pod/deim 4d-var for a finite-difference shallow water...

0.0.

POD/DEIM 4D-Var for a Finite-DifferenceShallow Water Equations Model

R. Stefanescu

North Carolina State University

[email protected]

R. Stefanescu

POD/DEIM 4D-Var for a Finite-Difference Shallow Water Equations Model 1/64

Part1 - POD/DEIM nonlinear model reduction

1 POD/DEIM justification and methodology

2 POD/DEIM as a discrete variant of EIM and their pseudo -algorithms

3 Application of DEIM to an ADI implicit scheme of the SWE ona rectangular domain

4 Numerical Results

5 Conclusion and future research

1.1. POD/DEIM justification and methodology

POD/DEIM 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.

R. Stefanescu



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 a discontinuous innerproduct to the ODE 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 is one of the most significant projection-based reductionmethods for non-linear dynamical systems.

It is also known as Karhunen - Loeve expansion, principalcomponent analysis in statistics, singular value decomposition(SVD) in matrix theory and empirical orthogonal functions(EOF) in meteorology and geophysical uid dynamics

Introduced in the field of turbulence by Lumley (1967) It wasSirovich (1987 a,b,c) that introduced the method of snapshotsobtained from either experiments or numerical simulation

Error formula for the POD basis of rank l∫ T

0‖y(t)−

l∑j=1

< y(t), vj >X vj ||2Xdt =∞∑

j=l+1

λj .

R. Stefanescu




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

.

R. Stefanescu


1.2. POD/DEIM as a discrete variant of EIM and their pseudo - algorithms

Discrete Empirical Interpolation Method (DEIM)

DEIM is a discrete variation of the Empirical Interpolationmethod proposed by Barrault et al. (2004) - Comptes Rendusde l’Academie des Sciences. The application was suggested byChaturantabut and Sorensen (2008, 2010, 2012).

Let f : D → Rn, D ⊂ Rn be a nonlinear function. IfU = ulml=1, ui ∈ Rn, i = 1, ..,m is a linearly independentset, for m ≤ n, then for τ ∈ D, the DEIM approximation oforder m for f (τ) in the space spanned by ulml=1 is given by

f (τ) ≈ Uc(τ), U ∈ Rn×m, c(τ) ∈ Rm. (2)

The basis U can be constructed effectively by applying thePOD method on the nonlinear snapshots f (τ ti ), i = 1, .., ns .

R. Stefanescu



Interpolation is used to determine the coefficient vector c(τ)by selecting m rows ρ1, .., ρm, ρi ∈ N∗, of the overdeterminedlinear system (2)

f1(τ)......

fn(τ)

︸︷︷︸

f (τ)∈Rn

=

u11 . . . u1m

... . . ....

... . . ....

un1 . . . unm

︸︷︷︸

U∈Rn×m

c1(τ)...

cm(τ)

︸︷︷︸

c(τ)∈Rm

.

to form a m-by-m linear system fρ1(τ)...

fρm(τ)

︸︷︷︸

f~ρ(τ)∈Rm

=

uρ11 . . . uρ1m... . . .

...uρm1 . . . uρmm

︸︷︷︸

U~ρ∈Rm×m

c1(τ)...

cm(τ)

︸︷︷︸

c(τ)∈Rm

.



In the short notation form

U~ρc(τ) = f~ρ(τ).

Lemma 2.3.1 in Chaturantabut (2008) proves that U~ρ isinvertible, thus we can uniquely determine c(τ)

c(τ) = U−1~ρ f~ρ(τ).

The DEIM approximation of F (τ) ∈ Rn is

f (τ) ≈ Uc(τ) = UU−1~ρ f~ρ(τ).

R. Stefanescu




U~ρ and f~ρ(τ) can be written in terms of U and f (τ)

U~ρ = PTU, f~ρ(τ) = PT f (τ)

whereP = [eρ1 , .., eρm ] ∈ Rn×m, eρi = [0, ..0, 1︸︷︷︸

ρi

, 0, .., 0]T ∈ Rn.

The DEIM approximation of f ∈ Rn becomes

f (τ) ≈ U(PTU)−1PT f (τ).

R. Stefanescu




By taking τ = y(t) ∈ Rn, the DEIM approximation for thenonlinear function

f (τ) = f (y(t)) = F(Vk y(t))

in the POD-Galerkin reduced system (1) is

F(Vk y(t)) ≈ U(PTU)−1PTF(Vk y(t)),

but since F evaluates componentwise at its input we have

F(Vk y(t)) ≈ U(PTU)−1F(PTVk y(t)).

R. Stefanescu



Thus, the DEIM approximation of the nonlinear POD-Galerkinterm N(y)

N(y) = V Tk︸︷︷︸

k×n

F(Vk y(t))︸︷︷︸n×1

isN(y) ≈ V T

k U(PTU)−1︸︷︷︸precomputed k×m

F(PTVk y(t))︸︷︷︸m×1

.

Using the DEIM approximation, the complexity for computingthe nonlinear term of the reduced system in each time step isnow independent of the dimension n of the original full-ordersytem.

The only unknowns need to be specified are the indicesρ1, ρ2, ..., ρm or matrix P.

DEIM: Algorithm for Interpolation Indices

INPUT: ulml=1 ⊂ Rn (linearly independent):

OUTPUT: ~ρ = [ρ1, .., ρm] ∈ Rm

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.

2 U = [u1], P = [eρ1 ], ~ρ = [ρ1].

3 For l = 2, ..,m do

a Solve (PTU)c = PTul for c

b r = ul − Uc

c [|ψ| ρl ] = max|r |

d U ← [U ul ], P ← [P eρl], ~ρ←

[~ρρl

]4 end for.

DEIM: Algorithm for Interpolation Indices

The term r can be viewed as the residual or the error betweenthe input basis ul and its approximation Uc from interpolatingthe basis u1, u2, .., ul−1 at the points xρ1 , xρ2 , .., xρl−1

.

The linear independence of the input basis ulml=1 guaranteesthat, in each iteration, r is a nonzero vector and the outputindices ρimi=1 are non - repeated.

An error bound for the DEIM approximation is provided inChaturantabut (2008).

A state space error analysis for POD-DEIM Nonlinear modelreduction applied to ODE systems arising from spatialdiscretizations of parabolic PDEs can be found inChaturantabut and Sorensen (2012).



The following example illustrates the efficiency of DEIM 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)sin

(πµ(x + 1)

)e−(1+x)µ,

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

R. Stefanescu




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 ].

R. Stefanescu




0 5 10 15 20 25 30 35 40 45 50−16

−14

−12

−10

−8

−6

−4

−2

0

2Singular values of 50 Snapshots

logarithmic scale

−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

1

1.2DEIM points and the first 6 POD basis functions

PODbasis1PODbasis2PODbasis3PODbasis4PODbasis5PODbasis6DEIM pointsExact function

Figure 1: Singular eigenvalues using logarithmic scale and thecorresponding first 6 POD basis functions with DEIM points ofsnapshots, µ = 1.38

R. Stefanescu


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

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0DEIM#1

u1

current point

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

−0.15

−0.1

−0.05

0

0.05

0.1

0.15DEIM#2

u2

r=u2−Uc

current pointprevious point

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2DEIM#3

u3

r=u3−Uc

current pointprevious points

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

−0.15

−0.1

−0.05

0

0.05

0.1

0.15DEIM#4

u4

r=u4−Uc


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

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5DEIM#5

u5

r=u5−Uc


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

−0.3

−0.2

−0.1

0

0.1

0.2

0.3DEIM#6

u6

r=u6−Uc


Figure 2: The selection process of DEIM interpolation 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

1

1.2DEIM#1

exactDEIM approx

−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

1

1.2DEIM#2

exactDEIM approx

−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

1

1.2DEIM#3

exactDEIM approx

−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

1

1.2DEIM#4

exactDEIM approx.

−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

1

1.2DEIM#5

exactDEIM approx.

−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

1

1.2DEIM#6

exactDEIM approx.

Figure 3: DEIM approximation for different values of m



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

−0.2

0

0.2

0.4

0.6

0.8

1

1.2The Exact function and its DEIM approximation for µ=1.38

Exact fuctionDEIM solution

0 5 10 15 20 25 30 35 40 45 50−16

−14

−12

−10

−8

−6

−4

−2

0

2

logarithmic scale

m (Reduced dimension)

Error in Euclidian Norm

DEIM errorPOD error

Figure 4: The DEIM approximate function for m = 20 compared with theexact function of dimension n = 101 at µ = 1.38 (left); Comparison ofthe spatial errors for POD and DEIM approximations (right)

R. Stefanescu


1.3. Application of DEIM to an ADI implicit scheme of the SWE on a rectangular domain

Application of DEIM to an ADI implicit scheme of theSWE on a rectangular domain

We consider a 2-D shallow-water(SWE) equations model on aβ-plane solved using an alternating direction fully implicitfinite-difference scheme (Gustafsson 1971, Fairweather andNavon 1980, Navon and De Villiers 1986, Kreiss and Widlund1966) on a rectangular domain.

The scheme was shown to be unconditionally stable for thelinearized equations. The discretization yields a number ofnonlinear systems of algebraic equations.

R. Stefanescu



Application of DEIM to an ADI implicit scheme of theSWE on a rectangular domain

Next we use a proper orthogonal decomposition to reduce thedimension of the S-W model. Due to nonlinearities, thecomputational complexity of the reduced model still dependson the number of variables of the nonlinear full shallow -water equations model. By employing the discrete empiricalinterpolation method (DEIM) we reduce the computationalcomplexity and regain the full model reduction expected fromthe POD model.

R. Stefanescu


SWE model

∂w

∂t= A(w)

∂w

∂x+ B(w)

∂w

∂y+ C (y)w , (3)

0 ≤ x ≤ L, 0 ≤ y ≤ D, t ∈ [0, tf ],

where w = (u, v , φ)T , u, v are the velocity components in the xand y directions, respectively, h is the depth of the fluid, g is theacceleration due to gravity and φ = 2

√gh.

The matrices A, B and C are expressed

A = −

u 0 φ/20 u 0φ/2 0 u

, B = −

v 0 00 v φ/20 φ/2 v

C =

0 f 0−f 0 00 0 0

,

f = f +β(y−D/2) (Coriolis force), β =∂f

∂y,with f and β constants.


SWE model

We assume periodic solutions in the x-direction

w(x , y , t) = w(x + L, y , t),

while in the y−direction we have

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

The initial conditions are derived from the initial height-fieldcondition No. 1 of Grammelvedt (1969), i.e.

h(x , y) = H0+H1+tanh

(9D/2− y

2D

)+H2sech

2

(9D/2− y

2D

)sin

(2πx

L

)

R. Stefanescu



SWE model

The initial velocity fields were derived from the initial heightfield using the geostrophic relationship

u =

(−gf

)∂h

∂y, v =

(g

f

)∂h

∂x .

The constants used were:

L = 6000km g = 10ms−2

D = 4400km H0 = 2000m

f = 10−4s−1 H1 = 220mm

β = 1.5 · 10−11s−1m−1 H2 = 133m.

R. Stefanescu



The nonlinear Gustafsson ADI finite difference implicitscheme

First we introduce a network of Nx · Ny equidistant points on[0, L]× [0,D], with dx = L/(Nx − 1), dy = D/(Ny − 1). Wealso discretize the time interval [0, tf ] using NT equallydistributed points and dt = tf /(NT − 1).

Next we define vectors of unknown variables of dimensionnxy = Nx · Ny containing approximate solutions such as

u(t) ≈ u(xi , yj , t), v(t) ≈ v(xi , yj , t),φ ≈ φ(xi , yj , t) ∈ Rnxy

R. Stefanescu


For tn+1, the Gustafsson nonlinear ADI difference scheme isdefined by

I. First step - get solution at t(n + 12 )

u(tn+ 12) +

∆t

2F11

(u(tn+ 1

2),φ(tn+ 1

2)

)= u(tn)− ∆t

2F12

(u(tn), v(tn)

)+

∆t

2[f , f , .., f︸︷︷︸

Nx

]T ∗ v(tn),

v(tn+ 12) +

∆t

2F21

(u(tn+ 1

2), v(tn+ 1

2)

)+

∆t

2[f , f , .., f︸︷︷︸

Nx

]T ∗ u(tn+ 12) = v(tn)−

∆t

2F22

(v(tn),φ(tn)

),

φ(tn+ 12) +

∆t

2F31

(u(tn+ 1

2),φ(tn+ 1

2)

)= φ(tn)− ∆t

2F32

(v(tn),φ(tn)

),

(4)

with ”*” denoting MATLAB componentwise multiplicationand the nonlinear functions F11,F12,F21,F22,F31,F32 : Rnxy×Rnxy → Rnxy are defined as follows


The nonlinear Gustafsson ADI finite difference implicitscheme

F11(u,φ) = u ∗ Axu +1

2φ ∗ Axφ,

F12(u, v) = v ∗ Ayu,F21(u, v) = u ∗ Axv ,

F22(v ,φ) = v ∗ Ayv +1

2φ ∗ Ayφ,

F31(u,φ) =1

2φ ∗ Axu + u ∗ Axφ,

F32(v ,φ) =1

2φ ∗ Ayv + v ∗ Ayφ,

where Ax ,Ay ∈ Rnxy×nxy are constant coefficient matrices fordiscrete first-order and second-order differential operators whichtake into account the boundary conditions.

R. Stefanescu


II. Second step - get solution at t(n + 1)

u(tn+1) +∆t

2F12

(u(tn+1), v(tn+1)

)− ∆t

2[f , f , .., f︸︷︷︸

Nx

]T ∗ v(tn+1) = u(tn+ 12)−

∆t

2F11

(u(tn+ 1

2),φ(tn+ 1

2)

),

v(tn+1) +∆t

2F22

(v(tn),φ(tn)

)= v(tn+ 1

2)− ∆t

2F21

(u(tn+ 1

2), v(tn+ 1

2)

)−

∆t

2[f , f , .., f︸︷︷︸

Nx

]T ∗ u(tn+ 12),

φ(tn+1) +∆t

2F32

(v(tn+1),φ(tn+1)

)= φ(tn+ 1

2)−

∆t

2F31

(u(tn+ 1

2),φ(tn+ 1

2)

).

(5)


The Quasi-Newton Method

The nonlinear systems of algebraic equations (4) and (5) arewritten in the form

g(α) = 0.

where α is the vector of unknown.

Due to the fact that no more than two variables are coupledto each other on the left-hand side of equations (4) and (5),we first solve system (4) for u = [u1, u2, ..., unxy ] andφ = [φ1, φ2, ..., φnxy ] and define

α = (u1, φ1, u2, φ2, ..., unxy , φnxy ) ∈ R2nxy .

R. Stefanescu




The Newton method is given by

α(m+1) = α(m) − J−1(α(m))g(α(m)), (6)

where the superscript denotes the iteration and J ∈ R2nxy×2nxy

is the Jacobian

J =∂g

∂α.

Owing to the structure of the Gustafsson algorithm for theSWE, the Jacobian matrix is either block cyclic tridiagonal orblock tridiagonal.

R. Stefanescu




J−1g in (6) is solved by first applying an LU decomposition toJ. Then it is computed by backsubstitution in two stages.First z is solved from

Lz = g ,

and then J−1g is obtained from

U(J−1g) = z .

In the quasi-Newton method, the computationally expensiveLU decomposition is performed only once every M − thtime-step, where M is a fixed integer.

The quasi-Newton formula is

α(m+1) = α(m) − J−1(α(m))g(α(m)), where

J = J(α(0)) + O(dt).

R. Stefanescu




The method works when M, the number of time-stepsbetween successive updating of the LU decomposition of theJacobian matrix J, is a relatively small number, in our case,M = 6 or M = 12.

The second part of the system (4) is solved forv = [v1, v2, ..., vnxy ] by employing the same quasi-Newtonmethod. Thus α is defined as

α = (v1, v2, vnxy ) ∈ Rnxy .

In order to obtain the SWE numerical solution at t(n + 1) weapplied the same quasi-Newton technique for system (5). Thistime the variables coupled first were

α = (v1, φ1, v2, φ2, ..., vnxy , φnxy ) ∈ R2nxy ,

while u was solved from the remaining equations.

R. Stefanescu



The POD version of SWE model

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.

Let Y = [u1, u2, .., uNT ] ∈ Rnxy×NT , be a snapshot set, i.e.the numerical solution obtained with ADI implicit FD schemeat t1, t2, .., tNT , of the horizontal component of the velocity.

R. Stefanescu




The POD basis U ∈ Rnxy×k is determined by solving the eigenvalue problem

Y TY ui = λi ui , i = 1, 2, ..,NT ,

and retaining the set of right singular vectors of Ycorresponding to the k largest singular values, i.e.U = uiki=1, ui = 1

λiY ui .

We determined the POD basis in this way by taking inconsideration the relationship NT nxy .

Similarly, let V ,Φ ∈ Rnxy×k be the POD basis matrices of thevertical component of the velocity and geopotential,respectively. Now we can approximate u, v and φ as following

u ≈ Uu, v ≈ V v , φ ≈ Φφ, u, v , φ ∈ Rk .

R. Stefanescu




The POD reduced-order system is constructed by applying theGalerkin projection method to ADI FD discrete model (4) and(5) by first replacing u, v,φ with their approximation Uu, V v ,Φφ, respectively, and then premultiplying the correspondingequations by UT , V T and ΦT .

R. Stefanescu


The resulting POD reduced system for the first step (tn+ 12) of

the ADI FD scheme is

u(tn+ 12) +

∆t

2UT F11

(u(tn+ 1

2), φ(tn+ 1

2)

)= u(tn)− ∆t

2UT F12

(u(tn), v(tn)

)+

∆t

2UT

([f , f , .., f︸︷︷︸

Nx

]T ∗ V v(tn)

),

v(tn+ 12) +

∆t

2V T F21

(u(tn+ 1

2), v(tn+ 1

2)

)+

∆t

2V T

([f , f , .., f︸︷︷︸

Nx

]T ∗ Uu(tn+ 12)

)

= v(tn)− ∆t

2V T F22

(v(tn), φ(tn)

),

φ(tn+ 12) +

∆t

2ΦT F31

(u(tn+ 1

2), φ(tn+ 1

2)

)= φ(tn)− ∆t

2ΦT F32

(v(tn), φ(tn)

),

(7)

where F11, F12, F21, F22, F31, F32 : Rk× Rk → Rk are defined by



F11(u, φ) = (Uu) ∗ (AxU︸︷︷︸ u) +1

2(Φφ) ∗ (AxΦ︸︷︷︸ φ),

F12(u, v) = (V v) ∗ (AyU︸︷︷︸ u), F21(u, v) = (Uu) ∗ (AxV︸︷︷︸ v),

F22(v , φ) = (V v) ∗ (AyV︸︷︷︸ v) +1

2(Φφ) ∗ (AyΦ︸︷︷︸ φ),

F31(u, φ) =1

2(Φφ) ∗ (AxU︸︷︷︸ u) + (Uu) ∗ (AxΦ︸︷︷︸ φ),

F32(v , φ) =1

2(Φφ) ∗ (AyV︸︷︷︸ v) + (V v) ∗ (AyΦ︸︷︷︸ φ).

(8)

R. Stefanescu




By defining A1,A2 ∈ Rnxy×k such as

A1(:, i) = [f , f , .., f︸︷︷︸Nx

]T∗V (:, i), A2(:, i) = [f , f , .., f︸︷︷︸Nx

]T∗U(:, i), i = 1, .., k ,

the linear terms in (7), ∆t2 UT

([f , f , .., f︸︷︷︸

Nx

]T ∗ V v(tn)

)and

∆t2 V T

([f , f , .., f︸︷︷︸

Nx

]T ∗ Uu(tn+ 12)

)can be rewritten as

∆t2 UTA1︸︷︷︸ v(tn) and ∆t

2 V TA2︸︷︷︸ u(tn+ 12) respectively.

R. Stefanescu



The coefficient matrices UTA1, VTA2 ∈ Rk×k defined in the

linear terms of the POD reduced system as well as thecoefficient matrices in the nonlinear functions (i.e. AxU,AyU,AxV ,AyV ,AxΦ,AyΦ ∈ Rn×k grouped by the curly braces)can be precomputed, saved and re-used in all time steps.

However, performing the componentwise multiplications in (8)and computing the projected nonlinear terms in (7)

UT︸︷︷︸k×nxy

F11(u, φ)︸︷︷︸nxy×1

,UT F12(u, v),V T F21(u, v),

V T F22(v , φ),ΦT F31(u, φ),ΦT F32(v , φ),

(9)

still have computational complexities depending on thedimension nxy of the original system from both evaluating thenonlinear functions and performing matrix multiplications toproject on POD bases.


The DEIM version of SWE model

DEIM is used to remove this dependency.

The projected nonlinear functions can be approximated byDEIM 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 indices from DEIM must beevaluated at each time step.

DEIM approximation is applied to each of the nonlinearfunctions F11, F12, F21, F22, F31, F32 defined in (8).

R. Stefanescu




Let UF11 ∈ Rnxy×m, m ≤ n, be the POD basis matrix of rankm for snapshots of the nonlinear function F11 (obtained fromADI FD scheme).

Using the DEIM algorithm we select a set of m DEIM indicescorresponding to UF11 , denoting by [ρF11

1 , .., ρF11m ]T ∈ Rm. The

DEIM approximation of F11 is

F11 ≈ UF11(PTF11

UF11)−1Fm11,

so the projected nonlinear term UT F11(u, φ) in the PODreduced system (7) can be approximated as

UT F11(u, φ) ≈ UTUF11(PTF11

UF11)−1︸︷︷︸E1∈Rk×m

Fm11(u, φ)︸︷︷︸m×1

,

where Fm11(u, φ) = PT

F11F11(u, φ).

R. Stefanescu


Since F11 is a pointwise function, Fm11 : Rk ×Rk → Rm can be defined as

Fm11(u, φ) = (PT

F11Uu) ∗ (PT

F11AxU︸︷︷︸ u) +

1

2(PT

F11Φφ) ∗ (PT

F11AxΦ︸︷︷︸ φ)

Similarly we obtain the DEIM approximation for the rest of the projectednonlinear terms in (9)

UT F12(u, v) ≈ UTUF12(PTF12

UF12)−1︸︷︷︸E2∈Rk×m

Fm12(u, v)︸︷︷︸m×1

,

V T F21(u, v) ≈ V TUF21(PTF21

UF21)−1︸︷︷︸E3∈Rk×m

Fm21(u, v)︸︷︷︸m×1

,

V T F22(v , φ) ≈ V TUF22(PTF22

UF22)−1︸︷︷︸E4∈Rk×m

Fm22(v , φ)︸︷︷︸m×1

,

ΦT F31(u, φ) ≈ ΦTUF31(PTF31

UF31)−1︸︷︷︸E5∈Rk×m

Fm31(u, φ)︸︷︷︸m×1

,

ΦT F32(v , φ) ≈ ΦTUF32(PTF32

UF32)−1︸︷︷︸E6∈Rk×m

Fm32(v , φ)︸︷︷︸m×1

,



Fm12(u, v) = (PT

F12V v) ∗ (PT

F12AyU︸︷︷︸ u),

Fm21(u, v) = (PT

F21Uu) ∗ (PT

F21AxV︸︷︷︸ v),

Fm22(v , φ) = (PT

F22V v) ∗ (PT

F22AyV︸︷︷︸ v) +

1

2(PT

F22Φφ) ∗ (PT

F22AyΦ︸︷︷︸ φ),

Fm31(u, φ) = (PT

F31Φφ) ∗ (PT

F31AxU︸︷︷︸ u) + (PT

F31Uu) ∗ (PT

F31AxΦ︸︷︷︸ φ),

Fm32(v , φ) =

1

2(PT

F32Φφ) ∗ (PT

F32AyV︸︷︷︸ v) + (PT

F32V v) ∗ (PT

F32AyΦ︸︷︷︸ φ).

(10)

R. Stefanescu


Each of the k ×m coefficient matrices grouped by the curlybrackets in (10), as well as Ei , i = 1, 2, .., 6 can beprecomputed and re-used at all time steps, so that thecomputational complexity of the approximate nonlinear termsare independent of the full-order dimension nxy . Finally, thePOD-DEIM reduced system for the first step of ADI FD SWEmodel is of the form

u(tn+ 12) +

∆t

2E1F

m11

(u(tn+ 1

2), φ(tn+ 1

2)

)= u(tn)− ∆t

2E2F

m12

(u(tn), v(tn)

)+

∆t

2UTA1v(tn),

v(tn+ 12) +

∆t

2E3F

m21

(u(tn+ 1

2), v(tn+ 1

2)

)+

∆t

2V TA2u(tn+ 1

2)

= v(tn)− ∆t

2E4F

m22

(v(tn), φ(tn)

),

φ(tn+ 12) +

∆t

2E5F

m31

(u(tn+ 1

2), φ(tn+ 1

2)

)= φ(tn)− ∆t

2E6F

m32

(v(tn), φ(tn)

).

(11)

1.4. Numerical Results

Numerical Results

The domain was discretized using a mesh of 121× 89 points.Thus the dimension of the full-order discretized model is10769. The integration time window was 24h.

ADI FD scheme proposed by Gustafsson (1971) was firstemployed in order to obtain the numerical solution of theSWE model.

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

R. Stefanescu


Numerical Results

18000 18000

18500

18500

19000

19000

19500

19500

20000

20000

20500

20500

21000

210002150

0

21500

22000 22000

Contour of geopotential from 22000 to 18000 by 500

y(km)

x(km)0 1000 2000 3000 4000 5000 6000

0

500

1000

1500

2000

2500

3000

3500

4000

0 1000 2000 3000 4000 5000 6000 7000−500

0

500

1000

1500

2000

2500

3000

3500

4000

4500Wind field

x(km)

y(km)

Figure 5: Initial condition: Geopotential height field for theGrammeltvedt initial condition (left). Wind field (arrows are scaled by afactor of 1km) calculated from the geopotential field by using thegeostrophic approximation (right).


Numerical Results

Our first numerical experiment was done using a time step∆t = 480s.

The Courant–Friedrichs–Levy criterion CFL=√gH ∆t

∆x wasdetermined

CFL = 1.3576

The nonlinear algebraic systems obtained from ADI FD SWEwere solved with the Quasi-Newton method, were the LUdecomposition was performed only once every 6− th timestep.

R. Stefanescu



Numerical Results

18000 18000

1850018500

19000 1900019500

19500

20000

20000

20500

20500

21000

2100021500

21500

2200022000

Contour of geopotential at time tf = 24h

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

0 1000 2000 3000 4000 5000 6000 7000−500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Wind field at time tf = 24h

x(km)

y(km

)Figure 6: The geopotential field (left) and the wind field (arrows arescaled by a factor of 1km) at t = tf = 24h obtained using the ADI FDSWE scheme for ∆t = 480s.

R. Stefanescu



Numerical Results

The POD basis vectors were constructed using 181 snapshotsobtained from the numerical solution of the full - order ADIFD SWE model at equally spaced time steps in the interval[0 24h].

The dimension of the POD bases for each variable was taken40, capturing more than 99.9% of the system energy.

R. Stefanescu



Numerical Results

0 20 40 60 80 100 120 140 160 180 200−20

−15

−10

−5

0

5

10Singular Values of Snapshots Solution

Number of snapshots

log

arit

hm

ic s

cale

uvφ

0 20 40 60 80 100 120 140 160 180 200−25

−20

−15

−10

−5

0Singular Values of Nonlinear Snapshots

Number of snapshots

log

arit

hm

ic s

cale

FF11FF12FF21FF22FF31FF32

Figure 7: The decay around the singular values of the snapshots solutionsfor u, v , φ and nonlinear functions (∆t = 480s).

R. Stefanescu


Numerical Results

We applied the DEIM 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

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40

DEIM POINTS for FF31

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

1

234 5

6

7

8

9

10

11

12

13

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30

31

3233

34

35

36

37

38

39

40

DEIM POINTS for FF32

x(km)

y(km

)

Figure 8: First 40 points selected by DEIM for the nonlinear functionsFF31 (left) and FF32 (right)


Numerical Results

Once the dimension of DEIM (no of points selected by DEIMalgorithm) reaches 40 the approximation errors from thePOD-DEIM and POD reduced systems are indistinguishable.

We emphasize the performances of POD - DEIM method incomparison with the POD approach using the numericalsolution of the ADI FD SWE model. Next three slides depictthe space error behaviors between POD/POD - DEIMsolution and ADI FD SWE solution.

R. Stefanescu



Numerical Results

Geopotential POD errors hPOD

−hADI FD

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

0.5

1

1.5

2

2.5

3

3.5

x 10−4 Geopotential POD −DEIM errors h

DEIM−h

ADI FD

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

0.5

1

1.5

2

2.5

3

3.5

x 10−4

Figure 9: Errors between the geopotential calculated with POD/POD-DEIM and geopotential determined with the ADI FD SWE model att = 24h (∆t = 480s). The number of DEIM points was taken 40.

R. Stefanescu



Numerical Results

u POD errors uPOD

−uADI FD

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

−1.5

−1

−0.5

0

0.5

1

1.5

x 10−4 u POD−DEIM errors u

DEIM−u

ADI FD

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

−1.5

−1

−0.5

0

0.5

1

1.5

x 10−4

Figure 10: Errors between u calculated with POD/POD -DEIM and udetermined with the ADI FD SWE model at t = 24h (∆t = 480s). Thenumber of DEIM points was taken 40.

R. Stefanescu



Numerical Results

v POD errors vPOD

−vADI FD

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

−8

−6

−4

−2

0

2

4

6

8

10

x 10−5 v POD − DEIM errors v

DEIM−v

ADI FD

x(km)

y(km

)

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

−8

−6

−4

−2

0

2

4

6

8

10

x 10−5

Figure 11: Errors between v calculated with POD/POD -DEIM and vdetermined with the ADI FD SWE model at t = 24h (∆t = 480s). Thenumber of DEIM points was taken 40.

R. Stefanescu


Numerical Results

Using the following norms

1

tf

tf∑i=1

||yADI FD(:, i)− yADI POD(:, i)||2||yADI FD(:, i)||2

1

tf

tf∑i=1

||yADI FD(:, i)− yADI POD DEIM(:, i)||2||yADI FD(:, i)||2

we obtained the following average errors.

ADI FD ADI POD ADI POD-DEIM

CPU time 88.06 10.13 0.981

φ - 2.12243 · 10−5 3.77796 · 10−5

u - 1.1881 · 10−3 1.9106 · 10−3

v - 1.8588 · 10−3 1.9106 · 10−3

Table 1: CPU time gains and the average errors for each of the modelvariables. The POD bases dimensions were taken 40 capturing more than99.9% of the system energy. 40 DEIM points were chosen.

Numerical Results

If the POD bases dimensions are larger than 20 a 10−1 CPUtime reduction is gained when employing the POD -DEIMmethod in comparison with the POD approach.

0 5 10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

100

101

102

103 CPU time

POD dimension

time(seconds)

DEIM40DEIM50DEIM100PODFULL

0 5 10 15 20 25 30 35 4010

−5

10−4

10−3

10−2 Average Relative Error of Geopotential

POD dim

Error

DEIM40DEIM50DEIM100POD

Figure 12: CPU time of the full system, POD reduced sytem and POD -DEIM reduced system (left); Average relative error of φ from thePOD-DEIM reduced system compared with the one from the PODreduced system (right)

1.5. Conclusion and future research

Conclusion and future research

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

In the case of solving the POD-DEIM 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 DEIM points determined bythe DEIM indexes algorithm, m nxy .

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

R. Stefanescu


Conclusion and future research

To prove the efficiency of DEIM we used the ADI FD SWEmodel. The CPU time was reduced by a factor of 10 for theADI POD - DEIM SWE. Also we noticed that theapproximation errors from the POD-DEIM and POD reducedsystems are indistinguishable when the dimension of DEIMreached 40.

Application to optimization and uncertainty quantification.

Adaptive ROMs based on DEIM points.

POD/DEIM reduced-order strategies for efficient fourdimensional variational data assimilation, Journal ofComputational Physics, Volume 295, 15 August 2015, Pages569-595

References

G. Fairweather , I. M. Navon, A linear ADI method for theshallow water equations , Journal of Computational Physics ,37 (1980) 1-18.

B. Gustafsson, An alternating direction implicit method forsolving the shallow water equations, Journal of ComputationalPhysics ,7 (1971) 239-254.

I. M. Navon, R. De Villiers, GUSTAF: A Quasi-NewtonNonlinear ADI FORTRAN IV Program for Solving theShallow-Water Equations with Augmented Lagrangians,Computers and Geosciences,12, No. 2 (1986) 151-173.

S . Chaturantabut, D .C. Sorensen, A state space errorestimate for POD-DEIM nonlinear model reduction ,SIAMJournal on Numerical Analysis , 50, 1 (2012) 46-63.

S . Chaturantabut, D.C. Sorensen, Nonlinear model reductionvia discrete empirical interpolation, SIAM Journal onScientific Computing, 32 , 5 (2010) 2737-2764.

References

S. Chaturantabut, Master’s Thesis: Dimension Reduction forUnsteady Nonlinear Partial Differential Equations viaEmpirical Interpolation Methods, November 2008.

S . Chaturantabut, D.C. Sorensen, Application of POD andDEIM on dimension reduction of non-linear miscible viscousfingering in porous media, Mathematical and ComputerModelling of Dynamical Systems , 17 , 4 (2011) 337-353 .

A. R. Kellems, S. Chaturantabut, D. C. Sorensen , S. J. Cox,Morphologically accurate reduced order modeling of spikingneurons . Journal of Computational Neuroscience. 28,Number 3 (2010), 477-494.

M. Barrault,Y. Maday , N.C. Nguyen; A.T. Patera An’empirical interpolation’ method: application to efficientreduced-basis discretization of partial differential equations,Comptes Rendus Mathematique, 339,9, (2004) 667-672.

References

M. Hinze , M. Kunkel, Discrete Empirical Interpolation inPOD Model Order Reduction of Drift-Diffusion Equations inElectrical Networks, Scientific Computing in ElectricalEngineering SCEE 2010 Mathematics in Industry, 16, Part5,(2012) 423-431.

O. Lass, S. Volkwein, POD Galerkin schemes for nonlinearelliptic-parabolic systems,Konstanzer Schriften in Mathematik,Nr. 301, Marz 2012, ISSN 1430-3558

Part2 - POD 4-D VAR of the limited area FEM SWE usingDual-Weighting and Trust Region method

pod/deim 4d-var for a finite-difference shallow water...

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