Starting from an initial velocity set Im with {−1, 1} ∈ Im we are able to calculate
the m−dimensional solution and compare it with the true one by interpolation (Step 1). If ∆Im is smaller then a given ∆tol, we say that our solution is satis-
factorily accurate and the Greedy algorithm stops (Step 2). Otherwise we have to extend the velocity basis set by adding an element µ ∈ CIm to Im, (Step 8).
In this section we introduce a dimension reduction of Problem P (Im), (Step 4), paving the way for efficient calculable errors ∆µ, (Step 5), on which the update of
Im (Step 7) will be based on.
Optimal Control Problem P (I
µm)
For a given set Im and arbitrary µ ∈ CIm, we define the optimization Problem
P (Im
µ). The restriction system of coupled differential equation is now based on
the set Iµm, i.e.
TmµΨmµ = (Amµ + Σmµ − Km µ)(x)Ψ m µ = Q m µ, (5.12)
with given extension of operators (5.7) in the following way:
Let sk,i(x) = σs(x)s(x, νik), i, k = 1, . . . , m + 1 with µm+1 = µ. Then we notate
the extended operators
Amµ, Σmµ, Kmµ : Zm+1 → Zm+1, by
Amµ(x) = diag(µ1, . . . , µm, µ)∂x,
Σmµ(x) = σt(x) · Idm+1,
Kmµ(x) with Kmk,i = ωi· sk,i(x).
(5.13)
The extended optimal control problem P (Iµm) then reads min Ψm µ,Qmµ J (Ψmµ, Qmµ), subject to TmµΨmµ − Qm µ = 0, 0 ≤ Qmµ(x) ≤ qmax ∀ x ∈ Z. (P (Iµm))
5.3 Model Reduction and Update of Im 113
Model Reduction of (P (I
µm))
Let the set Im and the corresponding solution ((Qm)∗, (Ψm)∗) of P (Im) with ∆Im > ∆tol be given. Our objective is to add a velocity direction µ ∈ CIm to
our existing set Im such that ∆
Im decreases.
We would like to extend our current velocity basis set Im by an element ˜µ ∈ CIm, such that
∆Im ˜
µ ≤ ∆Iµm, ∀µ ∈ CI
m.
The direct error calculation of (5.1) would be very expensive since size of IK is large. Thus we derive for any µ ∈ CIm a reduced solution ( bΨm
µ)
∗ of P (Im
µ). This
is achieved by reducing the transport model dimension of (5.12).
Reduced Velocity Basis
The approach we present is based on the hypothesis, that for any µi ∈ Im the
corresponding state solution ψ∗i, part of solution (Ψm µ)
∗, differs just marginally
from ψ∗i, part of (Ψm)∗. Further we assume, that ψm+1 = ψµ can be expressed as linear combination of ψi, i = 1, . . . , m, i.e. there exist γi ∈ R, i = 1, . . . , m, such
that ψµ= m X i=1 γiψi∗.
Based on this assumption we define a basis transformation matrix.
Definition 5.5 (Basis). Let ((Ψm)∗, (Qm)∗) be the optimal solution of P (Im). Further let each component ψk be normalized, i.e. kψkkZ = 1. The basis transfor-
mation Bm corresponding to the optimal solution of P (Im) is defined by
Bm:= ψ1 · · · 0 0 · · · 0 . .. ... ψm 0 · · · 0 0 · · · 0 ψe1 · · · ψem ∈ Z(m+1)×2m.
The components eψ1, . . . , eψlare an orthonormalized system of ψ1, · · · , ψm.
The defined basis transformation matrix will contain of completely new ele- ments in each iteration step since ψi are the solution of the recently calculated
114 § 5 Reduced Velocity Method
drawback that the information of the earlier elements will be lost in the process of calculating the error estimator ∆µ in the next step. Therefore we define a second
basis transformation matrix in order keep all informations about existing states. Definition 5.6 (Hierarchical Basis). Let ((Ψm
Im)∗, (QmIm)∗) be the optimal solution
of P (Im). Further let each component ψkIm be normalized, i.e. kψkImkZ = 1. The
basis transformation Bm
h corresponding to the optimal solution of P (Im) is then
defined by Bm h:= ψ1I2 0 ψ1I3 · · · 0 ψ1Im · · · 0 0 · · · 0 0 ψ2I2 0 . .. 0 0 . .. 0 0 ... 0 0 0 0 . .. ψI(m−1)(m−1) 0 . .. 0 0 ... 0 0 0 0 · · · 0 0 · · · ψmIm 0 · · · 0 0 0 0 · · · 0 0 · · · 0 ψeI m 1 · · · ψeI m m .
The components eψ1, . . . , eψlare again an orthonormalized system of ψ1, · · · , ψm. It
holds Bhm ∈ Z(m+1)×M, where M = m(m+1)
2 + (m − 1).
We define further the basis Bh,jm, where 0 ≤ j ≤ (m − 2) holds. Here, we take just the last j solution (Ψm
Ik)
∗ with k = m, . . . , m − j into account.
Note first of all, that we started already with a set of two elements as suggested by the choice of the quadrature rule. It also holds Bm
h,0 = Bm. Further note, that
we still assume the solution of the the new velocity solution bψµ to be a linear
combination just of the last solution elements. The choice of basis transformation matrix Bhm adds a second level of hierarchy of subspaces to the Greedy algorithm. The dimension of the space for each velocity mode bψi is increasing by one in each
iteration step and thus hierarchically subspaces are constructed for each bψi.
Remark 5.7 (Stability). The basis orthonormalization in Definition 5.5 and 5.6 is necessary to gain stability of the numerical solution. Unfortunately, we are orthonormalizing with respect to L2(Z). It would even be more efficient if we
would orthonormalize with respect to the norm induced by the operator Tmµ. But since this one changes for every µ, we would have to reorthogonolize the basis for each reduction.
Reduction of (P (I
µm))
For a set Im
µ we define the operator Rmµ by multiplication of Bm and its transpose
with the operator Tmµ Rm
5.3 Model Reduction and Update of Im 115
Thus, the reduced transport model of system (5.12) is then given for coefficients γµ, ρµ ∈ R2m by
Rm
µγµ = ρµ. (5.14)
Note, that the reduced transport model as well as all following statements can be achieved by applying Bm
h except, that the dimension changes from 2m to M .
This model is now independent of the space dimension Z and depends only on the chosen velocity space discretization (m + 1) and in particular it differs for every µ. If the matrix Rmµ ∈ R2m×2m is invertible then for any given right hand sight ρµ there exist a unique solution γµ. Applying the coefficient vector γµ and
ρµ on the basismatrix Bm we obtain bΨmµ = Bmγµ and bQmµ = Bmρµ, both elements
of Z(m+1). We define the cost functional related to transformation basis Bm by
jBm(γµ, ρµ) := J (Bmγµ, Bmρµ) = J ( bΨmµ, bQmµ). (5.15)
In this notation we set up the reduced optimization problem for a set Im µ, min γµ,ρµ jBm(γµ, ρµ), subject to Rm µγµ = ρµ. ( bP (Im µ))
Note that ( bP (Iµm)) is a finite dimensional optimization problem.
Corollary 5.8. Let Rmµ be invertible. Since J is convex, there exists a unique optimal solution (γµ∗, ρ∗µ) ∈ (R2m)2.
Remark 5.9 (Box Constraints). Unfortunately, the constraint 0 ≤ bQmµ ≤ qmax
has to be neglected since we loose positivity of Bm
i in the orthogonalisation process.
We could keep the possibility by keeping eψi positive and claim then 0 ≤ ρ ≤ qmax
which would cause stability loss.
Let Im be given, then we obtain for every µ ∈ CIm a different reduced
optimization problem (P (Iµm)). The optimal solution of ( bP (Iµm)) is given by (γµ∗, ρ∗µ) ∈ (R2m)2. The element b Qm µ ∗ ,Ψbmµ ∗ ∈ Z(m+1)2 is called reduced optimal solution of the extended OCP (P (Im
µ)).
In the following, a hat indicates a solution related to the reduced optimal solution ( bP (Iµm))
116 § 5 Reduced Velocity Method
∆
µError Calculation
In order to safe computational time and to avoid unnecessary computations, the error calculation (Step 5 in the Greedy Algorithm) is based on the reduced optimal solution of bP (Im
µ ). The main question is now, which norm has to be applied to
calculate the error ∆µbetween the reduced optimal solution and the true one. This
error has to be defined with respect of the mismatch of dimensions and is part of this subsection. We present several ways to compute the error ∆µ on which the
decision for the new element µ of the current velocity basis set Im will be based on.
True Solution
We apply the presented Greedy algorithm in order to avoid solving the high di- mensional OCP P (IK) in general. Unfortunately is this work a feasible study in order to determine, if further investigation in error estimators is worthwhile.
For a given dictionary IK, we set up the optimal control problem P (IK) and calculate the true control (QK)∗ ∈ ZK. Its optimal state (ΨK)∗ ∈ ZK leads to
an optimal Dose D∗ = DK (ΨK)∗ and also optimal objective functional value J∗ = J (ΨK)∗, (QK)∗.
Unfortunately, we will also have to solve at least once the true transport model TKΨ = Q for given Q ∈ ZK to obtain the accurate dose which is actually delivered to the patient. But since in a general optimization process the transport model has to be solved many times, we still safe computational time.
Reduced Solution
For any reduced velocity set Im and µ ∈ CIm, we set up the reduced model ( bP (Im
µ)). We obtain the optimal solution (γ ∗ µ, ρ
∗
µ) and by applying the reduced
velocity basis Bm we yield the reduced control and state
b
Qµ:= Bmρ∗µ and Ψbµ := Bmγµ∗,
respectively. The reduced dose is given by bDµ:= DmΨbµ and the reduced objective functional value consequently by bJµ := J ( bQµ, bΨµ) = j(γµ∗, ρ
∗ µ).
In order to compare this reduced solution with the true solution we have to investigate some effort to negotiate the dimension mismatch.
5.3 Model Reduction and Update of Im 117
Control Error
Since bQµ ∈ Zm, we apply interpolation operators (5.10) and obtain the interpo-
lated control
b
Qintµ := PKint( bQµ). (5.16)
Thus, we can define the first error calculator,
∆1µ:= k bQintµ − (QK)∗kZK.
A second way to get over the dimension mismatch is to evaluate the error just in the reduced velocity basis set dimension. Since Im ⊂ IK holds, we might just
evaluate the true solution at the actual basis set points. Therefore we define PIm
by restriction on the reduced basis set Im,
PIm : ZK→ Zm : (QK)∗ 7→ (QKIm)∗ = PIm(QK)∗,
and define the low dimensional error
∆2µ= k bQµ− (QKIm)∗kZm.
State Error
To obtain a state solution based on the reduced state solution bΨµin the true space,
we apply the interpolation operator and obtain b
Ψintµ := PKint( bΨµ).
On the other hand, we compute out of an interpolated control (5.16) the state by the inverse operator of TK. This leads to a second reduced state
b
Ψintµ,K := (TK)−1Qbintµ . These two reduced states are used to calculate the errors
∆3µ= k bΨintµ − (ΨK)∗k
ZK and ∆4µ= k bΨµ,Kint − (ΨK)∗kZK.
Further, we obtain low dimensional errors by applying the restriction operator (ΨKIm)∗ := PIm( bΨK)∗ or applying the inverse operator of Tm and obtain,
(ΨmIm)∗ := (Tm)−1(QKIm)∗.
Thus, we define the low dimensional state errors,
118 § 5 Reduced Velocity Method
Dose Error
One further value of interest is the dose. To calculate the reduced dose we apply the dose operator (5.9) to all interpolated and reduced states. This procedure leads to four dose distributions,
b
Dµint= DK( bΨintµ ) and Dbintµ,K= DK( bΨintµ,K) and DIKm = Dm (Ψ K Im) ∗ and DImm = Dm((ΨmIm) ∗ ) .
These reduced dose distributions bDµ are compared with the true dose D∗K and
therefore the errors are defined by
∆7µ= k bDintµ − D∗KkZ and ∆8µ = k bDintµ,K− D∗KkZ and
∆9µ= k bDµ− DIKmkZ and ∆10µ = k bDµ− DmImkZ.
Objective Functional Error
Since J depends on control and state, one could think in first instance about combining all reduced controls and states, but we only pick states with their cor- responding control, i.e. controls, that lead to these states. Therefore we obtain for the following errors for the reduced objective functional,
∆11µ = |J ( bΨintµ , bQintµ,m) − JK∗|, ∆12µ = |J ( bΨintµ,K, bQintµ,m) − JK∗|.
Error Estimator ∆
µAfter introducing numerous ways to calculate the error, we combine these errors to one final error calculator. This one is going to be applied in the Greedy algorithm to determine the element µ ∈ CIK added to the current reduced velocity set. We define ∆µ by ∆µ= 12 X i=1 b βi∆iµ, βbi ≥ 0 (5.17)