Lectures on Model Reduction (Ref: Introduction to Model Order Reduction,Wil Schilders in Model Order Reduction:Theory, Research Aspects and Applications in computational science and engineering. Springer,2008)
• Method of snapshots and POD algorithm Motivation The ever increasing demand for realistic simulations of complex
products places a heavy burden on the shoulders of mathematicians
and, more generally, researchers working in the area of
Realistic simulations imply that the errors of the virtual models
should be small, and that different aspects of the product must be
taken into account.
The former implies that care must be taken in the numerical
treatment and that, for example, a relatively fine adaptively
determined mesh is necessary in the simulations.
The latter explains the trend in coupled simulations, for example
combined mechanical and thermal behaviour, or combined
mechanical and electromagnetic behaviour.
An important factor in enabling the complex simulations carried
out today is the increase in computational power. Computers and
chips are getting faster, Moore’s law predicting that the speed will
double every 18 months (see Figure 2).
This increase in computational power appears to go hand-in-
hand with developments in numerical algorithms.
Iterative solution techniques for linear systems are mainly
responsible for this speed-up in algorithms, as is shown in
Figure 3.
Important contributions in this area are the conjugate
gradient method , preconditioned conjugate gradient
methods (ICCG, biCGstab) and multigrid methods.
Dynamical Systems
To place model reduction in a mathematical context, we
need to realize that many models developed in
computational science consist of a system of partial and/or
ordinary differential equations, supplemented with boundary
conditions.
Important examples
• The Navier-Stokes equations in computational fluid
dynamics (CFD), and
• the Maxwell equations in electromagnetics (EM).
When partial differential equations
are used to describe the behaviour, one often encounters
the situation that the independent variables are space and
time.
Thus, after (semi-) discretising in space, a system of
ordinary differential equations is obtained in time.
Therefore, we limit the discussion to ODE’s and consider
the following explicit finite-dimensional dynamical system
(following book of Antoulas):
Here, u is the input of the system, y the output, and x the so-called
state variable.
The dynamical system can thus be viewed as an input-output
system, as displayed in Figure 7.
The complexity of the system is characterized by the number of its
state variables, i.e. the dimension n of the state space vector x.
It should be noted that similar dynamical systems can also be
defined in terms of differential algebraic equations,
in which case the first set of equations in (1) is replaced by
Model order reduction can now be viewed as the task of
reducing the dimension of the state space vector, while
preserving the character of the input‐output relations.
In other words, we should find a dynamical system of the form
where the dimension of x is much smaller than n.
In order to provide a good approximation of the original input-
ou s should be satisfied: tput system, a number of condition
• the approximation error is small,
• preservation of properties of the original system, such as
stability and passivity
• the reduction procedure should be computationally efficient.
A special case is encountered if the functions f and g are
linear, in which case the system reads
Here, the matrices A,B,C,D can be time-dependent, in which case
we have a linear time-varying (LTV) system, or time-independent,
in which case we speak about a linear time-invariant (LTI) system.
For linear dynamical systems, model order reduction is equivalent
to reducing the matrix A, but retaining the number of columns of B
and C.
Approximation by Projection
Although there are ways of approximating input-output systems,
there is a unifying feature of the approximation methods that is
worthwhile discussing briefly: projection.
Methods based on this concept truncate the solution of the original
system in an appropriate basis.
To illustrate the concept, consider a basis transformation T that
maps the original n-dimensional state space vector x into a vector
that we denote by
where x is k-dimensional.
The basis transformation T can then be written as
and its inverse as
is an oblique projection along the kernel of W ∗ onto the
k-dimensional subspace that is spanned by the columns of the
matrix V .
If we substitute the projection into the dynamical system (1), the
first part of the set of equations obtained is
Note that this is an exact expression.
The approximation occurs when we would delete the terms
involving x , in which case we obtain the reduced system
For this to produce a good approximation to the original system,
the neglected term T1 x must be sufficiently small. This has
implications for the choice of the projection Π. In the following
sections, various ways of constructing this projection are discussed
POD model reduction methods application to
geosciences and 4-D VAR data assimilation
1. Introduction Interest in reduced cost of implementation of 4-D VAR data
assimilation in the geosciences motivated research efforts aimed
towards reducing dimension of control space without significantly
compromising quality of the final solution.
IntroductionProper Orthogonal Decomposition
POD Model Reduction of Large ScaleGeophysical Models
Ionel M. Navon
Department of Scientific ComputingFlorida State University
Tallahassee, Florida
I.M. Navon
IntroductionProper Orthogonal Decomposition
Outline
1 IntroductionIntroductionMotivation
2 Proper Orthogonal DecompositionProper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
I.M. Navon
IntroductionProper Orthogonal Decomposition
Outline
1 IntroductionIntroductionMotivation
2 Proper Orthogonal DecompositionProper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Introduction
• Karl Pearson(1901) paper:"On Lines and Planes of Closest Fit to A System of Points" seems to beat origin of a general approach to dimensionality reduction. Alsodeveloped independently by Hotelling(1933).
• Idea of principal component analysis is to reduce dimensionality of adata set in which there are a large number of interrelated variables,while retaining as much as possible of the variation present in the dataset.
• The reduction is achieved by transforming to a new set of variables-theprincipal components which are uncorrelated, and which are ordered.So that first, few retain most of the variation present in data set.
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Introduction
• Karl Pearson(1901) paper:"On Lines and Planes of Closest Fit to A System of Points" seems to beat origin of a general approach to dimensionality reduction. Alsodeveloped independently by Hotelling(1933).
• Idea of principal component analysis is to reduce dimensionality of adata set in which there are a large number of interrelated variables,while retaining as much as possible of the variation present in the dataset.
• The reduction is achieved by transforming to a new set of variables-theprincipal components which are uncorrelated, and which are ordered.So that first, few retain most of the variation present in data set.
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Introduction
• Karl Pearson(1901) paper:"On Lines and Planes of Closest Fit to A System of Points" seems to beat origin of a general approach to dimensionality reduction. Alsodeveloped independently by Hotelling(1933).
• Idea of principal component analysis is to reduce dimensionality of adata set in which there are a large number of interrelated variables,while retaining as much as possible of the variation present in the dataset.
• The reduction is achieved by transforming to a new set of variables-theprincipal components which are uncorrelated, and which are ordered.So that first, few retain most of the variation present in data set.
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Motivation
• In many fields of science and engineering such as fluid dynamics,geophysical fluid dynamics, large-scale systems used to be simulated,optimized and controlled. Since they are solved using discretizations ofnonlinear PDEs, they yield high dimensional problems.
• The number of degrees of freedom for simulation kept pace withincreasing computing power.
• In particular optimal design, control PDE constrained optimization, theproblems become too large to be tackled with standard techniques.
• Hence need for model reduction techniques to reduce computationalcost and storage requirements.
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Motivation
• In many fields of science and engineering such as fluid dynamics,geophysical fluid dynamics, large-scale systems used to be simulated,optimized and controlled. Since they are solved using discretizations ofnonlinear PDEs, they yield high dimensional problems.
• The number of degrees of freedom for simulation kept pace withincreasing computing power.
• In particular optimal design, control PDE constrained optimization, theproblems become too large to be tackled with standard techniques.
• Hence need for model reduction techniques to reduce computationalcost and storage requirements.
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Motivation
• In many fields of science and engineering such as fluid dynamics,geophysical fluid dynamics, large-scale systems used to be simulated,optimized and controlled. Since they are solved using discretizations ofnonlinear PDEs, they yield high dimensional problems.
• The number of degrees of freedom for simulation kept pace withincreasing computing power.
• In particular optimal design, control PDE constrained optimization, theproblems become too large to be tackled with standard techniques.
• Hence need for model reduction techniques to reduce computationalcost and storage requirements.
I.M. Navon
IntroductionProper Orthogonal Decomposition
IntroductionMotivation
Motivation
• In many fields of science and engineering such as fluid dynamics,geophysical fluid dynamics, large-scale systems used to be simulated,optimized and controlled. Since they are solved using discretizations ofnonlinear PDEs, they yield high dimensional problems.
• The number of degrees of freedom for simulation kept pace withincreasing computing power.
• In particular optimal design, control PDE constrained optimization, theproblems become too large to be tackled with standard techniques.
• Hence need for model reduction techniques to reduce computationalcost and storage requirements.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Proper Orthogonal Decomposition
• It is a projection method where dynamical system is projected onto asubspace of original phase space.
• In combination with Galerkin projection provides powerful tool toderive surrogate models for high dimensional dynamical systems.
• Requires only standard matrix computation despite its application tononlinear problems.
• Solves: Find a subspace approximating a given set of data in an optimalleast squares sense.
• Uses data points given by sampling from experiments or by trajectoriesof the physical system extracted from simulations of the full nonlinearmodel.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Proper Orthogonal Decomposition
• It is a projection method where dynamical system is projected onto asubspace of original phase space.
• In combination with Galerkin projection provides powerful tool toderive surrogate models for high dimensional dynamical systems.
• Requires only standard matrix computation despite its application tononlinear problems.
• Solves: Find a subspace approximating a given set of data in an optimalleast squares sense.
• Uses data points given by sampling from experiments or by trajectoriesof the physical system extracted from simulations of the full nonlinearmodel.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Proper Orthogonal Decomposition
• It is a projection method where dynamical system is projected onto asubspace of original phase space.
• In combination with Galerkin projection provides powerful tool toderive surrogate models for high dimensional dynamical systems.
• Requires only standard matrix computation despite its application tononlinear problems.
• Solves: Find a subspace approximating a given set of data in an optimalleast squares sense.
• Uses data points given by sampling from experiments or by trajectoriesof the physical system extracted from simulations of the full nonlinearmodel.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Proper Orthogonal Decomposition
• It is a projection method where dynamical system is projected onto asubspace of original phase space.
• In combination with Galerkin projection provides powerful tool toderive surrogate models for high dimensional dynamical systems.
• Requires only standard matrix computation despite its application tononlinear problems.
• Solves: Find a subspace approximating a given set of data in an optimalleast squares sense.
• Uses data points given by sampling from experiments or by trajectoriesof the physical system extracted from simulations of the full nonlinearmodel.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Proper Orthogonal Decomposition
• It is a projection method where dynamical system is projected onto asubspace of original phase space.
• In combination with Galerkin projection provides powerful tool toderive surrogate models for high dimensional dynamical systems.
• Requires only standard matrix computation despite its application tononlinear problems.
• Solves: Find a subspace approximating a given set of data in an optimalleast squares sense.
• Uses data points given by sampling from experiments or by trajectoriesof the physical system extracted from simulations of the full nonlinearmodel.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Construction of POD basis
Consider dynamical system described by PDEs. We restrict ourselves tofinite dimensions and start with vector space V and a given data in V.SetV = Rn and set of sampled data y = {y1(t), ..., ym(t)} given by trajectories
yi(t) ∈ Rn, i = 1, . . . , t ∈ [0, T]
Use principal component analysis of data to find subspace Vd ⊂ Vapproximating data in some least squares sense, i.e. we seek orthogonalprojection:
∏d : V −→ Vd of fixed rank d that minimizes total least-squares
distance
‖y−∏
d
y‖2 =m∑
i=1
∫ T
0‖yi(t)−
∏
d
yi(t)‖2 dt
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Construction of POD basis
Solution of this problem relies on introduction of the correlationmatrixK ∈ Rn×n defined by
K =m∑
i=1
∫ T
0yi(t)yi(t)∗ dt
where star(∗) stands for transpose of vector or matrix.K is by definition symmetric positive definite matrix with real non-negativeeigenvalues λ1 ≥ λ2 ≥ · · ·λn ≥ 0.Let uj denote corresponding eigenvectors
Kuj = λuj, j = 1, . . . , n
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Construction of POD basis
Due to the special structure of K we can choose them in fact as anorthogonal basis of V.Thus for POD an optimal subspace Vd of dimension d representing the datais given by
Vd = span{u1, u2, . . . , ud}
Vectors uj are called POD modes.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Construction of POD basis
TheoremLet K be correlation matrix of data defined above and letλ1 ≥ λ2 ≥ · · ·λn ≥ 0 be the ordered eigenvalues of K. Then it holds
minVd‖y−
∏
d
y‖ =m∑
j=m−d+1
λj
where min is taken over all subspaces Vd of dimension d.Further, the optimal orthogonal projection
∏
d
: V −→ Vd
with∏
d
∏∗d = I is given by
∏
d
=d∑
j=1
uju∗j
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Construction of POD basis
Each data vector yi(t) ∈ V can be written as
yi(t) =m∑
j=1
yij(t)uj
where yij =< yi(t), uj >.It then holds that
∏
d
yi(t) =d∑
j=1
uju∗j (n∑
l=1
yil(t)ul) =d∑
j=1
yij(t)uj
since < ui, uj >= δij
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Choosing the dimension
In terms of a dynamical system we look at eigenvalues of K. Largeeigenvalues correspond to main characteristics of the system, while smalleigenvalues provide only small perturbations of overall dynamics.We aim atchoosing d small enough while the relative information content
I(d) =
∑dj=1 λj∑nj=1 λj
i.e.
d = argmin{I(d) : I(d) ≥ p100
}
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Choosing the dimension
• POD modes are not constructed to be the modes approximating thedynamics generating the given data set.
Consider new techniques
• POD coupled with balanced-truncation• Dual weighted (goal-oriented POD)• POD works naturally with finite-element discretization
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Choosing the dimension
• POD modes are not constructed to be the modes approximating thedynamics generating the given data set.
Consider new techniques
• POD coupled with balanced-truncation• Dual weighted (goal-oriented POD)• POD works naturally with finite-element discretization
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Choosing the dimension
• POD modes are not constructed to be the modes approximating thedynamics generating the given data set.
Consider new techniques
• POD coupled with balanced-truncation• Dual weighted (goal-oriented POD)• POD works naturally with finite-element discretization
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
Choosing the dimension
• POD modes are not constructed to be the modes approximating thedynamics generating the given data set.
Consider new techniques
• POD coupled with balanced-truncation• Dual weighted (goal-oriented POD)• POD works naturally with finite-element discretization
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
POD and Snapshots
In practical applications we have dimensions of 106 -1010. Calculation ofPOD modes requires solution of a full matrix K ∈ Rn×n, i.e. maybeinfeasible.Sirovich(1987) proposed the method of snapshots.Instead of solving eigensystem for K ∈ Rn×n one considers only a matrix inRm×m, where m is number of snapshots considered. Snapshots areconstructed from trajectories of dynamical system by evaluating them atdiscrete time instances t1, t2, . . . , tm ∈ [0, T]. We here consider a newcorrelation matrix K defined by
K =m∑
i=1
y(ti)y(ti)∗
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
POD and Snapshots
How many snapshots should one choose?Consider matrix y = (y(t1), . . . , y(tm)) ∈ Rn×n i.e. columns of the snapshots,then y∗y ∈ Rm×m and solving eigenvalue problem
y∗yvj = λvj, j = 1, . . . , m, vj ∈ Rm
Choosing orthogonal basis of eigenvectorsv1, . . . , vm POD modes are givenby
uj =1√λj
yvj, j = 1, . . . , m
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
POD and SVD
Approximating POD basis should contain as much information as possible.We can write the problem of approximating the snapshot vectors yi by asingular vector u as constrained optimization problem
maxm∑
i=1
| < yj, u > |2s.t.|u| = 1
Using Lagrangian formalism a necessary condition for this problem is givenby the eigenvalue problem
yy∗u = σ2u
Singular value analysis yields that u1 solves this eigenvalue problem withfunctional value σ2
1 .
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
POD and SVD
Iterate this procedure and derive that ui, i = 1, . . . , d solves
maxm∑
j=1
| < yi, u > |2 s.t.|u| = 1 and < ui, uj >= 0, j = 1, . . . , i−1 >
and the value of the functional is given by σ2i . By construction it is clear that
for every d ≤ m the approximation of the columns y = (y1, y2, . . . , ym) bythe first d singular vectors {ui}d
i=1 is optimal in the least − squares senseamong all rank d approximations of the columns of y.
I.M. Navon
IntroductionProper Orthogonal Decomposition
Proper Orthogonal DecompositionConstruction of POD basisPOD and SnapshotsPOD and SVD
POD and SVD
This allows practical determination of a POD basis.If m ≤ n holds, one cancompute m eigenvalues vi corresponding to the largest eigenvalues ofy∗y ∈ Rn×n. These relate to POD basis as follows:
ui =1σi
yvi, i = 1, . . . , d
I.M. Navon
Recent advances
1. A dual weighted trust-region adaptive POD 4D-Var applied to a
Finite-Element shallow-water Equations Model
• We consider a limited-area finite-element discretization of the shallow-water
equations model. Our purpose in this paper is to solve an inverse problem
for the above model controlling its initial conditions in presence of
observations being assimilated in a time interval (window of assimilation).
• We then attempt to obtain a reduced-order model (ROM) of above inverse
problem, based on proper orthogonal decomposition (POD), referred to as
POD 4-D Var.
• Different approaches of POD implementation of the reduced inverse
problem are compared, including a dual-weighed method for snapshot
selection coupled with a trust-region POD approach.
• Numerical results obtained point to an improved accuracy in all metrics
tested when dual-weighing choice of snapshots is combined with POD
adaptivity of the trust-region type. Results of ad-hoc adaptivity of the POD
4-D Var turn out to yield less accurate
results than trust-region POD when compared with high-fidelity model. Directions
of future research are finally outlined.
2. A POD reduced order unstructured mesh ocean modelling method for
moderate Reynolds number flows
• Herein a new approach to enhance the accuracy of a novel Proper
Orthogonal Decomposition (POD) model applied to moderate Reynolds
number flows (of the type typically encountered in ocean models) is
presented.
• This approach develops the POD model of Fang et al. [Fang, F., Pain, C.C.,
Navon, I.M., Piggott, M.D., Gorman, G.J., Allison, P., Goddard, A.J.H.,
2008. Reduced-order modelling of an adaptive mesh ocean model.
International Journal for Numerical Methods in Fluids] used in conjunction
with the Imperial College Ocean Model (ICOM), an adaptive, non-
hydrostatic finite element model.
• Both the velocity and vorticity results of the POD reduced order model
(ROM) exhibit an overall good agreement with those obtained from the full
model.
• The accuracy of the POD-Galerkin model with the use of adaptive meshes is
first evaluated using the Munk gyre flow test case with Reynolds numbers
ranging between 400 and 2000.
• POD models using the L2 norm become oscillatory when the Reynolds
number exceeds Re = 400.
• This is because the low order truncation of the POD basis inhibits generally
all the transfers between the large and the small (unresolved) scales of the
fluid flow.
• Accuracy is improved by using the H1 POD projector in preference to the
L2 POD projector. The POD bases are constructed by incorporating
gradients as well as function values in the H1 Sobolev norm.
• The accuracy of numerical results is further enhanced by increasing the
number of snapshots and POD bases.
• Error estimation was used to assess the effect of truncation (involved in
the POD-Galerkin approach) when adaptive meshes are used in conjunction with
POD/ROM.
• The RMSE of velocity results between the full model and POD-Galerkin
model is reduced by as much as 50% by using the H1 norm and increasing
the number of snapshots and POD bases.
3.A POD reduced-order 4D-Var adaptive mesh ocean modelling
Approach
• A novel proper orthogonal decomposition (POD) inverse model, developed
for an adaptive mesh ocean model (the Imperial College Ocean Model,
ICOM), is presented here.
• The new POD model is validated using the Munk gyre flow test case, where
it inverts for initial conditions.
• The optimized velocity fields exhibit overall good agreement with those
generated by the full model. The correlation between the inverted and the
true velocity is 80–98% over the majority of the domain.
• Error estimation was used to judge the quality of reduced-order adaptive
mesh models.
• The cost function is reduced by 20% of its original value, and further by
70% after the POD bases are updated.
• In this study, the reduced adjoint model is derived directly from the
discretized reduced forward model.
• The whole optimization procedure is undertaken completely in reduced
space. The computational cost for the four-dimensional variational (4D-Var)
data assimilation is significantly reduced (here a decrease of 70% in the test
case) by decreasing the dimensional size of the control space, in both the
forward and adjoint models.
• Computational efficiency is further enhanced since both the reduced forward
and adjoint models are constructed by a series of time-independent sub-
matrices. The reduced forward and adjoint models can be run repeatedly
with negligible computational costs.
• An adaptive POD 4D-Var is employed to update the POD bases as
minimization advances and loses control, thus adaptive updating of the POD
bases is necessary.
• Previously developed numerical approaches Fang et al. (Int. J. Numer. Meth.
Fluids 2008) are employed to accurately represent the geostrophic balance
and improve the efficiency of the POD simulation.
4. A POD goal-oriented error measure for mesh optimisation
• A novel approach for designing an error measure to guide an adaptive
meshing algorithm using a POD adjoint or goal-based method, is presented
here. In this work the multi-field error measure was applied to each field
type at each time level between two points in time where mesh adaptivity is
applied.
• The aim is to obtain a new mesh that can resolve all the fields at all time
levels, to an optimal (with respect to the functional) level of accuracy.
• These goal based methods solve both forward and adjoint equations to form
the overall error norms or metric tensors.
• Once metric tensors are obtained (the focus of this work) the tetrahedral
elements are then optimised so that they have unit element lengths and good
quality aspect ratios when measured with respect to the metric tensors.
Forward and adjoint evolutionary equations are solved to obtain forward and
adjoint solutions which together with their equation residuals are used to
form the error norms.
• This is the first attempt to apply the POD approach to form the overall error
indicators for optimizing meshes which can resolve all the fields during the
simulation period, to an optimal (with respect to a prescribed functional)
level of accuracy. The POD approach facilitates efficient integration
backwards in time and yields the sensitivity analyses necessary for the goal
based error estimates.
• The accuracy of both the primal and adjoint reduced models are thus
optimised (through the use of anisotropic mesh adaptivity) (for example,
here, the RMSE of velocity results between the POD results and the true
ones is reduced by half in the case of the inversion for initial conditions).
• The introduction of the POD approach in the error measure ensures that the
additional overhead associated with the use of finite elements coupled to a
mesh adaptivity algorithm is less than the overhead incurred from the use of
a uniform grid, composed of elements of the finest resolution required.
• Importantly, the goal functional for optimising meshes is consistent with that
for 4-D Var data assimilation.
• The error indicators developed here can also be used to implement a dual-
weighted POD method for order reduction and 4D-Var data assimilation in
future work.
3.2.1 Improved POD Bases
The POD provides the optimal basis for interpolating the input collection, but they are not
necessarily the best basis on which to build a dynamical system. Searching
for new bases for model reduction is an active research area. The primary emphasis of this
research will be to seek bases that are i.) well suited to ocean flows, e.g. are able to identify
dynamically relevant structures and ii.) have advantages when used within an optimization
setting, e.g. are robust to perturbations in model parameters and can be efficiently updated
when the flow evolves such that it is not well represented by the basis. The following are
potentially fruitful research directions.
Principal Interaction Pattern (PIP) Analysis The goal is to find a limited number
of structures in seemingly very complicated physical scenarios. These vary according to
nonlinear equations determining the interaction between the different structures. By minimizing
a suitably chosen error function, calculated by comparing a PIP model with observed
or synthetic data sets, both the structures and their interaction coefficients are determined
simultaneously. This may serve as a useful tool for identifying basic structures where POD
encounters difficulties in 3D.
Principal Interval Decomposition (PID) The principal interval decomposition
provides a natural complement to many reduced-basis techniques. The PID is a natural
way to select time intervals on which to perform basis selection. Although it can provide a
mechanism to build reduced-order models with specified error estimates, it usually does not
do this efficiently. Our interest in this approach lies in the promise of finding structures that
are dynamically important but only persist over short periods of time. Averaging methods
such as the POD will miss these structures since they do not persist for a long period of
time. This may lead to models that are better suited to predict complex phenomena, such
as the behavior of the system near bifurcation points.
3.2.2 Closure Modeling ( Iliescu, Borggaard)
• From the earliest stages of POD, it was recognized that a simple Galerkin
truncation will generally produce inaccurate results, even if the modes
retained capture most of the system's energy .
• Thus, closure modeling (i.e., modeling the effect of the discarded modes on
the modes retained in the system) has always played a central role in POD
reduced-order modeling (POD ROM) strategies.
• The closure modeling has developed into two main directions(that often
overlap):
• (i) improving the numerical stability and
(ii) improving the physical accuracy.
• The research proposed here is concerned with the latter. Although there are
several closure modeling strategies (e.g., the novel approach of Noack et al.
that uses a finite-time thermodynamics
formalism), our approach is different in that it addresses explicitly the following
well-known drawbacks of the POD Galerkin truncation:
• The proposed closure modeling strategies target: (i) a wide range of
Reynolds numbers, and (ii) time intervals longer than those over which
snapshots were collected. Both features are essential for the reduced-order
modeling data assimilation strategies relevant to realistic ocean flows.
Proposed Research:
• We will develop POD closure models based on ideas from two state-of the-
art LES modeling approaches and compare them with state-of-the-art closure
modeling strategies, such as those developed by Noack's group .
• POD Eddy Viscosity (POD-EV) models:
• Our approach builds on the pioneering work
in (see also) and employs the concept of energy cascade. That is, the effect
of the discarded POD modes is modeled by adding a dissipative term to the ROM,
which is equivalent to increasing the effective viscosity in the POD-ROM
The validity of the energy cascade concept in a POD setting was confirmed in a
recent numerical study .
This study, together with the long list of recent reports in which variations of the
simple mixing length eddy viscosity model were successfully used clearly
indicate that turbulence modeling ideas could be used in the development of
closure models for POD.
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