4. The Reduced Basis Method
4.1. The Reduced Basis Method for Finite Volume Solvers
Most of the publications in the RBM focus on for finite element discretised equation systems, cf. e.g. [PR07, GMNP07]. In [HO08] Haasdonk and Ohlberger present an approach for linear parabolic equation systems which are discretised by the FVM. To execute the basis generation for time dependent problems the POD-Greedy algorithm is introduced there for the first time, cf. Section 4.3. Before that the time was treated as an additional parameter and included into the “standard” Greedy algorithm [PR07].
For that we refer e.g. to the Ph.D. thesis by Grepl [Gre05].
Truth Approximation
The analytical solution to a parameterised partial differential equation system is called
“exact” solution. In many applications an analytical solution is not available, in partic-ular, for the present problem (3.1) to (3.5). The solution to the discretised system (3.10) is called “true” solution. The RBM diminishes the number of unknowns in comparison to the FV model via Galerkin projection. It is assumed that the non-measurable error between the “exact” and the “true” solution is very small in comparison to the error between the “reduced” and the “true” solution. In the present work we do not consider the discretisation error but the error between the “reduced” and the “true” solution.
Further we estimate the error when the FV solution takes the place of our “true” solu-tion and the RB solusolu-tion represents our “reduced“ solusolu-tion. If we refer to the RB model always the reduced order model (ROM) constructed by the RBM is meant.
Approach
We consider a parameterised partial differential equation system and are interested in the solution to many different parameters. The idea is to generate a basis for a convenient reduced space with a certain number of FV solutions and to construct RB solutions as a linear combination. More precisely: the acceptable parameter range is denoted byD. This parameter set is discretised in a proper way and denoted byDtrain. To compute the FV solution to these parameter sets, much effort is needed.1 As a consequence merely the FV solutions for “chosen” parameter sets are computed. With the information of
1Since we consider in the present work a one-dimensional equation system, the numerical effort is far away from a three-dimensional one, cf. Section 3.4.
these solutions we compute the so called “basis vectors”. When computing the RB solution to a parameter µ of the parameter set D, we assume that this solution is a linear combination of the basis vectors.
The computation of the RBM can be decomposed into a computationally expensive
“offline” and cheap “online” part. In the “offline” part the reduced basis is generated and so via the POD-Greedy algorithm selected FV solutions are computed. Further all computations which have to be done for all RB solutions is put into this part. In the
“online” part the RB solutions to the requested parameters are computed.
There are a few open issues: How to choose the parameters for computing the FV solutions for generating a basis of the RB space? How to ensure that the accuracy of the RB solution is “good enough”? Does an RB solution exist? Is it appropriate to assume that an RB solution is a linear combination of the basis vectors?
The Greedy algorithm is a search algorithm to choose the “right” parameters for the basis vectors (c.f. Algorithm 4.2): The error between the FV and the RB solution of the discretised parameter set is estimated/computed. The FV solution to the parameter where this error attains its maximum is added to the basis vector portfolio. The “new”
RB solution “knows” the FV solution at this parameter set. Therefore the error with respect to this parameter set is expected to be diminished significantly. This step is repeated until the given accuracy of the RB solution in comparison to the FV solution is reached on the discrete training set Dtrain.
For time dependent problems as the present problem, the POD-Greedy algorithm is used (c.f. Algorithm 4.3): The computed FV solution of the Greedy algorithm is reduced in time by the POD method, see e.g. [Vol12]. In this algorithm there are two loops. The
“outer” loop is similar to the Greedy algorithm: we add the snapshot matrix u(µ) ∈ RNx×Nt, where Nx ∈ N is the number of control volumes in spatial and Nt ∈ N is the number of discretisation points in time. For the chosen parameter µ∈Dtrain⊂RNP the error between the FV and the RB solution is the biggest. In the “inner” loop we apply the POD method to the snapshot matrix u(µ)∈RNx×Nt and get an Nx×ℓ-dimensional system. Then we estimate the projection error between the two matrices in the POD algorithm, which is described in more detail below.
The Greedy algorithm needs an a posteriori error estimator which estimates the error between the FV and RB solutions to avoid expensive computations of the FV solutions to the discretised parameter set Dtrain. An error estimator for linear problems is examined in [HO08]. We will give a sketch of it in Section 4.5.
When the RB procedure is applied to systems which have been discretised using the finite elements method the weak formulation of the problem has to be considered. To this end the problem has to be written in the following form: find au∈H
a(u, v;µ)=f(v;µ),∀v∈H, (4.1) where a(⋅,⋅;µ) is a symmetric, coercive, affine parameter dependent and continuous bilinear form, f is an affine parameter dependent linear form and H an appropriate Hilbert space. The strategy for finite element discretised problems can be found in [PR07] for example.
4.1. The Reduced Basis Method for Finite Volume Solvers
If the partial differential equation system is not in a “standard form” (4.1), i.e. linear and affine parameter dependent, it should be approximated so that the reduced computation gets efficient, for example with the empirical interpolation method. The idea of this method is to evaluate the nonlinearities at special points, the so called “magic points”
and approximate the nonlinearities with a linear function which is a linear combination of the functions evaluated at the “magic points”. We refer to [BNMP04], [DHO12] and to [GMNP07] for the treatment with non “standard” equations.
We follow the approach from [HO08]. A parameter dependent elliptic partial differential equation system which is discretised by the FV method can be formulated as an N -dimensional equation system: for µ∈D⊂RNP find uN(µ)∈RN such that
FN(uN(µ);µ)=0. (4.2)
Keep in mind that the discretised coupled elliptic-parabolic partial differential equation system can be written in the same way, cf. (3.10).
By XN we denote theN-dimensional space of the FV solutions. The coefficient vector uN(µ)∈RN represents the element uN(µ)∈XN, cf. Section 3.3.
The above equation (4.2) can now be formulated as
FN(uN(µ);µ)∶=A(µ)⋅uN(µ)+bN(µ)+GN(uN(µ);µ)=! 0,
where A(µ) ∈ Mat(R,N) is an N ×N-matrix which should represent the linear part of FN(⋅;µ)∶RN →R, bN ∈RN the inhomogeneous part and GN(⋅;µ)∶RN →RN the nonlinear part.
Forµ∈D the problem to determine a function uN(µ)∈XN satisfying FN(uN(µ);µ)=0,
is equivalent to the problem to find a function uN(µ)∈XN so that
⟨FN (uN(µ);µ), v⟩W =0,∀v∈XN.
Using the Galerkin projection we approximate the N-dimensional space XN by an N -dimensional one,XN. We assume that there is anN-dimensional basis Ξ=(ξ1, . . . , ξN)∈ RN×N of the spaceXN ⊂XN, withN ≪N and that the RB solution can be written as
uN(µ)=∑N
j=1
uNcoeff,j(µ)ξj =Ξ⋅ucoeff(µ)∈RN,
whereucoeff=(uNcoeff,1, . . . , uNcoeff,N)T ∈RNis the unknown coefficient for the basis vectors.
The evaluation of the following inner products determines the reduced equation system
as well as the RB solution: Letµ∈D be arbitrary, then
⟨FN(Ξ⋅ucoeff(µ);µ), ξj⟩W
= ⟨A(µ)⋅(∑N
n=1
ξnuNcoeff,n(µ))+bN(µ)+GN(∑N
n=1
ξnuNcoeff,n(µ);µ), ξj⟩
W
=ξjT ⋅W ⋅A(µ)⋅(∑N
n=1
ξnuNcoeff,n(µ))+ξjT ⋅W ⋅bN(µ)+ξjT ⋅W ⋅GN(∑N
n=1
ξnuNcoeff,n(µ);µ) summing up them-th entries of the weighted scalar product:
=∆x
N m∑=1
ΞTj,m⎛
⎝
N
p∑=1{A(µ)}m,p⋅(∑N
n=1
Ξp,nuNcoeff,n(µ))+{bN(µ)}m +{GN(∑N
n=1
ξnuNcoeff,n(µ);µ)}
m
) for all j∈{1, . . . , N},
where⟨⋅,⋅⟩W denotes the weighted scalar product inRN withW ∶=∆xdiag(ones(N,1),1) and vm denotes the m-th column of a vector v∈RN, m∈{1, . . . ,N}. Thus we have
ΞT ⋅W ⋅(A(µ)⋅(Ξ⋅ucoeff)+bN(µ)+GN(Ξ⋅ucoeff;µ))=∶FN(ucoeff). (4.3) Remark 4.1. To apply the Newton method we need an initial value. For time dependent systems we use the value of the previous time step. Hence to compute the coefficient vector to the second time step we need the coefficient vector of the first one at time step t1=0. The coefficient vector uNcoeff(t1;µ) is given by
uNcoeff(t1;µ)=ΞTW uN⋅,1(µ), (4.4) where uN⋅,1(µ) is the initial value for the FV system.
RBM Applied on a Coupled System
In the sequel it is shown how to apply the RBM to the present problem (3.10). Especially how to deal with the coupling of the concentration cand the potential φ.
The function F1N ∶ RN ×RN ×D → RN, (c, φ;µ) ↦ F1N(c, φ;µ) denotes equation (3.1a) with the corresponding boundary and interface conditions discretised with the FV method, analogously F2N ∶RN ×RN ×D →RN, (c, φ;µ)↦F2N (c, φ;µ)for the FV discretised equation (3.1b) with the corresponding boundary and interface conditions.
To apply the Greedy algorithm to a coupled problem first we need a common training set for cand φ. The training set is the discretised parameter set.
Dtrain={µ1, . . . , µNP}.
If we need the concentration to a certain parameter we also have to compute the electrical potential and the other way around. The parameter choice in the Greedy algorithm can differ for c and φ, so we have to compute two FV solutions in this case. Under the