is a nonlinear functional which can be of thefollowing form

[Nh(y(t))]i =

∫Ω

∂ψi (x)

∂xF (Ψ(x)y(t))dΩ, i = 1, ..n.

[Nh(y(t))]i =

∫Ωψi (x)F (Ψ(x)y(t))dΩ, i = 1, ..n.




Using the Galerkin projection with basis Φ(x) = Ψ(x)Uk ,Φ(x) ∈ R1×k , Uk ∈ Rn×k calculated via POD, for k n, i.e.y(t, x) ≈ Φ(x)y(t), y(t) ∈ Rk we apply the following innerproduct

< x , y >Mh= xTMhy .

One obtains the corresponding discretized reduced ordermodel:

UTk MhUk︸︷︷︸I∈Rk×k

d

dty(t) = UT

k KhUk︸︷︷︸k×k

y(t) + UTk Nh(y(t))︸︷︷︸

N(y(t))

. (3)


The projected nonlinear term still depends on the dimensionof the original system

N(y(t)) = UTk︸︷︷︸

k×n

Nh(y(t))︸︷︷︸n×1

.

[Nh(y(t))]i =

∫Ωψi (x)F (Φ(x)y(t))dΩ, i = 1, ..n.

The Empirical Interpolation Method (EIM) approximation ofthe nonlinear function F (Φ(x)y(t)) is given by

F (Φ(x)y(t)) ' Q(x)ρ(t) = Q(x)(Q(z))−1F (Φ(z)y(t)),

Q(x) = [q1(x), ..., qm(x)], z = [z1, ..., zm], m n

Q(z) ∈ Rm×m, Φ(z) ∈ Rm×k ,

F (Φ(z)y(t)) ∈ Rm×1 − F is applied componentwise


Thus

Nh(y(t)) '∫

ΩΨ(x)TQ(x)dΩ︸︷︷︸

n×m

(Q(z))−1︸︷︷︸m×m

F (Φ(z)y(t))︸︷︷︸m×1

Now we are able to separate the unknown y(t) from theintegrals allowing us the precomputation of the integralswhich then can be used in all of the time steps.

N(y(t)) ' UTk︸︷︷︸

k×n

∫Ω

Ψ(x)TQ(x)dΩ(Q(z))−1︸︷︷︸n×m

F (Φ(z)y(t))︸︷︷︸m×1

.

2. Description of EIM algorithm

Empirical Interpolation Method (EIM)

Empirical Interpolation method was proposed by Barrault et al.(2004).

M Barrault, Y Maday, NC Nguyen, AT Patera. An ’empiricalinterpolation’ method: application to efficient reduced basisdiscretization of partial differential equations. ComptesRendus Acad. Sci. Paris Series I, 339 (2004).

Grepl, Maday, Nguyen, Patera. Efficient reduced-basistreatment OF nonaffine and nonlinear partial differentialequations. Modelisation Mathematique et AnalyseNumerique, Vol.41 No.3 (2007).

Maday, Nguyen, Patera, Pau. A GENERAL MULTIPURPOSEINTERPOLATION PROCEDURE: THE MAGIC POINTS.Communications on Pure and Applied Analysis Vol.8 No.1(2009).





A rigorous a posteriori error estimation may be found in

JL Eftang, MA Grepl, and AT Patera. A Posteriori ErrorBounds for the Empirical Interpolation Method. CR Acad.Sci. Paris Series I, 348 (2010).





Let f (x ;µ) be a nonlinear parameterized real-valued function,x ∈ Ω ⊂ Rd and µ ∈ D ⊂ Rp.

The approximation

f (x ;µ) =m∑j=1

qj(x)cj(µ),

where qjmj=1 form a basis whose span gives a goodapproximation to

spanf (·;µ) : µ ∈ D

and cj(µ)mj=1 are obtained from the coefficient functionapproximation using a set of pre-specified points, calledinterpolation points zimi=1 ∈ Ω which are expected tocapture the parameter variation.





Two ingredients are needed for constructing f .

A set of basis functions qjmj=1. The original paper for EIMuses this basis set constructed from a greedy selection processon the set of snapshots. Here the basis set is constructedfrom POD and EIM algorithm for interpolation points.

A set of interpolation points zimi=1 (EIM algorithm forinterpolation points) used in coefficient functionapproximation for cj(µ). For a fixed value of µ,c1(µ), .., cm(µ) satisfies

f (zi ;µ) = f (zi ;µ) =m∑j=1

qj(zi )cj(µ), i = 1, 2, ..m.





Introducing the matrix vector notations

Q(x) = [q1(x), .., qm(x)], z = [z1, .., zm]T , c = [c1, .., cm]T

the approximation f of the function f can be written as:

f (x ;µ) = Q(x)c(µ)

and in particular

f (z ;µ) = Q(z)c(µ).





EIM algorithm for interpolation points chooses to use basisQ(x) and interpolation points z such that Q(z) is invertible.

Then the coefficient can be written as a function of parameterµ explicitly as follows:

c(µ) = (Q(z))−1f (z;µ).

The final function approximation has the following form

f (x ;µ) = Q(x)(Q(z))−1f (z;µ).



EIM: Algorithm for interpolation points

INPUT: ulml=1 ⊂ Rn (linearly independent generated by PODfrom the snapshots of f ):

OUTPUT: z and Q(x)

1 z1 = arg ess supx∈Ω|u1(x)|2 q1(x) = u1(x)

u1(z1)

3 z = [z1], Q(x) = [q1(x)].

4 For l = 2, ..,m do

a Solve Q(z)c = ul(z) for c

b rl(x) = ul(x)−Q(x)c

c zl = arg ess supx∈Ω|rl(x)|d ql(x) = rl (x)

rl (zl )

e z←[

zzl

], Q(x)← [Q(x) ql(x)].

5 end for.

EIM: Algorithm for interpolation points -discrete variant

INPUT: ulml=1 ⊂ Rn (linearly independent generated by PODfrom the snapshots of f ):

OUTPUT: z and Q(x)

1 [|ψ| ρ1] = max |u1|, ψ ∈ R and ρ1 is the component positionof the largest absolute value of u1, with the smallest indextaken in case of a tie. z1 = x(ρ1)

2 q1(x) = u1(x)u1(z1)

3 z = [z1], Q(x) = [q1(x)].4 For l = 2, ..,m do

a Solve Q(z)c = ul(z) for c

b rl(x) = ul(x)−Q(x)c

c [|ψ| ρl ] = max |rl |, zl = x(ρl)

d ql(x) = rl (x)rl (zl )

e z←[

zzl

], Q(x)← [Q(x) ql(x)].

5 end for.



The following example illustrates the efficiency of EIM inapproximating a highly nonlinear function defined on adiscrete 1D spatial domain. Consider a nonlinearparameterized function s : Ω×D→ R defined by

s(x ;µ) = (1− x)cos

(πµ(x + 1)

)e−(1+x)µ,

where x ∈ Ω = [−1, 1] and µ ∈ D = [0, π2 ] ⊂ R.





Let [x1, x2, ..., xn] ∈ Rn, xi ∈ R being equally distributed in Ω,for i = 1, 2, .., n, n = 101. We introduce f : D→ Rn asfollows

f (µ) = [s(x1;µ), s(x2;µ), .., s(xn;µ)T ] ∈ Rn, µ ∈ D

We used 50 snapshots f (µj)50j=1 to construct POD basis

ulml=1 with µj equidistantly points in [0, π2 ].





0 5 10 15 20 25 30 35 40 45 50−40

−35

−30

−25

−20

−15

−10

−5

0

5Singular values of 50 snapshots

log

arit

hm

ic s

cale

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2EIM Points and the first 6 basis functions

q1(x)q2(x)q3(x)q4(x)q5(x)q6(x)Eim ptsExact

Figure 1: Singular eigenvalues using logarithmic scale and thecorresponding first 6 basis functions with interpolation points generatedby EIM algorithm, µ = 1.13



−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2EIM#1

q1(x)current point

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1EIM#2

q2(x)r=u2(x)−Q(x)ccurrent pointprevious points

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1EIM#3


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1EIM#4


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1EIM#5


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1EIM#6


Figure 2: The selection process of EIM interpolation points

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2EIM#1

exactEIM approx.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2EIM#2

exactEIM approx.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2EIM#3

exactEIM approx.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2EIM#4

exactEIM approx.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2EIM#5

exactEIM approx.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2EIM#6

exactEIM approx.

Figure 3: EIM approximation for different values of m



−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2 The Exact function and its EIM approximation for µ =1.13

Exact functionEIM approximation

0 5 10 15 20 25 30 35 40 45 50−35

−30

−25

−20

−15

−10

−5

0

5

m (Reduced dimension)

log

arit

hm

ic s

cale

Error in Euclidian Norm

EIM errorPOD error

Figure 4: The EIM approximate function for m = 20 compared with theexact function of dimension n = 101 at µ = 1.13 (left); Comparison ofthe spatial errors for POD and EIM approximations (right)



3. Limited - Area SWE

Limited - Area SWE

The shallow - water equations model on a β−plane

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+∂φ

∂x− fv = 0

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+∂φ

∂y+ fu = 0

∂φ

∂t+

∂

∂x(uφ) +

∂

∂x(vφ) = 0

(x , y) ∈ Ω = [0, L]× [0,D], t > 0

where L and D are the dimensions of a rectangular domain ofintegration, u and v are the velocity components in the x andy axis respectively, φ = gh is the geopotential height, h is thedepth of the fluid and g is the acceleration of gravity.




Limited - Area SWE

The scalar function f is the Coriolis parameter.

f = f + β

(y − D

2

), β =

∂f

∂y

The f is the Coriolis frequency

f = f + β

(y − D

2

)The Coriolis parameter

f = 2Ωv sin θ

is defined at a mean latitude θ0, where Ωv is the angularvelocity of the earth’s rotation and θ is latitude.




Limited - Area SWE

We impose initial conditions

w(x , y , 0) = ϕ(x , y), where state variables are

w = w(x , y , t) = (u(x , y , t), v(x , y , t), φ(x , y , t)) ,

with periodic boundary conditions are assumed in thex-direction:

w(0, y , t) = w(L, y , t)

whereas solid wall boundary condition are used in y -direction:

v(x , 0, t) = v(x ,D, t) = 0.




Finite Element Limited - Area SWE

Finite element method (FEM) is used to obtain the discretemodel of the limited SWE

We employed linear piecewise polynomials on triangularelements in the formulation of Galerkin finite-elementshallow-water equations model, in which the global matrix wasstored into a compact matrix.




Finite Element Limited - Area SWE

A time-extrapolated Crank-Nicholson time differencingscheme was applied for integrating in time the system ofordinary differential equations.

The Galerkin finite-element boundary conditions were treatedusing the approach suggested by Payne and Irons (1963) andmentioned by Huebner (1975), i.e. modifying the diagonalterms of the global matrix associated with the nodal variablesby multiplying them by a large number, say 1016, while thecorresponding term in the right-hand vector is replaced by thespecified boundary nodal variable multiplied by the same largefactor times the corresponding diagonal term.




Variational form of the Limited - Area SWE

The variational form is obtained by multiplying each of themodel equations byw ∈ H1(Ω), w(0, y) = w(L, y), w(x , 0) = w(x ,D) = 0, andrequiring < w ,Ei >L2(Ω)= 0, i = 1, 2, 3 i.e.∫

Ω

∂u

∂twdΩ+

∫Ωu∂u

∂xwdΩ+

∫Ωv∂u

∂ywdΩ+

∫Ω

∂φ

∂xwdΩ−

∫ΩfvwdΩ = 0

∫Ω

∂v

∂twdΩ+

∫Ωu∂v

∂xwdΩ+

∫Ωv∂v

∂ywdΩ+

∫Ω

∂φ

∂ywdΩ+

∫ΩfuwdΩ = 0

∫Ω

∂φ

∂twdΩ−

∫Ωuφ∂w

∂xdΩ +

∫Ωvφ∂w

∂ydΩ = 0




Galerkin FE Limited - Area SWE

Let Ψ = [ψ1, .., ψN ] be a set of continuous and piecewiselinear functions which are formed by a sum of linear shapefunctions Ne

i , i = 1, 2, 3 i.e.

Nei (x , y) =

ai + bix + ciy

2∆

defined for each triangular element e of a N = nx × ny mesh.x , y are the coordinates of the mesh points.

ai = xjym − xmyj , bi = yj − ym, ci = xm − xj , i , j ,m = 1, 2, 3.

Each ψi , i = 1,N are constrained to satisfy the periodicboundary conditions on the x directions and the solidboundary conditions in the y directions.





By assuming that the SWE solutions have the following form

u 'N∑j=1

uj(t)ψj(x , y); v 'N∑j=1

vj(t)ψj(x , y); φ 'N∑j=1

φj(t)ψj(x , y)

and taking w = ψi , i = 1, 2, ..,N we obtain the Galerkin FEdiscrete SWE model.

Assume also

f ' fI ≡N∑j=1

fjψj .





N∑j=1

u′j

∫ΩψiψjdΩ+

N∑j=1

uj

N∑p=1

up

∫Ωψiψj

∂ψp

∂xdΩ+

N∑j=1

vj

N∑p=1

up

∫Ωψiψj

∂ψp

∂ydΩ

+N∑j=1

φj

∫Ωψi∂ψj

∂xdΩ−

N∑j=1

fj

N∑p=1

vp

∫ΩψiψjψpdΩ = 0

N∑j=1

v′j

∫ΩψiψjdΩ+

N∑j=1

uj

N∑p=1

vp

∫Ωψiψj

∂ψp

∂xdΩ+

N∑j=1

vj

N∑p=1

vp

∫Ωψiψj

∂ψp

∂ydΩ

+N∑j=1

φj

∫Ωψi∂ψj

∂ydΩ +

N∑j=1

fj

N∑p=1

up

∫ΩψiψjψpdΩ = 0





N∑j=1

φ′j

∫ΩψiψjdΩ−

N∑j=1

uj

N∑p=1

φp

∫Ω

∂ψi

∂xψjψpdΩ

−N∑j=1

vj

N∑p=1

φp

∫Ω

∂ψi

∂yψjψpdΩ = 0.





We introduce the following notation

Mij =

∫Ωψiψj , i , j = 1, ..,N,

A1ijp =

∫Ωψiψj

∂ψp

∂xdΩ, i , j , p = 1, ..,N,

A2ijp =

∫Ωψiψj

∂ψp

∂ydΩ, i , j , p = 1, ..,N,

Mdxij =

∫Ωψi∂ψj

∂x, i , j = 1, ..,N,





A3ijp =

∫ΩψiψjψpdΩ, i , j , p = 1, ..,N,

Mdyij =

∫Ωψi∂ψj

∂y, i , j = 1, ..,N,

A4ijp =

∫Ω

∂ψi

∂xψjψpdΩ, i , j , p = 1, ..,N,

A5ijp =

∫Ω

∂ψi

∂yψjψpdΩ, i , j , p = 1, ..,N,





Using the Silvestre formula∫e(Ne

1 )m(Ne2 )n(Ne

3 )lde =m!n!l!

(m + n + l + 2)!2∆,

the global matrices defined previous are calculated.




The Matrix Formulation

M︸︷︷︸N×N

u′︸︷︷︸N×1

+ uT︸︷︷︸1×N

A1︸︷︷︸:,N×N

u︸︷︷︸N×1

+ vT︸︷︷︸1×N

A2︸︷︷︸:,N×N

u︸︷︷︸N×1

+Mdx︸︷︷︸N×N

φ︸︷︷︸N×1

− fT︸︷︷︸N×1

A3︸︷︷︸:,N×N

v︸︷︷︸N×1

= 0,

M︸︷︷︸N×N

v′︸︷︷︸N×1

+ uT︸︷︷︸1×N

A1︸︷︷︸:,N×N

v︸︷︷︸N×1

+ vT︸︷︷︸1×N

A2︸︷︷︸:,N×N

v︸︷︷︸N×1

+Mdy︸︷︷︸N×N

φ︸︷︷︸N×1

+ fT︸︷︷︸N×1

A3︸︷︷︸:,N×N

u︸︷︷︸N×1

= 0,

M︸︷︷︸N×N

φ′︸︷︷︸N×1

+ uT︸︷︷︸1×N

A4︸︷︷︸:,N×N

φ︸︷︷︸N×1

+ vT︸︷︷︸1×N

A5︸︷︷︸:,N×N

φ︸︷︷︸N×1

= 0.




The Matrix Formulation

With the following notation

K11(u) = uT︸︷︷︸1×N

A1︸︷︷︸:,N×N

∈ RN×N,

K12(v) = vTA2, K13 = fTA3,

K31(u) = uTA4, K32(v) = vTAt ,

we finally obtain

Mu′ + K11(u)u + K12(v)u + Mdxφ− K13v = 0,

Mv′ + K11(u)v + K12(v)v + Mdyφ + K13u = 0,

Mu′ − K31(u)φ− K32(v)φ = 0.




Crank - Nicolson scheme

The time-extrapolated Crank Nicolson method was used forintegrating in time the system of ordinary differentialequations resulting from the application of the Galerkin FEMto the SWE model.

This method was previously used by Douglas and Dupont,Hinsman, Navon and Muller etc.

An average is taken at time levels n and n + 1 of expressionsinvolving space de rivatives, while the non-linear advectiveterms are quasi- linearized by estimating them at time leveln + 1

2 using the following second-order approximation in time:

un+ 12 = u∗ =

3

2un − 1

2un−1 + O(∆t2),

vn+ 12 = v∗ =

3

2vn − 1

2vn−1 + O(∆t2).





The algebraic systems of equations from bellow were solved byemploying the Gauss-Seidel iterative method. Let∆φ = φn+1 − φn, ∆u = un+1 − un, ∆v = vn+1 − vn[

2

∆tM − K31(u∗)− K32(v∗)

]∆φ = 2

[K31(u∗) + K32(v∗)

]φn,[

2

∆tM + K11(u∗) + K12(v∗)

]∆u = −2

[K11(u∗) + K12(v∗)

]un

−Mdx

(φn+1 + φn

)+ K13v∗,[

2

∆tM + K11(un+1) + K12(v∗)

]∆v = −2

[K11(un+1) + K12(v∗)

]vn

−Mdy

(φn+1 + φn

)− K13un+1.



4. Generation of POD using FEM

Generation of POD using FEM

Proper orthogonal decomposition provides a technique forderiving low order model of dynamical systems. It can bethought of as a Galerkin approximation in the spatial variablebuilt from functions corresponding to the solution of thephysical system at specified time instances. These are calledsnapshots.

LetU = [u1, u2, .., uns ] ∈ RN×ns ,

V = [v1, v2, .., vns ] ∈ RN×ns ,

Φ = [φ1, φ2, .., φns ] ∈ RN×ns ,

be the snapshot sets, i.e. the numerical solution obtained withFE SWE at different time levels t1, t2, .., tns .





We introduce

Ψ(x , y) = [ψ1(x , y), ..., ψN(x , y)]

a 1- by -N matrix of FE basis functions.

The l th snapshots at time tl is of the form

u(x , y , tl) = Ψ(x , y)u(tl),

v(x , y , tl) = Ψ(x , y)v(tl),

φ(x , y , tl) = Ψ(x , y)φ(tl).





The POD bases of dimensions ki , i = u, v , φ, of the snapshots U,V , Φ can be constructed as follows

Compute the following matrices

Lu = UTMU, Lv = V TMV , Lφ = ΦTMΦ,

Lu, Lv , Lφ ∈ Rns×ns .

Find ku eigenvectors of Lu corresponding to the first kulargest eigenvalues

Luwi = wiλi , wi ∈ Rns×1, i = 1, 2, .., ku.

Then each POD basis function uPODi is given by

uPODi (x , y) =

1

λi

ns∑l=1

(wi )lu(x , y , tl), i = 1, 2, .., ku.





To write each POD basis function uPODi in terms of the FE

basis Ψ(x , y) = [ψ1(x , y), ..., ψN(x , y)] in a matrix vectorform, let LuW = WΛ be the eigenvalue decomposition of Lu,where Λ = diag(λ1, ..λns) with λ1 ≥ ...λns ≥ 0 andW = [w1, ..,wns ] ∈ Rns×ns containing all the ns eigenvectorsof Lu.Denote by Wk the first ku columns of W and

Λku = diag(λ1, .., λku) ∈ Rku×ku

.The 1 -by-ku matrix of POD basis functions,UPOD(x , y) = [uPOD

1 (x , y), .., uPODku

(x , y)] can be written as

UPOD(x , y) = Ψ(x , y) U︸︷︷︸N×ns

Wku︸︷︷︸ns×ku

Λ−1/2ku︸︷︷︸

ku×ku

= Ψ(x , y)Uk .





In a similar way we obtain the POD basis corresponding to vand φ

V POD(x , y) = Ψ(x , y)Vk , ΦPOD(x , y) = Ψ(x , y)Φk ,

Vk ∈ RN×kv , Φk ∈ RN×kφ .



5. POD FEM SWE Model

POD FEM SWE Model

By using the POD bases UPOD , UPOD , ΦPOD theapproximate solution can be written as

u(x , y , t) = UPOD(x , y)u(t), v(x , y , t) = V POD(x , y)v(t),

φ(x , y , t) = ΦPOD(x , y)φ(t), u ∈ Rku , v ∈ Rkv , φ ∈ Rkφ .

From the weak form of the SWE model, instead of using thelarger set of finite element basis functions, we now use the

uPODi (x , y)kui=1, vPOD

i (x , y)kvi=1, φPODi (x , y)kφi=1 as the

test functions.




POD FEM SWE Model

Using the relationship between the reduced bases and the FEbasis we obtain the discretized system corresponding to FESWE model

UTk︸︷︷︸

ku×N

M︸︷︷︸N×N

Uk︸︷︷︸N×ku

d

dtu + UT

k︸︷︷︸ku×N

N11(u)︸︷︷︸N×1

+ UTk︸︷︷︸

ku×N

N12(u, v)︸︷︷︸N×1

+ UTk︸︷︷︸

ku×N

Mdx︸︷︷︸N×N

Φk︸︷︷︸N×kφ

φ− UTk︸︷︷︸

ku×N

K13︸︷︷︸N×N

Vk︸︷︷︸N×kv

v = 0.

V Tk︸︷︷︸

kv×N

M︸︷︷︸N×N

Vk︸︷︷︸N×kv

d

dtv + V T

k︸︷︷︸kv×N

N21(u, v)︸︷︷︸N×1

+ V Tk︸︷︷︸

kv×N

N22(v)︸︷︷︸N×1

+ V Tk︸︷︷︸

kv×N

Mdy︸︷︷︸N×N


φ + V Tk︸︷︷︸

kv×N

K13︸︷︷︸N×N

Uk︸︷︷︸N×ku

u = 0.




POD FEM SWE Model

ΦTk︸︷︷︸

kφ×N

M︸︷︷︸N×N


d

dtφ + ΦT

k︸︷︷︸kφ×N

N31(u, φ)︸︷︷︸N×1

+ ΦTk︸︷︷︸

kφ×N

N32(v, φ)︸︷︷︸N×1

= 0.

where

[N11(u)]i =

∫Ωψi (x , y)Nf11(UPOD(x , y)u)dΩ = uT︸︷︷︸

1×ku

UTk︸︷︷︸

ku×N

A1︸︷︷︸i ,N×N

Uk︸︷︷︸N×ku

u︸︷︷︸ku×1

.

[N12(u, v)]i =

∫Ωψi (x , y)Nf12(UPOD(x , y)u,V POD(x , y)v)dΩ =

vT︸︷︷︸1×kv

V Tk︸︷︷︸

kv×N

A2︸︷︷︸i ,N×N

Uk︸︷︷︸N×ku

u︸︷︷︸ku×1

.




POD FEM SWE Model

[N21(u, v)]i =

∫Ωψi (x , y)Nf21(UPOD(x , y)u,V POD(x , y)v)dΩ =

uT︸︷︷︸1×ku

UTk︸︷︷︸

ku×N

A1︸︷︷︸i ,N×N

Vk︸︷︷︸N×kv

v︸︷︷︸kv×1

.

[N22(v)]i =

∫Ωψi (x , y)Nf22(V POD(x , y)v)dΩ = vT︸︷︷︸

1×kv

V Tk︸︷︷︸

kv×N

A2︸︷︷︸i ,N×N

Vk︸︷︷︸N×kv

v︸︷︷︸kv×1

.

[N31(u, φ)]i =

∫Ω

∂

∂xψi (x , y)Nf31(UPOD(x , y)u,ΦPOD(x , y)φ)dΩ =

uT︸︷︷︸1×ku

UTk︸︷︷︸

ku×N

A4︸︷︷︸i ,N×N


φ︸︷︷︸kφ×1

.




POD FEM SWE Model

[N32(v, φ)]i =

∫Ω

∂

∂yψi (x , y)Nf32(V POD(x , y)v,ΦPOD(x , y)φ)dΩ =

vT︸︷︷︸1×kv

V Tk︸︷︷︸

kv×N

A5︸︷︷︸i ,N×N


φ︸︷︷︸kφ×1

.

Nf11(u) = u∂u

∂x, Nf12(u, v) = v

∂u

∂y,

Nf21(u, v) = u∂v

∂x, Nf22(v) = v

∂v

∂y,

Nf31(u, φ) = uφ, Nf22(v , φ) = vφ.




POD FEM SWE Model

Next we introduce

Mk = UTk︸︷︷︸

ku×N

Muk︸︷︷︸N×ku

,

K k11(u) = UT

k︸︷︷︸ku×N

uTUTk A1Uk︸︷︷︸

N×ku

, K k12(v) = UT

k︸︷︷︸ku×N

vTV Tk A2Uk︸︷︷︸N×ku

,

Mdxk = UTk︸︷︷︸

ku×N

MdxΦk︸︷︷︸N×kφ

, K k13 = UT

k︸︷︷︸ku×N

K13Vk︸︷︷︸N×kv

.




POD FEM SWE Model

K k21(u) = V T

k︸︷︷︸kv×N

uTUTk A1Vk︸︷︷︸

N×kv

, K k22(v) = V T

k︸︷︷︸kv×N

vTV Tk A2Vk︸︷︷︸

N×kv

,

Mdyk = V Tk︸︷︷︸

kv×N

MdyΦk︸︷︷︸N×kφ

, K k23 = V T

k︸︷︷︸kv×N

K13Uk︸︷︷︸N×ku

.

K k31(u) = ΦT

k︸︷︷︸kφ×N

uTUTk A4Φk︸︷︷︸

N×kφ

, K k32(v) = ΦT

k︸︷︷︸kφ×N

vTV Tk A5Φk︸︷︷︸

N×kφ

.




Matrix form of the POD FEM SWE

Mkd

dtu + K k

11(u)u + K k12(v)u + Mdxkφ− K k

13v = 0,

Mkd

dtv + K k

21(u)v + K k22(v)v + Mdykφ− K k

13u = 0,

Mkd

dtφ− K k

31(u)φ− K k32(v)φ = 0.





Let us define

un+ 12 = u∗ =

3

2un − 1

2un−1 + O(∆t2),

vn+ 12 = v∗ =

3

2vn − 1

2vn−1 + O(∆t2),

∆φ = φn+1 − φn, ∆u = un+1 − un, ∆v = vn+1 − vn

than the Crank - Nicolson scheme corresponding to PODFEM SWE model is





[2

∆tMk − K k

31(u∗)− K k32(v∗)

]∆φ = 2

[K k

31(u∗) + K k32(v∗)

]φn,[

2

∆tMk + K k

11(u∗) + K k12(v∗)

]∆u = −2

[K k

11(u∗) + K k12(v∗)

]un

−Mdxk

(φn+1

+ φn)

+ K k13v∗,[

2

∆tMk + K21(un+1) + K k

22(v∗)

]∆v = −2

[K k

21(un+1) + K k12(v∗)

]vn

−Mdyk

(φn+1

+ φn)− K k

13un+1.




The POD/EIM version of SWE model

In POD FEM SWE model we still have computationalcomplexities depending on the dimension N of the originalsystem from both evaluating the nonlinear functions andperforming matrix multiplications to project on POD bases.EIM is used to remove this dependency.The projected nonlinear functions can be approximated byEIM in a form that enables precomputation so that thecomputational cost is decreased and independent of theoriginal system.Only a few entries of the nonlinear term corresponding to thespecially selected interpolation points from EIM must beevaluated at each time step.EIM approximation is applied to each of the nonlinearfunctions Nf11,Nf12,Nf21,Nf22,Nf31,Nf32 defined in PODreduced model.





Let UNf11 ∈ RN×m, m ≤ N, be the POD basis matrix of rankm for snapshots of the nonlinear function Nf11 (obtained fromFE SWE model).

Using the EIM algorithm we select a set of m EIM pointscorresponding to UNf11 , denoting byzNf11 = [zNf11

1 , .., zNf11m ]T ∈ Rm and generate the EIM basis

QNf1111 (x , y). The EIM approximation of Nf11 is

Nf11(x , y) ≈ QNf1111 (x , y) QNf11

11 (zNf11) Nf11(UPOD(zNf11)u),

whereQNf11

11 (x , y) = [qNf111 (x), ..., qm(x)Nf11 ],

Q(zNf11) ∈ Rm×m, UPOD(zNf11) ∈ Rm×ku ,

Nf11(UPOD(zNf11)u) ∈ Rm×1





Now the projected nonlinear term UTk N11(u) in the POD

reduced system can be approximated as

UTk N11(u) =

K k−EIM11︷︸︸︷

UTk︸︷︷︸

ku×N

∫Ω

Ψ(x , y)TQNf1111 (x , y)dΩ︸︷︷︸

N×m

·QNf1111 (zNf11)︸︷︷︸

m×m

·Nf11(UPOD(zNf11)u)︸︷︷︸m×1

.

Similarly we obtain the EIM approximation for the rest of theprojected nonlinear terms




UTk N12(u, v) =

K k−EIM12︷︸︸︷

UTk︸︷︷︸

ku×N

∫Ω

Ψ(x , y)TQNf1212 (x , y)dΩ︸︷︷︸

N×m

·QNf1212 (zNf12)︸︷︷︸

m×m

·

Nf12(UPOD(zNf12)u,V POD(zNf12)v)︸︷︷︸m×1

,

V Tk N21(u, v) =

K k−EIM21︷︸︸︷

V Tk︸︷︷︸

kv×N

∫Ω

Ψ(x , y)TQNf2121 (x , y)dΩ︸︷︷︸

N×m

·QNf2121 (zNf21)︸︷︷︸

m×m

·

Nf21(UPOD(zNf21)u,V POD(zNf21)v)︸︷︷︸m×1

,


V Tk N22(v) =

K k−EIM22︷︸︸︷

V Tk︸︷︷︸

kv×N

∫Ω

Ψ(x , y)TQNf2222 (x , y)dΩ︸︷︷︸

N×m

·QNf2222 (zNf22)︸︷︷︸

m×m

·

Nf22(V POD(zNf22)v)︸︷︷︸m×1

,

ΦTk N31(u, φ) =

︷︸︸︷ΦTk︸︷︷︸

kφ×N

∫Ω

∂

∂xΨ(x , y)

T

QNf3131 (x , y)dΩ︸︷︷︸

N×m

·QNf3131 (zNf31)︸︷︷︸

m×m

·

K k−EIM31

Nf31(UPOD(zNf31)u,ΦPOD(zNf31)φ)︸︷︷︸m×1

,



ΦTk N32(v, φ) =

K k−EIM32︷︸︸︷

ΦTk︸︷︷︸

kφ×N

∫Ω

∂

∂yΨ(x , y)

T

QNf3232 (x , y)dΩ︸︷︷︸

N×m

·QNf3232 (zNf32)︸︷︷︸

m×m

·

Nf32(V POD(zNf32)v,ΦPOD(zNf32)φ)︸︷︷︸m×1

.



Matrix form of the POD FEM SWE

Each of the K k−EIM11 , K k−EIM

12 , K k−EIM21 , K k−EIM

22 , K k−EIM31 ,

K k−EIM32 matrices can be precomputed and re-used at all time

steps, so that the computational complexity of theapproximate nonlinear terms are independent of the full-orderdimension N .

the POD/EIM FE SWE reduced system is

Mkd

dtu + K k−EIM

11 Nf11(UPOD(zNf11)u) + K k−EIM12 ·

Nf12(UPOD(zNf12)u,V POD(zNf12)v) + Mdxkφ− K k13v = 0,

Mkd

dtv+K k−EIM

21 Nf21(UPOD(zNf21)u,V POD(zNf21)v)+K k−EIM22 ·

Nf22(V POD(zNf22)v) + Mdykφ− K k−EIM13 u = 0,

Mkd

dtφ+K k−EIM

31 Nf31(UPOD(zNf31)u,ΦPOD(zNf31)φ)+K k−EIM32 ·

Nf32(V POD(zNf32)v,ΦPOD(zNf32)φ) = 0.

6. Numerical Results

Numerical Results

The domain was discretized using a mesh of 30× 23 points.Thus the dimension of the full-order discretized model is 690.The integration time window was 24h.

We employed linear piecewise polynomials on triangularelements in the formulation of Galerkin Finite-elementshallow-water equations model, in which the global matrix wasstored into a compact matrix.

The initial condition were derived from the geopotentialheight formulation introduced by Grammelvedt (1969) usingthe geostrophic balance relationship.




Numerical Results

18000 18000 18000

18500 18500

18500

1850019000

1900019000

19500

1950019500

20000

2000020000

20500

20500 20500

21000

21000 2100021500

21500

21500 21500

22000 22000 22000

Contour of geopotential from 22000 to 18000 by 500

y(k

m)

x(km)0 1000 2000 3000 4000 5000

0

500

1000

1500

2000

2500

3000

3500

4000

0 1000 2000 3000 4000 5000 6000−500

0

500

1000

1500

2000

2500

3000

3500

4000

4500 Wind field

y(k

m)

x(km)

Figure 5: Initial condition: Geopotential height field for theGrammeltvedt initial condition (left). Wind field calculated from thegeopotential field by using the geostrophic approximation (right).




Numerical Results

18000

18000 18000

18500

18500

18500

18500

19000

19000

19000

19500

19500

19500

20000

20000

20000

2050020500

20500

21000 21000

2100021500

21500

21500

2200022000

22000


y(k

m)

x(km)0 1000 2000 3000 4000 5000

0

500

1000

1500

2000

2500

3000

3500

4000

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500 Wind field

y(k

m)

x(km)

Figure 6: The geopotential field (left) and the wind field at t = tf = 24hobtained using the FE SWE scheme for ∆t = 75s.




Numerical Results

The POD basis vectors were constructed using 1152snapshots obtained from the numerical solution of the full -order FE SWE model at equally spaced time steps in theinterval [0 24h].

The dimension of the POD bases for each variable was taken50, capturing more than 99.9% of the system energy.




Numerical Results

0 50 100 150 200 250 300 350 400−30

−20

−10

0

10

20

30 Singular Values of Snapshots Solution

Number of snapshots

log

arit

hm

ic s

cale

uvΦ

0 50 100 150 200 250 300 350 400−50

−40

−30

−20

−10

0

10

20 Singular Values of Nonlinear Snapshots Solution

Number of snapshots

log

arit

hm

ic s

cale

Nf11Nf12Nf21Nf22Nf31Nf32

Figure 7: The decay around the singular values of the snapshots solutionsfor u, v , φ and nonlinear functions (∆t = 75s).



Numerical Results

We applied the EIM algorithm for interpolation indices toimprove the efficiency of the POD approximation and toachieve a complexity reduction of the nonlinear terms with acomplexity proportional to the number of reduced variables.

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

1

23 4

5

6

7

8

9

10

11

1213

1415

16

17

18

19

20

21

22

23

24

25

26

2728

2930

31

32

33

34

3536

3738

39

40

EIM POINTS for NF31

x(km)

y(km

)

1

23 4

5

6

7

8

9

10

11

1213

1415

16

17

18

19

20

21

22

23

24

25

26

2728

2930

31

32

33

34

3536

3738

39

40

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

1234

5

67

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

2324

25

2627

28

29

30

31

32

33

3435

36

37

38

39

40

EIM POINTS for Nf32

x(km)

y(km

)

1234

5

67

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

2324

25

2627

28

29

30

31

32

33

3435

36

37

38

39

40

Figure 8: First 40 points selected by EIM for the nonlinear functions Nf31

(left) and Nf32 (right)


Numerical Results

The dimension of EIM bases was chosen to be equal with 140

We emphasize the performances of POD - EIM method incomparison with the POD approach using the numericalsolution of the full FE SWE model. Next three slides depictthe space error behaviors between POD/POD - EIM solutionand FE SWE solution.




Numerical Results

POD errors φPOD

−φFEM

x(km)

y(km

)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

−8

−6

−4

−2

0

2

4

6

8

10

x 10−6 POD/EIM errors φ

POD/EIM−φ

FEM

x(km)

y(km

)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

−4

−2

0

2

4

6

x 10−3

Figure 9: Errors between the geopotential calculated with POD/POD-EIM and geopotential determined with the FE SWE model at t = 24h(∆t = 75s). The number of DEIM points was taken 140.




Numerical Results

POD errors uPOD

−uFEM

x(km)

y(km

)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

−1.5

−1

−0.5

0

0.5

1

x 10−6 POD/EIM errors u

POD/EIM−u

FEM

x(km)

y(km

)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−4

Figure 10: Errors between u calculated with POD/POD -EIM and udetermined with the FE SWE model at t = 24h (∆t = 75s). Thenumber of EIM points was taken 140.




Numerical Results

POD errors vPOD

−vFEM

x(km)

y(km

)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

−10

−8

−6

−4

−2

0

2

x 10−6 POD/EIM errors v

POD/EIM−v

FEM

x(km)

y(km

)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

Figure 11: Errors between v calculated with POD/POD -EIM and vdetermined with the FE SWE model at t = 24h (∆t = 75s). Thenumber of DEIM points was taken 140.




Numerical Results

FD SWE POD SWE POD-EIM SWE

CPU time 40.938 10.933 0.8785

φ - 1.8522 · 10−6 1.6532 · 10−3

u - 5.2427 · 10−7 7.0129 · 10−5

v - 1.2073 · 10−6 6.20619 · 10−4

Table 1: CPU time gains and the root mean square errors for each of themodel variables. The POD bases dimensions were taken 50 capturingmore than 99.9% of the system energy. 140 EIM points were chosen.



7. Reduced order POD 4-D VAR

Reduced order POD 4-D VAR

The reduced-order cost functional can be expressed as

JPOD(yPOD

0

)=

1

2

(yPOD

0 − yb)T

B−1(yPOD

0 − yb)

+

1

2

k=n∑k=0

(Hky

PODk − yok

)TR−1k

(Hky

PODk − yok

)B background error covariance matrix

Rk observation error covariance matrix at time level k

Hk observation operator at time level k

yPOD0 control variables vector represented by POD basis

yPODk vector of variables obtained from the reduced-order

model at the time level k





The initial value yPOD0 and the reduced-order model solution

yPODk can be expressed as

yPOD0 =

∑i=Mi=1 αi (0)φi = Φα0

yPODk =

∑i=Mi=1 αi

(tk)φi = Φαk

where Φ =φ1, φ2, . . . , φM

is an ensemble of POD basis.





We can rewrite the reduced - order cost functional as follows

JPODα (α0) =

1

2

(Φα0 − yb

)TB−1

(Φα0 − yb

)+

1

2

k=n∑k=0

(Hk (Φαk)− yok )T R−1k (Hk (Φαk)− yok )

The reduced model can be written as

αk = MPOD0→k (α0) ,∀k

αk = MPODk−1→k (αk−1) = MPOD

k (αk−1) , ∀k

and by recurrence

αk = MPODk · · ·MPOD

1 α0, ∀k





The reduced-order cost functional JPODα (α0) can be divided

into two components

JPODα = JPOD,b

α + JPOD,oα

and more,

JPODα = JPOD,b

α +n∑

k=0

JPOD,oα,k

where JPOD,oα,k = (Hk (Φαk)− yok )T dk and dk denotes the

’normalized departure’

dk = R−1k (Hk (Φαk)− yok ) .





Hence the gradient of the POD reduced-order cost functionalw.r.t. α0 is written as

∇α0JPODα = ΦTB−1

(Φα0 − yb

)+

n∑k=0

(MPOD

1

)T. . .(

MPODk

)TΦTHT

k dk

where(MPOD

k

)Tis the POD reduced-order adjoint model at

time step k .



Pseudo - Algorithmic form

1 Initialize reduced-order adjoint variables α∗ at final time tozero: α∗n = 0

2 For each step k − 1, adjoint variables α∗k−1 are obtained by

adding reduced-order adjoint forcing term ΦTHTk dk to α∗kand

performing reduced-order adjoint integration by multiplying

result by(MPOD

k

)T, i.e. α∗k−1 =

(MPOD

k

)T (α∗k + ΦTHT

k dk)

3 At the end of recurrence, the value of adjoint variableα∗0 = Joα0

yields the gradient of the observational costfunctional

4 Compute

∇α0JPOD,bα = ΦTB−1

(Φα0 − yb

)obtaining

∇α0JPODα = ∇α0J

POD,bα +∇α0J

POD,oα .

8. Trust Region POD

Trust region POD optimal control approach

The Trust - Region algorithm hooks direction of descent andstep-size simultaneously. It approximate a certain region, thetrust region (a sphere in Rn of the objective function with aquadratic model function

mk (p) = fk +∇f Tk +

1

2pTBkp, where

fk = f (xk) , ∇fk = ∇f (xk) and Bk is an approximation to the Hessian.

We seek a solution of

min mk (p) =fk +∇f Tk +

1

2pTBkp

s.t ‖p‖ ≤ δk ,

where δk > 0 is the trust-region radius.



8. Trust Region POD

Trust region POD optimal control approach

The trust-region radius δk at each iteration is determined byanalyzing the following ratio

ρk =f (xk)− f (xk + pk)

mk (0)−mk (pk).

If ρk < 0, the new objective value is greater than the currentvalue so that the step must be rejected.

If ρk is close to 1, there is good agreement between theapproximate model mk and the object function fk over thisstep, so it is safe to expand the trust region radius for thenext iteration

If ρk is positive but not close to 1, we do not alter the trustregion radius, but if it is close to zero or negative, we shrinkthe trust region radius.



9. T-R POD algorithm

Trust region POD 4D-VAR algorithm I

Let 0 < η1 < η2 < 1 , 0 < γ1 < γ2 < 1≤ γ3 and y(0)0 , δ0 be given,

set k = 0

1 Compute snapshot set YSNAPk based on initial condition y

(k)0

2 Compute the POD basis Φ(k) and build up the correspondingPOD based control model based on the initial conditionα

(0)0 =

⟨y

(0)0 ,Φ(0)

⟩3 Compute the minimizer sk of

min mk

(α

(k)0 + s

)= JPOD

α

(α

(k)0 + s

)subject to ‖s‖ ≤ δk




Trust region POD 4D-VAR algorithm II

4 Compute the new J(

Φ(k−1)(α

(k)0 + sk

))of the full model

and

ρk =J(

Φ(k−1)α(k)0

)− J

(Φ(k−1)

(α

(k)0 + sk

))mk

(α

(k)0

)−mk

(α

(k)0 + sk

)5 Update the trust-region radius:

If ρk ≥ η2: implement outer projection

y(k+1)0 = y + Φ(k−1)

(α

(k)0 + sk

)and increase trust-region

radius δk+1 = γ3δk and GOTO 1

If η1 < ρk < η2: implement outer iteration

y(k+1)0 = y + Φ(k−1)

(α

(k)0 + sk

)and decrease trust-region

radius δk+1 = γ2δk and GOTO 1




Trust region POD 4D-VAR algorithm III

If ρk ≤ η1: set y(k+1)0 = y

(k)0 and decrease trust-region radius

δk+1 = γ1δk and GOTO 3



10. Illustration

Numerical Results

18000 18000 18000

18500 18500

18500

19000 19000

19000

19500 19500

1950020000

20000

2000020500

20500

205002100021000

21000

21500

21500 21500

22000 22000 22000


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

Wind field calculated from the geopotential field by geostrophic approximation

Figure 12: Initial condition:(a) Geopotential field for the Grammeltvedtinitial condition. (b) Wind field calculated from the geopotential field bythe geostrophic approximation.



Numerical Results

We employed linear piecewise polynomials on triangularelements in the formulation of Galerkin finite-elementshallow-water equations model, in which the global matrix wasstored into a compact matrix.

A time-extrapolated Crank-Nicholson time differencingscheme was applied for integrating in time the system ofordinary differential equations.

The Galerkin finite-element boundary conditions were treatedusing the approach suggested by Payne and Irons (1963) andmentioned by Huebner (1975), i.e. modifying the diagonalterms of the global matrix associated withthe nodal variablesby multiplying them by a large number, say 1016, while thecorresponding term in the right-hand vector is replaced by thespecified boundary nodal variable multiplied by the same largefactor times the corresponding diagonal term.

Numerical Results

We applied a 1% uniform random perturbations on the initialconditions in order to provide twin-experiment “observations”.

The data assimilation was carried on a 48 hours window usingthe ∆t = 1800s in time and a mesh of 30× 24 grid points inspace with ∆x = ∆y = 200km.

We generated 96 snapshots by integrating the fullfinite-element shallow-water equations model forward in time,from which we choose 10 POD bases for each of theu(x , y),v(x , y),and φ(x , y) to capture over 99.9% of theenergy.

The dimension of control variables vector for thereduced-order 4-D Var is 10× 3 = 30.

Numerical Results

The Polak Ribiere nonlinear conjugate gradient (CG)technique was employed for high-fidelity full model 4-D VARand all variants of ad-hoc POD 4-D Var, while thesteepest-descent strategy was used in the trust-region POD4-D Var.

In the ad-hoc POD 4-D Var, the POD bases are re-calculatedwhen the value of the cost function cannot be decreased bymore than 10−1 for ad-hoc POD 4-D Var and 10−2 for ad-hocDWPOD 4-D Var between the consecutive minimizationiterations.

In the trust-region 4-D Var, the POD bases are re-calculatedwhen the ratio ρk is larger than the trust-region parameter η1

in the process of updating the trust-region radius.

0 10 20 30 40 50 60 70 80−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

iterations

log(

cost

/cos

t 0)

Full 4D−VarUW ad−hoc POD 4D−VarDW ad−hoc POD 4D−VarUW TRPOD 4D−VarDW TRPOD 4D−Var

Figure 13: Comparison of the performance of minimization of costfunctional in terms of number of iterations for ad-hoc POD 4-D Var,ad-hoc dual weighed POD 4-D Var, trust-region POD 4-D Var,trust-region dual weighed POD 4-D Var and the full model 4-D Var.

To quantify the performance of the dual weighted trust-region4-D Var, we use two metrics namely the root mean squareerror (RMSE) and correlation of the difference between thePOD reduced-order simulation and high-fidelity model.

POD 4-D Var ADPOD DWAHPOD TRPOD DWTRPOD Full

Iterations 22 42 46 57 80

Outer projections 2 6 10 12 N/A

Error 10−1 10−2 10−5 10−8 10−10

CPU time (s) 15.2 38.7 121.2 142.8 222.6

Table 2: Comparison of iterations, outer projections, RMSE andCPU time for ad-hoc POD 4-D Var, ad-hoc dual weighed POD 4-DVar, trust-region POD 4-D Var, trust-region dual weighed POD 4-DVar and the full model 4-D Var.

Next image depicts the errors between the retrieved initialgeopotential and true initial geopotential applying dualweighted trust-region POD 4-D Var to the 5% uniformrandom perturbations of the true initial conditions taken asinitial guess.

17954

17954

17954

17954

18475

18475

18997

18997

19518

19518

20039

20039

20560

20560

21081

21081

21602

21602

22124 2212422124

22124

x−axis

y−ax

is

The contour of 5% perturbation of true initial geopotential

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−332

−332

−332

−115

−115

−115

−115

−115

−115

−115

−115

−115

−115

−115

−115

−115

−115−115

−115

−115

−115

−115

−115 −115

−115

102

102

102102

102

102

102

102

102

102

102

102

102 102102

102

102

102

102102

102

102

102

102

102

102102

102102319

319

319

319

319

319

319

319

319

319

319

319

319

319

319

x−axis

y−ax

is

The contour of difference between 5% perturbation of true initial geopotential

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1821718217

18666

18666

19115

19115

19565

19565

20014

20014

20463

20463

20913

209132136

2

21362

2181121811

x−axis

y−ax

is

The contour of retrieved initial geopotential(Window = 2(days), dt = 1800s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−66

−66

−66

−66

−66

−66

−66

−66

−66

−66

−66

−22

−22

−22

−22

−22

−22

−22

−22

−22−22

−22

−22

−22

−22

−22

−22−22

−22

−22

−22

−22

−22

−22−

22

−22

−22

−22

−22

−22

−22

22

22

22

2222

22

22

2222

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

22

6666

66

66

66

66

66

66

66

66

66

6666

66

66

66

66

66

66

x−axis

y−ax

is

The contour of difference between retrieved initial geopotential and true initial geopotential

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Numerical Results

The correlation coefficient r used to evaluate quality of theinversion simulation is defined below

ri =cov i12

σi1σi2

,

where

σi1 =

j=N∑j=1

(Ui ,j − U j

)2, σ2 =

j=N∑j=1

(UPODi ,j − UPOD

j

)2, i , j = 1, . . . , n

cov12 =

j=N∑j=1

(Ui ,j − U j

) (UPODi ,j − UPOD

j

), i , j = 1, . . . , n

with U j and UPODj are the means over the simulation period

[0,T ] obtained from the full model and ones obtained byoptimal POD reduced-order model at node j , respectively.

10. Illustration

Numerical Results

0 20 40 60 80 1000.993

0.994

0.995

0.996

0.997

0.998

0.999

1

1.001

time steps

Cor

rela

tion

CORRELATION of geopotential before the DWTRPODCORRELATION of geopotential after the DWTRPOD

0 20 40 60 80 1000.982

0.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

1

1.002

time steps

Cor

rela

tion

CORRELATION of velocity before the DWTRPODCORRELATION of velocity after the DWTRPOD

Figure 14: Comparison of the correlation between the full model and theROM before and after data assimilation applying dual weightedtrust-region POD 4-D Var to the 5% uniform random perturbations ofthe true initial conditions serving as initial guess: geopotential (left),wind field (right).



11. Conclusions

Conclusion and future research

To obtain the approximate solution from both POD andPOD-EIM reduced systems, one must store POD or POD-EIMsolutions of order O(kNT ) and POD matrices of order O(Nk),k being the POD bases dimension, NT the number of timesteps in the integration window and N the space dimension.

The coefficient matrices that must be retained while solvingthe POD reduced system are of order of O(k2) for projectedlinear terms and O(Nk) for the nonlinear term.



11. Conclusions

Conclusion and future research

In the case of solving the POD-EIM reduced system thecoefficient matrices that need to be stored are of order ofO(k2) for projected linear terms and O(mk) for the nonlinearterms, where m is the number of EIM points, m N.

EIM therefore improves the efficiency of the PODapproximation and achieves a complexity reduction of thenonlinear term with a complexity proportional to the numberof reduced variables.



11. Conclusions

Conclusions

We compared several variants of POD 4-D Var, namelyunweighted ad-hoc POD 4-D Var, dual-weighed ad-hoc POD4-D Var, unweighted trust-region POD 4-D Var anddual-weighed trust-region POD 4-D Var, respectively.

We found that the ad-hoc POD 4-D Var version yielded theleast reduction of the cost functional compared with thetrust-region 4-D VAR . We assume that this result may beattributed to lack of feedbacks from the high-fidelity model.

The trust-region POD 4-D Var version yielded a sizably betterreduction of the cost functional, due to inherent properties ofTRPOD allowing local minimizer of the full problem to beattained by minimizing the TRPOD sub-problem. Thustrust-region 4-D Var resulted in global convergence to thehigh fidelity local minimum starting from any initial iterates.



11. Conclusions

Burger Equations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Finite Element Solution of 1D Burger Equation

y(x,1)y(x,2)y(x,3)y(x,5)y(x,51)

0 2 4 6 8 10 12 14 16 18−40

−35

−30

−25

−20

−15

−10

−5

0

5 Singular Values of Snapshots Solution

loga

rithm

ic s

cale

Number of Snapshots

y(x,t)

Figure 15



11. Conclusions

Burger Equations

0 2 4 6 8 10 12 14 16 18−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2 Singular Values of Nonlinear Snapshots

loga

rithm

ic s

cale

Number of Snapshots

0.5*y(x,t)2

5 10 15 20 25 30 35 40 45−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 EIM points for 0.5*y(x,t)2

EIM points

Figure 16



11. Conclusions

Burger Equations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6x 10

−4 POD error and POD/EIM error

|yPOD(25,x)−yFEM(25,x)||yPOD/EIM(25,x)−yFEM(25,x)||yPOD(51,x)−yFEM(51,x)||yPOD/EIM(51,x)−yFEM(51,x)|

0 5 10 150

0.05

0.1

0.15

0.2

0.25 Euclidian Error of Burger Solution at t=0.48

POD dimension

Eu

clid

ian

No

rm

PODEIM5EIM25EIM45

Figure 17



11. Conclusions

Burger Equations

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 CPU time

POD dimension

tim

e(se

con

ds)

PODEIM5EIM25EIM45

0 5 10 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09 Euclidian Error of Burger Solution at t=1

POD dimension

Eu

clid

ian

No

rm

PODEIM5EIM25EIM45

Figure 18



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