parameter estimation within a kinematic
TRANSCRIPT
Parameter Estimation within a Kinematic
Magnetohydrodynamics Framework
Evan Rajbhandari1, Nathan Gibson2, and C. Rigel Woodside3
1Department of Mathematics, Oregon State University, [email protected] andNational Energy and Technology Lab, Albany, OR
2Department of Mathematics, Oregon State University3National Energy and Technology Lab, Albany, OR
Nov 1, 2021
Abstract. We discuss the well-posedness of the forward problem for the magnetohydrodynamicsystem with the inclusion of the ion-slip parameter. We also demonstrate the convergence of aparameter estimation scheme. Focusing on power-generation, we implement and the validate anumerical model with an engineering multi-physics software, COMSOL, using ideal-power equa-tions. We conclude with a demonstration of the parameter estimation scheme, demonstratingthe degree of convergence as a function of noise-level, with multiple runs of randomly generatednoise.
1 Introduction
Magnetohydrodynamics (MHD) is the study of an electrically-conductive medium flowing througha magnetic field [20]. It is a multi-physics problem, governing the behavior of fluid flow, elec-tric fields and currents, magnetic fields, and their interactions. MHD has applications in manydifferent areas of study, such as astrophysics [24], medicine [17] and power generation [20]. Inthis paper, we focus on only power-generation. By artificially generating a plasma and apply-ing a magnetic field, an electric field and current arise. Using electrodes and placing loads onthe channel results in an extraction of power [12]. This particular type of generator is calledan MHD generator. Maxwell’s equations, fluid dynamic equations and the generalized Ohm’slaw govern this multi-physics problem. Coupling these results in a non-linear system of partialdifferential equations. Assuming a steady-state problem and assigning a fluid flow leads to aparticular version of the MHD equations, denoted the kinematic MHD system [18].
Of these equations, one of great import is the generalized Ohm’s law. This equation can befurther simplified under neglibility assumptions of the Hall and ion slip parameters. Modelsthat neglect one or both of these parameters can be seen in [20] [7] and [11]. However, as theseparameters play a crucial role in determining the theroetical power output of an MHD generator[20], the theoretical power outputs are exaggerated when they are neglected. Understandingthis exaggeration is crucial for optimal design and operation of an MHD generator, and thus isexplored in the following work.
The Hall parameter, which is the most often included of the two, accounts for the electron-driftfrom the fluid flow and results in a tilting of the electric field. A similar but almost alwaysneglected term, the ion-slip parameter, accounts for the ion-drift from the fluid flow, and will beshown to lower the magnitude of the electric potential, and thus the available power within the
generator. The ion-slip effect has been studied in general on the MHD equations [9], but notspecific to MHD generators. With a lack of working theory specific to this situation, it is notknown what the theoretical impact of non-negligible ion mobility will be on power generation.Thus, the following theory is presented, establishing in a theoretical Faraday MHD generatorthe impact of the ideal ion-slip parameter. Furthermore, a numerical model using COMSOL [1]is established. Agreement between the ideal power and numerical power is seen when ion-slipis negligible, validating the model. A new theoretical ideal power for non-negligible ion-slip isthen presented. Following this, the same COMSOL model is compared to these ideal equations.Agreement is shown between the new ideal-power equation and the numerical model undervarious conditions. With the forward problem established, we move onto the inverse problem.
Optimization of the operation of MHD generators can be done at run-time, by changing the angleof connnected electrodes, as well as the load being placed on the channel. It has been understoodthat under working conditions including a non-zero Hall parameter, there is an optimal angleand load to be placed on the generator, resulting in peak efficiency of power extraction [20].However, the state parameters of the system (e.g. fluid flow, conductivity, electron-mobility,and ion-mobility), cannot be directly measured. They must be recovered from measurements ofthe electric potential and current. Therefore, in the real-time optimization of the generator, onemust be able to reliably recover these parameters. Thus, the second portion of this paper dealswith the theoretical problem and numerical implementation of parameter estimation. Althoughthis concept has been explored previously with regards to many other systems, [23], and even theMHD system itself, [19], numerical parameter estimation has not yet been appled to the MHDgenerator system. We utilize theory from Banks [3] to analytically determine when parameterrecovery can be expected. We then implement a parameter estimation scheme using tools fromMatlab [14] to perform the recovery under various noise-levels, showcasing the robustness of ourmodel.
2 Model Formulation
We begin with the governing equations for our application: Maxwell’s equations, coupled withthe generalized Ohm’s law, as given in Rosa [20] section 2.7. First, for simplicity, we prescribethe fluid flow and assume a steady-state system, which allows the reduction of the completeMHD system to the kinematic MHD system. We also assume that all other state parameters:the conductivity, σ, electron-mobility, µe, and ion-mobility, µi are prescribed. Furthermore, weassume that the applied magnetic field, B, is given, and that this field dominates the inducedfield [4]. Let D ⊂ R3 denote the spatial domain, open with compact closure, and with boundarydenoted as ∂D. Then the kinematic MHD system for J and E, the electric current density andelectric field respectively, is described by
∇×E = 0, on D, (1a)
J = σ(E + u×B) +βe||B||
(J×B) +βi||B||2
((J×B)×B
), on D, (1b)
∇ · J = 0, on D. (1c)
Here, βe and βi, the Hall parameter and ion-slip parameter respectively, are defined as
βe(x) = µe(x)||B(x)||l2 , and βi(x) = µe(x)µi(x)||B(x)||2l2 .
We complete the system of equations with appropriate boundary conditions for power generation,perfectly electrically insulating boundary conditions. For J, we have
J · n = 0, on ∂D, (2a)
while E satisfiesE× n = 0, on ∂D. (2b)
Here, n represents a vector normal to ∂D.
With some algebra, the generalized Ohm’s Law can be rewritten in an explicit form, where theright-hand-side depends on E as the only unknown. A simpler form is presented in Rosa [20]with the Hall parameter included, but lacking the ion-slip parameter. The following sectiondemonstrates how to include βi.
An Explicit Generalized Ohm’s Law
To rewrite (1b) in an explicit form, first consider that the cross product can be written as amatrix applied to a vector, e.g.
J×B =
0 Bz −By
−Bz 0 Bx
By −Bx 0
JxJyJz
= [B]×J,
where [B]× denotes the cross-product matrix operator. Clearly, [B]2×J denotes the vector tripleproduct between J,B,B, as [B]2×J = [B]×(J ×B) = (J ×B) ×B. Using this, Ohm’s law canbe rewritten as
J = σ(E + u×B) + µe(J×B) + µi((J×B)
= σ(E + u×B) +βe||B||
[B]×J +βi||B||2
[B]2×J
(I − βe||B||
[B]× −βi||B||2
[B]2×)J = σ(E + u×B).
Noting that the determinant of (I − βe||B|| [B]× − βi
||B||2 [B]2×) is given by 1 + β2i + β2
e + 2βi and
since βi > 0 by definition, the LHS is always invertible, implying that
J = (I − βe||B||
[B]× −βi||B||2
[B]2×)−1σ(E + u×B)
Defining σ as the conductivity tensor σ = (I − βe||B|| [B]× − βi
||B||2 [B]2×)−1σ yields the following
explicit generalized Ohm’s law,Ji = σ(E + u×B). (3)
We now turn to using this to manipulate (1) into a mixed-Poisson form to show well-posedness.
3 Forward Problem: Well-Posedness
In the following section, we will establish the well-posedness of the kinematic MHD system byusing the Babushka-Brezzi-Kovalevskaya (BBK) theorem.
Theorem 1 (BBK theorem). Let A : V → V ′ and B : V → W ′ be continuous operators fromthe Hilbert spaces V,W to their duals. In addition, assume
• A is V-coercive on V , i.e. ∃α > 0 such that
Av(v) ≥ α||v||2V , v ∈ V,
• B obeys the following inf sup condition: ∃β > 0
infq∈W
supv∈V
|Bv(q)|||v||V ||q||W
≥ β.
Given these conditions, then ∀f ∈ V ′, ∀g ∈ W ′ there exists a unique pair v, w ∈ V ×W suchthat
Av +B′w = f ∈ V ′, (4a)
Bv = g ∈W ′, (4b)
which obey the following a priori estimates:
||v||V ≤1
α
(||f ||V ′ +
1
β(||A||L(V,V ′) + α)||g||W ′
), (5a)
||w||W ≤1
β
(||f ||V ′ + ||A||L(V,V ′)||g||W ′
). (5b)
A proof of BBK can be found in Boffi, Brezzi, and Fortin [8]1. To apply BBK directly, we needto convert the given system to a weak mixed-Poisson system, (10), and then to an equivalentdual operator form.
3.1 Mixed-Poisson Strong Form
Similar to the work shown in Mcgregor [15] and Rosa [20], define Ji to be
Ji = σE = σ∇V, (6)
which, coupled with the divergence-free condition, (1c), implies
0 = ∇ · J = ∇ ·(Ji + σ(u×B)
).
This of course implies∇ · Ji = −∇ ·
(σ(u×B)
).
We also need another form of E to transform (1) to a mixed-Poisson form. From (1a), and theassumption that D is a bounded open domain, it follows from Stoke’s theorem [22] that
E = ∇V. (7)
Substituting the definitions of Ji and V ((6) and (7) respectively) into (3) and combining the re-sulting equation with the divergence condition (3.1) yields the following mixed-Poisson systemm
σ−1Ji −∇V = 0, (8a)
−∇ · Ji = ∇ ·(σ(u×B)
). (8b)
Rather then solving (1) for J,E, we solve (8) for Ji,V. Using the definitions of Ji,V, we havethat J,E are described by
J = Ji + σ(u×B), (9a)
E = ∇V. (9b)
We now show that (8) is well-posed, by converting to an operator form and applying the BBKtheorem.
1Corollary 4.2.1.
3.2 Weak and Operator Forms
The perfectly-electrically insulating boundary conditions imply that we seek our voltage V inthe subspace of H1(D) defined by
V ∈W (D) := W 1,20 (D) =
f ∈ H1(D) : T (f) = 0
,
where T (f) is the trace of f on D. These same boundary conditions imply we seek Ji ∈ V (D),defined as
Ji ∈ V (D) :=
f ∈ (L2(D))3 : f · n = −(σ(u×B)
)· n on ∂D
.
Multiplying (8) by test functions in the appropriate spaces and integrating, we arrive at theweak form of the MHD equations, given as∫
Dσ−1Ji · φ−
∫D∇V · φ = 0 ∀φ ∈ V (D), (10a)
−∫D
Ji · ∇ψ =
∫D
(σu×B)∇ · ψ ∀ψ ∈W (D). (10b)
We now wish to define an operator form of (10). Thus, define the mapping from V → V ′ by
A(F)() :=
∫D
(σ−1F) · , (11)
and define the mapping from V →W ′ by
B(G)() := −∫D
G · ∇ . (12)
Note now that the dual of this operator is given by B′ : W → V ′,
B′(H)() = −∫D∇H · .
Using these operators, (10) can be written as(A(Ji) + B′(V)
)(φ) = 0 ∀φ ∈ V (D), (13a)
B(Ji)(ψ) =
∫D
(σu×B
)· ∇ψ ∀ψ ∈W (D). (13b)
For ease of notation, denote the operator g ∈W (D)→ R by
g(ω) =
∫D
(σu×B
)· ∇ω.
Then (13) is equivalent toA(Ji) + B′(V) = 0 ∈ V ′(D), (14a)
B(Ji) = g ∈W ′(D). (14b)
Clearly, (14) is equivalent to (10). We denote (14) the operator form.
3.3 Properties of the dual operators A,B
As can be seen in the assumptions, to apply Theorem 1, certain properties of the mixed-Poissonoperators must be established. The following lemmas give show that the operators A,B havethese necessary properties. Note that the domain of the functional spaces is assumed to be thegiven D ⊂ R3, open with compact closure, unless specifically stated otherwise.
Lemma 1. If B ∈ (L2(D))3, σ, βe and βi positive, bounded, real-valued functions on D, thenthe operator A is coercive and continuous.
Proof. Coercivity: A is coercive if and only if ∃α > 0 such that
A(F)(F) ≥ α||F||V ∀F ∈ V.
Let F ∈ V and consider that
A(F)(F) =
∫Dσ−1F · F
=
∫D
1
σ
(I − βe||B||l2
[B]× −βi||B||2
l2[B]2×
)F · F
=
∫D
1
σ
(F · F− βe
||B||l2(F×B) · F− βi
||B||2l2
((F×B)×B
)· F).
Given that σ is essentially positive and bounded, the function 1σ is also essentially positive and
bounded, and achieves a minimum value of 1||σ||L∞(D)
implying
A(F)(F) ≥ 1
||σ||L∞(D)
∫D
(F · F− βe
||B||l2(F×B) · F− βi
||B||2l2
((F×B)×B
)· F)
=1
||σ||L∞(D)
∫D
(F · F− βi
||B||2l2
((F×B)×B
)· F),
as (F × B) · F = 0 almost everywhere. Using properties of triple vector products and dotproducts, we have
1
||σ||L∞(D)
∫D
(F · F−
[ βi||B||2
l2
((B · F)B− (B ·B)F
)· F])
=1
||σ||L∞(D)
∫D
(F · F− βi
||B||2l2
((B · F)(B · F)− (B ·B)(F · F
))=
1
||σ||L∞(D)
∫D
(F · F− βi
||B||2l2
((B ·B)(F · F) cos2(θ)− (B ·B)(F · F
))=
1
||σ||L∞(D)
∫D
(F · F)(
1− βi||B||2
l2(B ·B)
(cos2(θ)− 1
))=
1
||σ||L∞(D)
∫D
(F · F)(
1 +βi||B||2
l2(B ·B) sin2(θ)
)=
1
||σ||L∞(D)
∫D
(F · F)(
1 + βi sin2(θ)).
Thus, finally
A(F)(F) ≥ 1
||σ||L∞(D)||F||V > 0.
We show continuity of A by showing that it is bounded. To see boundedness of A, it remainsto be seen that
|A(f)(g)| < c||f||V ||g||V ,
for some c ∈ R. Thus, let f,g ∈ V and consider the following.
|A(f)(g)| =∣∣∣ ∫
Dσ−1f · g
∣∣∣=∣∣∣ ∫
D
1
σ
(I − βe||B||l2
[B]× −βi||B||l2
[B]2×
)f · g
∣∣∣=∣∣∣ ∫
D
1
σ
(f · g− βe
||B||l2(f×B) · g− βi
||B||2l2
((f×B)×B
)· g) ∣∣∣
≤∫D
∣∣∣ 1σ
∣∣∣∣∣∣(f · g− βe||B||l2
(f×B) · g− βi||B||2
l2
((f×B)×B
)· g) ∣∣∣
≤∫D
∣∣∣ 1σ
∣∣∣(|f · g|+ ∣∣∣ βe||B||l2
(f×B) · g∣∣∣+∣∣∣ βi||B||2
l2
((f×B)×B
)· g∣∣∣)
≤ 1
ess inf σ
(∫D|f · g|︸ ︷︷ ︸I1
+
∫D
βe||B||l2
|(f×B) · g|︸ ︷︷ ︸I2
+
∫D
βi||B||2
l2|((f×B)×B
)· g|︸ ︷︷ ︸
I3
).
Rewriting the integrals I1, I2, I3 in terms of ||f||V ||g||V will yield the desired result. Considerfirst I1.
I1 =
∫D|f · g| ≤ ||f||V ||g||V by Cauchy-Schwartz.
Now, consider I2. First, note that ||(f×B)||l2 ≤ ||f||l2 ||B||l2 . As well, note that
|u · v| ≤ ||u||l2 ||v||l2
by Cauchy-Schwartz. Applying these two ideas to I2 yields
I2 =
∫D
βe||B||l2
|(f×B) · g|
≤∫D
βe||B||l2
||(f×B)||l2 ||g||l2 by Cauchy-Schwarz
≤∫D
βe||B||l2
||f||l2 ||B||l2 ||g||l2
= ||βe||L∞(D)
∫D||f||l2 ||g||l2
≤ ||βe||L∞(D)||f||V ||g||V by Cauchy-Schwartz.
By a similar argument, we have for I3
I3 =
∫D
βi||B||2
l2
∣∣∣((f×B)×B)· g∣∣∣
≤∫D
βi||B||2
l2||(f×B)×B||l2 ||g||l2
≤∫D
βi||B||2
l2||f||l2 ||B||2l2 ||g||l2
= ||βi||L∞(D)
∫D||f||l2 ||g||l2
≤ ||βi||L∞(D)||f||V ||g||V by Cauchy-Schwartz.
Combining all three inequalities yields
A(f)(g) ≤(
1 + ||βe||L∞(D) + ||βi||L∞(D)
)||f||V ||g||V ,
and as A is both linear and bounded, we have that A(f)() is continuous on V .
To see a similar result for B, an inequality that bounds the norm of a function by the norm ofits derivative must be utilized. The Poincare Inequality does so, and is thus presented next.
Lemma 2 (Poincare Inequality). Let 1 ≤ p < ∞ and let Ω be bounded on at least one side.Then ∃C > 0, depending only on p,Ω, such that ∀u ∈W 1,p
0 (Ω) where u is a zero-trace function,the following holds
||u||Lp(Ω) ≤ C||∇u||Lp(Ω).
This inequality is well-known, and complete proofs are presented in a variety of textbooks. Onesuch book is Rudin [22]. Now, to give the desired properties of B.
Lemma 3. The operator B : V →W ′ is continuous and obeys the following inf-sup condition
infg∈W
supf∈V
|B(f)(g)|||f||V ||g||W
≥ β > 0.
Proof. Fix g ∈W. This immediately implies that ∇g ∈ (L2(D))3, by definition of W . Therefore,
supf∈V|B(f)(g)|||f||V ||g||W
≥ |B(∇g)(g)|||∇g||L2(D)||g||W
=
∣∣∣ ∫D∇g · ∇g∣∣∣( ∫D∇g · ∇g
)1/2(||g||2
L2(D)+ ||∇g||2
L2(D)
)1/2
=||∇g||L2(D)(
||g||2L2(D)
+ ||∇g||2L2(D)
)1/2
≥||∇g||L2(D)(
(1 + Cp.f.)||∇g||2L2(D)
)1/2by Poincare’s Inequality,
=||∇g||L2(D)(
1 + Cp.f.
)1/2||∇g||L2(D)
=1
(1 + Cp.f.)1/2.
Here, Cp.f. represents the constant from the Poincare inequality, which has no dependence on g.This immediately implies that the supremum is bounded below independent of the choice of g,i.e.
infg∈W
supf∈V
|B(f)(g)|||f||V ||g||W
≥ 1
(1 + Cp.f.)1/2> 0,
and thus the inf-sup condition holds. To see continuity, recall that B is linear, and thus bound-edness implies continuity. Fix f ∈ V, g ∈W , and consider
|B(f)(g)|2 =∣∣∣ ∫
Df · ∇g
∣∣∣2≤∫D|f · ∇g|2
≤∫D||f||2l2 ||∇g||
2l2
≤ ||f||2V ||∇g||2L2(D)
≤ ||f||2V (||∇g||2L2(D) + ||g||2L2(D))
= ||f||2V ||g||2W .
Taking the square-root of both sides implies that B(f)(g) is bounded by the norms of f, g, whichin turn implies continuity.
With these properties of the operators established, BBK can now be applied.
3.4 Existence and Uniqueness of Solutions
The operators A, defined in eqn. (11), and B, defined in eqn. (12), have been shown to havethe desired properties to apply BBK in Lemmas 1 and 3. This leads to the first of three maintheorems, the existence and uniqueness of solutions to (14), which in turn implies existence anduniqueness of solutions to (10).
Theorem 2. Given u,B ∈ (L2(D))3, bounded, σ, βe and βi positive, bounded, and real-valuedon D, there exist unique solutions, Ji,V, to
A(Ji) + B′(V) = f ∈ V ′, (15a)
B(Ji) = g ∈W ′, (15b)
which obey the following a priori estimates:
||Ji||V ≤1
α||f ||V ′ +
1
αβ
(||A||L(V,V ′) + α
)||g||W ′ , (16)
||V||W ≤1
β
(||f ||V ′ + ||A||L(V,V ′)||g||V ′
), (17)
where α is the coercive constant for A and β is the bounding constant for B, i.e.,α = 1 + ||βe||L∞(D) + ||βi||L∞(D) and β = (1 + Cp.f.)
1/2.
Proof. Let A and B be defined as above. Then by Lemmas 1 and 3, A is coercive, B obeysthe inf sup condition, and both are continuous. Applying BBK implies both existence anduniqueness of the solutions Ji,V in their respective spaces. The estimates follow from BBK aswell.
Moreover, applying Theorem 2 when f = 0, g() =∫D∇ · σ(u ×B) gives that there exists a
pair of unique solutions to (14), in turn implying existence and uniqueness of solutions to (10).We now show that not only the solutions, Ji,V, to (10) exist uniquely, but that the solutionsdepend continuously upon the parameters, namely u,B, σ, βe, and βi.
3.5 Continuous Dependence
In the following section, continuous dependence of the solutions on a set of parameters is estab-lished. We show that perturbing the operators only causes small changes in the solutions, andthe difference of the perturbed and unperturbed operator can be bounded by some constant.Kato’s theorem, presented below, gives a useful bound for perturbing linear operators. First,however, we have a definition of a bounding operator.
Definition: 3.5.1. Let V,W be hilbert spaces, and let M be a linear mapping between them.Then M is bounding if there exists a M∗ > 0 such that
||Mv||W ≥M∗||v||V ∀v ∈ V. (18)
Thus, a bounding operator is an injective operator with a continuous inverse [8]. With thisdefinition in mind, we state Kato’s theorem.
Lemma 4 (Kato’s Theorem). Let V,W be Hilbert spaces, and let T1 and T2 be linear operatorsfrom V to W . If T1 is bounding, then there exists ε0 > 0 such that for all ε ∈ R with |ε| ≤ ε0,the perturbed operator T1 + εT2 is also bounding, and we have moreover
||T−11 − (T1 + εT2)−1||L(V,W ) ≤ C|ε|,
with C depending on ε0 but independent of ε.
A proof is presented in Fortin, [8]. Utilizing this theorem, it is now shown that the solutionsJi,V depend continuously on the parameter set.
Theorem 3. Solutions to (10) depend continuously on σ, βe and βi.
Proof. Fix u,B ∈ (L2(D))3, bounded, and let σ, βe, βi be positive and bounded functions on D.We show that the theorem holds for βe only, as a similar proof follows for βi, σ. Thus, considera small pertubation function to βe given by δβe such that βe + δβe ≥ 0 on D and consider thefollowing perturbed system:∫
D
1
σ
(I − (βe + δβe)[B]× − βi[B]2×
)J′i · φ−
∫D∇V ′ · φ = 0 ∀φ ∈ L2(D), (19a)
∫D
J′i · ∇ψ = −∫D∇ ·
(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1(u×B)ψ
)∀ψ ∈W. (19b)
Here J′i,V ′ is the unique solution as guaranteed by Theorem 2. Let Ji,V be the solution to theunperturbed system, i.e the solution to (10) and let δJi = J′i − Ji and δV = V ′ − V denote thedifferences between the solutions to the perturbed and unperturbed system. Subtracting (10)from (19) yields
0 =
(∫D
1
σ
(I − (βe + δβe)[B]× − βi[B]2×
)J′i · φ−
∫D
1
σ
(I − βe[B]× − βi[B]2×
)Ji · φ
)
−(∫
D∇V ′ · φ−
∫D∇V · φ
)=
∫D
1
σ
[(I − (βe + δβe)[B]× − βi[B]2×
)J′i −
(I − βe[B]× − βi[B]2×
)Ji
]· φ−
∫D
[∇V ′ −∇V
]· φ
=
∫D
1
σ
[(I − βe[B]× − βi[B]2×
)(J′i − Ji)− δβe[B]×J′i
]· φ−
∫D
[∇(V ′ − V)
]· φ
=
∫D
1
σ
[(I − βe[B]× − βi[B]2×
)(δJ)− δβe[B]×J′i
]· φ−
∫D∇δV · φ
=
∫Dσ−1(δJ) · φ−
∫D∇δV · φ−
∫D
1
σ(δβe[B]×J′i) · φ.
Rewriting this with the operators as defined in Section 3.2 yields
A(δJ)(φ) +B′(δV)(φ) =
∫D
1
σ(δβe[B]×J′i) · φ. (20)
Similarly, for the second equation, we have:∫D
J′i · ∇ψ −∫D
Ji · ∇ψ = −∫D∇ ·
(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1(u×B)ψ
)
+
∫D∇ ·(σ(u×B)ψ
)
∫DδJi · ∇ψ = −
∫D∇ ·
(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1(u×B)ψ − σ(u×B)ψ
)
B(δJi)(ψ) = −∫D∇ ·[(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1 − σ)
(u×B)]ψ.
Now, for notational simplicity, let
F1(φ) =
∫D
1
σ(δβe[B]×J′i) · φ
and
G1(ψ) = −∫D∇ ·[(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1 − σ)
(u×B)]ψ.
Then combining this with (20) yields the following system,
A(δJ)(φ) +B′(δV)(φ) = F1φ, (21a)
B(δJi)(ψ) = G1(ψ). (21b)
Applying Theorem 2 gives that δJi, δV must exist. Applying the estimates from Theorem 2 willgive the desired dependence result, but first, to simplify the estimate, we apply Green’s theoremto the operator G1 ∈W ′. For any ψ ∈W , this gives
G1(ψ) = −∫∂D
[(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1 − σ)
(u×B)]ψ · n.
Taking the absolute value of each side, we have
|G1(ψ)| =∣∣∣ ∫
∂D
[(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1 − σ)
(u×B)]ψ · n
∣∣∣≤∫∂D
∣∣∣[(σ(I − (βe + δβe)[B]× − βi[B]2×)−1 − σ
)(u×B)
]ψ · n
∣∣∣.Applying Cauchy-Schwarz and noting that ||n||l2 on ∂D yields
|G1(ψ)| ≤∣∣∣∣∣∣(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1 − σ)
(u×B)ψ∣∣∣∣∣∣L2(D)
,
≤∣∣∣∣∣∣(σ(I − (βe + δβe)[B]× − βi[B]2×
)−1 − σ)∣∣∣∣∣∣
V ′||(u×B)ψ||L2(D).
Noting now that σ is bounding, define T1 = σ−1, T2 = [B]×, ε = δβe. Applying Kato’s theoremyields ∃C > 0, not depending δβe such that∣∣∣∣∣∣(σ(I − (βe+δβe)[B]×−βi[B]2×
)−1−σ)∣∣∣∣∣∣
V ′||(u×B)ψ||L2(D) ≤ C||δβe||L∞(D)||(u×B)ψ||L2(D)
Applying this to the above inequality yields
|G1(ψ)| ≤ ||δβe||L∞(D)C||(u×B)ψ||L2(D).
For notational simplicity in this proof, let K = 1αβ
(||A||L(V,V ′) + α
). Then, using the a-priori
estimate from Theorem 2, we have
||δJi||V ≤1
α||F1||V ′ +K||G1||W ′ .
Now, as these operator norms are minimizations over the spaces V,W , where applicable, itfollows that for any φ ∈ V and ψ ∈W such that ||φ||V = ||ψ||W = 1, it holds that
||δJi||V ≤1
α
∣∣∣ ∫D
1
σ(δβe[B]×J′i) · φ
∣∣∣+KC||δβe||L∞(D)||(u×B)ψ||L2(D)
≤ ||δβe||L∞(D)
(1
α
∣∣∣∣∣∫D
1
σ([B]×J′i) · φ
∣∣∣∣∣+KC||(u×B)ψ||L2(D)
).
As φ, ψ were arbitrary, it follows that as as δβe → 0, δJi → 0. Given that V depends continuouslyon Ji, as apparent from (10), it follows that the solutions, Ji,V, depends continuously upon βe.Similar logic shows that the solutions also continuously depend on βi, σ.
Theorem 4. Solutions to (10) depend continuously on u and B.
Proof. We first show that the solutions to (10) depend continuously on B. Fix u,B ∈ (L2(D))3,bounded, and let σ, βe, βi be positive, bounded functions on D. Let δB be a small perturbationon B in any one direction and consider the following perturbed system∫
D
1
σ
(I − βe[B + δB]× − βi[B + δB]2×
)J′i · φ−
∫D∇V ′ · φ = 0, ∀φ ∈ V, (22a)
∫D
J′i · ∇ψ = −∫D∇ ·
(σ(I − βe[B + δB]×− βi[B + δB]2×
)−1(u×B)ψ
), ∀ψ ∈W, (22b)
where J′i,V ′ denote the solutions, guaranteed by Theorem 2. Let Ji,V denote the solutions tothe unperturbed system, (10), and let δJi, δV denote the differences between the perturbed andunperturbed solutions. Subtracting (10) from (22) yields∫D
1
σ
(I − βe[B + δB]× − βi[B + δB]2×
)J′i · φ−
∫D
1
σ
(I − βe[B]×−βi[B]2×
)Ji · φ
= −
(∫D∇V ′ · φ−
∫D∇V · φ
)∫D
1
σ
[(I − βe[B + δB]× − βi[B + δB]2×
)J′i −
(I − βe[B]× − βi[B]2×
)Ji
]· φ
= −
(∫D
[∇V ′ −∇V
]· φ
).
By linearity it is apparent that [B + δB]× = [B]×+ [δB]× and [B + δB]2× = [B]2×+ [B]×[δB]×+[δB]×[B]× + [δB]2×. Applying this to the above yields∫
D
[1
σ
[(I − βe[B]× − βe[δB]× − βi([B]2× + [B]×[δB]× + [δB]×[B]× + [δB]2×)
)J′i
−(I − βe[B]× − βi[B]2×
)Ji
]· φ
]= −
(∫D∇δV · φ
)∫D
[ 1
σ
(− βe[δB]× − βi
([B]×[δB]× + [δB]×[B]× + [δB]2×
))J′i − σ−1δJi
]· φ =
(∫D∇δV · φ
)
For simplicity, let χi denote a vector of indicator functions in the ith direction. Then
−∫D
δB
σ
(− βe[χi]× − βi
([B]×[χi]× + [χi]×[B]× + δB[χi]
2×))
J′i · φ = A(δJi)(φ)B′(δV)(φ).
For notational convenience, define
F2(φ) = −∫D
δB
σ
(− βe[χi]× − βi
([B]×[χi]× + [χi]×[B]× + δB[χi]
2×))
J′i · φ
For the second equation, we have∫DδJ′i · ∇ψ = −
∫D∇ ·
(σ(I − βe[B + δB]× − βi[B + δB]2×
)−1(u×B)ψ
)−∇ · σ(u×B)ψ
Writing this in the notation of Section 3.2, we have
B(δJi)(ψ) = −∫D∇ ·
((σ(I − βe[B + δB]× − βi[B + δB]2×
)−1− σ
)(u×B)ψ
).
Again, for notational convenience, define
G2(ψ) = −∫D∇ ·
((σ(I − βe[B + δB]× − βi[B + δB]2×
)−1− σ
)(u×B)ψ
).
Thus, we have the following system,
A(δJi)(φ) + B′(δV)(φ) = F2(φ) (23a)
B(δJi)(ψ) = G2(ψ). (23b)
By applying Theorem 2, δJi, δV both exist. Similar to the proof of Theorem 3, we apply Kato’stheorem to simplify the estimates given by Theorem 2. For ψ ∈W , we have
∣∣∣G2(ψ)∣∣∣ ≤ ∣∣∣∣∣
∣∣∣∣∣((
σ(I − βe[B + δB]× − βi[B + δB]2×
)−1− σ
)(u×B)ψ
)∣∣∣∣∣∣∣∣∣∣L2(D)
≤ ||δB||L∞(D)C||(u×B)ψ||L2(D)
for some C ∈ R. Utilizing the a priori estimates given in Theorem 2, we have
||δJi||V ≤1
α||F2||V ′ +
1
αβ
(||A||L(V,V ′) + α
)||G2||W ′
Using the definition of opertator norms, for any φ ∈ V and ψ ∈W such that ||φ||V = ||ψ||W = 1
||δJi||V ≤1
α
∣∣∣∣∣∫D
δB
σ
(βe[χi]× + βi
([B]×[χi]× + [χi]×[B]× + δB[χi]
2×))
J′i · φ
∣∣∣∣∣+1
αβ
(||A||L(V,V ′) + α
)||δB||L∞(D)C||(u×B)ψ||L2(D)
≤ ||δB||L∞(D)
[1
α
∣∣∣∣∣∫D
1
σ
(− βe[χi]× − βi
([B]×[χi]× + [χi]×[B]× + δB[χi]
2×))
J′i · φ
∣∣∣∣∣+
1
αβ
(||A||L(V,V ′) + α
)C||(u×B)ψ||L2(D)
].
Thus, it is clear that the solutions are continuously dependent upon B. Continuous depen-dence on u follows immediately from (10) and the continuous dependence of integrals on theirarguments.
With well-posedness of the forward problem established, we now turn the discussion to parameterestimation.
4 Parameter Estimation Theory
4.1 Parameter Estimation Scheme
Parameter estimation is an optimization problem in which if u is the solution to a system ofequations, and d represents provided data, the goal is to find parameter(s) q to minimize thedifference between the solution and data, i.e.
Minimize J(q) := ||u(·; q)− d||,
where q varies over some admissible parameter space, and J is defined in some appropriate norm.We will denote this problem for our system of equations the ID or identification problem. Tomatch the notation of Banks [3], we define the following:
U =
[JiV
], A(q) =
[A B′B 0
], F (q) =
[0G
],
where A,B, G are as defined in the above sections. Then the system of equations, (10), can bewritten as
AU = F. (24)
Clearly, U ∈ V ×W for any solution of (10), and we denote this space H := V ×W . The set ofunknowns for this specific parameter estimation problem is given by
q = (u,B, σ, βe, βi).
We now define a series of parameter spaces over which a solution to the inverse problem exists,following from the stipulations within Theorem 2. We begin with the most general, which isgiven by
Q :=(L2(D)
)3 × (L2(D))3 × L∞+ (D)× L∞+ (D)× L∞+ (D)
whereL∞+ (D) = f ∈ L∞(D) : f > 0 on D.
For any parameter set chosen within Q, there will be a solution to (24), guaranteed by Theorem2. For simplicity, denote the solution to the system with a given parameter set q as U(·; q). Fornotational purposes, let M = L∞(D). We now further refine the admissible parameter set toreflect the underlying physics of the problem that is not captured by the system of equations,i.e. the realistic admissible parameter set, which will be denoted by Q. Note that we choosethis subset to be compact. First, define
˜Q := q ∈ Q : ||u||M3 < umax, ||B||M3 < Bmax, σmin < ||σ||M < σmax,
||βe||M < βe,max, ||βi||M < βi,max,
and then characterize Q as the subset of˜Q where u,B both have uniform Lipschitz bounds,
denoted l1,i, l2,i for i = 1, 2, 3. Notationally, ∀ε > 0,
Q := q ∈ ˜Q : |ui(xi, ; )− ui(xi + ε, ; )| ≤ εl1,i, |Bi(xi, ; )−Bi(xi + ε, ; )| ≤ εl2,i, for i = 1, 2, 3.
Observe that this final restriction on u,B implies that they are continuous. As well, this impliesthat the identification problem that defines this parameter estimation is to minimize
(ID) J(q) := ||U(·; q)−U||H , nada
over q ∈ Q and for some given state U . Now, for both numerical and theoretical considerations,we define a finite-dimensional approximation to Q. Let D be partitioned into M subsets, anddefine
QM :=q ∈ Q : u
∣∣Dk∈ Pn, B
∣∣Dk∈ Pn, σ
∣∣Dk∈ P0, βe
∣∣Dk∈ P0, βi
∣∣Dk∈ P0 for k = 1, . . .M
,
for some n ∈ N. Here, Pn is the space of polynomials of degree at most n on Dk, and P0 is clearlythe space of constant functions. As well, note that as QM ⊂ Q, u,B will still be continuousfunctions for any q ∈ QM . Finally, we complete the statement of the parameter estimationproblem by turning the discussion to partial observational data.
Realistically, the data provided will not be known across the domain. Thus, for incomplete orpartial observational data, define xklk=1 ⊆ D to be the points at which the data, d, is given.Define the projection operator to be
C(F) = F(xk)lk=1,
where F denotes any function mapping D → R4. This implies that the restricted domain IDproblem is given by
(IDM ) JM (q) := ||C(U(·; q))−d||(L2)4 nada.
over q ∈ QM . Note now that the weak solution space, H, does not necessarily lie in the domainof C. Thus, we define a continuously imbedded Banach space into H by continuous functions,i.e.
H =((C(D)
)3 × C(D))∩(V ×W
).
Then define the domain of C to be:
C : H → Z = (Rl)4,
where l is determined by the number of observation points. This projection operator representswhere the data will be provided within the domain D.
We also seek our solutions to (10) in some finite-dimensional (FD) space. The particular FDapproximation to H we use, denoted HN , is defined as a space of linear functions. Define τto be a given Delauney finite element triangularization of D with N elements. Then clearlythe dimension of HN is finite, with a basis of linear functions, φj, where φj will be 1 onthe jth node, and 0 on every other node, vanishing at the endpoints. Similar to HN , define
HN = HN ×((C(D)
)3 × C(D))
. With this finite dimensional approximation space, we arrive
at the final version of the identification problem, analogous to (IDM ): find the minimum of
(IDNM ) JNM (q) := ||C(UN (·; q))−d||, nada
over q ∈ QM , with UN (·; q) ∈ HN (q).
With the identification problem now clear, we state the definition of a function space parameterestimation convergent or FSPEC set, as seen in [3]. This will in turn allow us to show that theinverse problem is well-posed.
Definition: 4.1.1. For notational convenience, let xk := xNkMk
where appropriate. Then, a set
consisting of a finite-dimensional approximation to the solutions space of (13), HN , a finite-dimensional approximation to the solutions of (13), UN (q), a projection operator, C and afinite-dimensional approximation to the admissible parameter space, QM is FSPEC if it satisfiesthe following.
i. For each N = 1, 2, . . . there exists a solution qNM ∈ QM of IDNM .
ii. Every convergent subsequence qk converges to a solution q∗ ∈ Q of ID.
iii. Jk(qk)→ J(q∗).
iv. ||UNk(·; qk)− U(·; q∗)||H(qk)
→ 0 .
v. There exists at least one subsequence satisfying ii.− iv.
We now go on to show that the identification problem as defined above satisfies the requirementsto be FSPEC.
4.2 Necessary Postulates
In the following section, several propositions are presented concerning the parameter estimationproblem. These are then utilized to construct a set which is function space parameter estimationconvergent, which implies that the inverse problem is well-posed. All of the following propositionsstem from similar postulates found in Banks [3], and are labeled to correspond accordingly. Webegin with properties of the solution Hilbert space, and its finite dimensional approximation.
Proposition 1 (HS). For each q in the metric space Q, ρ, the space H is a Hilbert space, His a Banach space continuously imbedded in H, and HN is a finite dimensional (closed) linearsubspace of H with dimension independent of q.
Proof. We define ρ to be the metric induced by H’s inner product, i.e.
〈·, ·〉q = 〈·, ·〉(L2(D))3 + 〈·, ·〉L2(D).
It immediately follows that H is complete, as any Cauchy sequence fk = (f1,k, f2,k) in H musthave the property that f1,k is Cauchy in L2(D), and by the Riesz-Fischer Theorem [21], this is acomplete space, and thus f1,k converges. A similar result holds for f2,k, given that W is a closedsubset of a complete space. Here, the closure property is immediate from W being defined asthe closure of the continuous test functions with compact support. Thus, H is a Hilbert space.
As the continuous functions are dense in L2(D), it immediately follows that H is a Banach spacecontinuously embedded in H, as the product of the dense subspace of factors is dense in theproduct space.
Finally, HN is finite-dimensional, as it has a finite basis. As well, HN is clearly closed undervector addition and scalar multiplication, as the compactness of D guarantees the closure. Thus,HN is a closed linear space. The dimension depends only on the number of partitions of τ , whichis not a parameter of q, and the claim holds.
Now, to present a classic theorem for compactness in L2 spaces. This theorem is analogous toArzela-Ascoli [22] for Lp spaces, and is necessary to characterize compactness in Q.
Theorem 5 (Frechet-Kolmogorov Theorem). Let F be a subset of Lp(D) with p ∈ [1,∞) andlet τhf denote the translation of f ∈ F by h, i.e., τhf = f(x − h). Then F is compact if andonly if
i. F is closed,
ii. F is equicontinuous, i.e. lim|h|→0
||τhf − f ||Lp = 0 uniformly, and
iii. F is equitight, or limr→∞
∫||x||>r
|f |p = 0 uniformly.
A proof can be found in [6]. We now apply this theorem to the first two subspaces of Q.
Lemma 5. The space of uniformly bounded and uniformly Lipschitz continuous functions, de-noted F , is a compact subset of L2(D).
Proof. We clearly would like to apply the Frechet-Kolmogorov theorem. Thus, we show thethree conditions listed stated in the theorem hold. Let α denote the uniform bound on F , andlet β denote the uniform Lipschitz condition.
i. Closed. From the Lipschitz condition, it follows that fk must be differentiable at all buta countable number of points for each k. Now, as the union of countable sets is againcountable, assume that these points of non-differentiability are the same for each fk. Denotethese points xj and consider any interval between such points. At most, f ′k ≤ β on each.
Suppose that fk → f . It then follows that f has at most the same points of non-differentiability as fk. As well, by continuity of the derivative operator, it follows thatf ′k → f ′. Therefore f ′ ≤ β as each f ′k is, and thus f is Lipschitz continuous with Lipschitzbound ≤ β.
Now, as continuous functions with bounded derivatives are compact by Arzela-Ascoli [22],it immediately follows that on each interval of differentiability, ||f ||L∞([xk,xk+1]) < α, and bycontinuity of f , it must be that ||f ||M < α.
ii. Equicontinuous. Let f ∈ F . The uniformly Lipschitz condition implies that
|τhf − f(x)| ≤ β|h|.
Applying this yields
||τhf − f ||2L2(D) =
∫D|τhf − f | dV ≤
∫Dβ|h| → 0,
and as f is arbitrary, F must be equicontinuous.
iii. Equitight. This follows immediately from D being compact and thus bounded. Thereforethe support of any f ∈ F is compact, and for any ball larger than the radius of D, theintegral of f outside of that ball is zero, and therefore F is equitight.
We now use the classical definition of compactness, totally bounded and complete [6], to showthat the latter three subspaces for σ, βe, βi are compact.
Lemma 6. The space of uniformly bounded functions, denoted G, is a compact subset of L∞(D).
Proof. We show that G must be complete by contradiction, as totally bounded follows imme-diately from uniformly bounded. Recall that L∞(D) is denoted by M for notational purposes.Let γ denote the uniform bound of G. To see that this space is complete, assume that gk isa Cauchy sequence of functions approaching some g, where ||g||M > γ. However, as gk ∈ G, wehave that ||gk||M ≤ γ. Considering this, we have
||g − gk||M ≥∣∣∣||g||M − ||gk||M∣∣∣
> 0 As gk being bounded uniformly by γ < ||g||M.
This is a contradiction as gk cannot approach g. Therefore G is complete, and it follows that Gmust be compact.
Combining these two lemmas, we see that our realistic admissible parameter set is compact.
Proposition 2 (HQ1). The set Q is a compact subset of the metric space (Q, ρ).
Proof. For simplicity, label each of the product spaces of Q as Qi so that
Q = Q1 × . . .× Q5.
Note that Q1 and Q2 are compact by Lemma 5. Also note that Q3, Q4, Q5 are compact fromLemma 6. As the product of compact spaces is compact, Q is compact in Q.
We now show that our approximations to the space pass convergent sequences to convergentsequences.
Proposition 3 (HQC). For any arbitrary sequence qM → q0 in Q, we have∣∣∣∣UN (·; qM )− U(·; q0)∣∣∣∣H(q)→ 0.
Proof. Consider the following.
||UN (·; qM )− U(·; q0)||H(q)
= ||UN (·; qM )− U(·; qM ) + U(·; qM )− U(·; q0)||H(q)
≤ ||UN (·; qM )− U(·; qM )||H(q)
+ ||U(·; qM )− U(·; q0)||H(q)︸ ︷︷ ︸
B
.
As it has been shown that the solution, U , depends continuously on the parameters (Theorems3 and 4), B must go to 0 as qM → q0. As well, consider that as UN satisfies the weak-form, i.e(14), it must also satisfy the minimization (Ritz) formulation, given by
||UN (; , qM )− U(; , qM )||H(q)≤ ||V − U(; , qM )||
H(q)∀V ∈ HN .
In particular, this is satisfied for the interpolant of U in HN , denoted UN
(q), and thus
||UN (; , qM )− U(; , qM )||H(qM )
≤ ||UN (; , qM )− U(; , qM )||H(q)
.
Now, define hK = diam(K) as the longest edge of element K ∈ τ and then define h :=maxK∈τ hk. Then from [5], it follows that
≤ Ch3||U(; , qM )||H(q)
,
for some constant C that does not depend on the chosen grid. Recall that τ is the Delauneyfinite-element triangularization of D. . Clearly, as N →∞, h→ 0 for a Delauney finite-elementmesh, and therefore the inequality holds.
We now need to show the existence of surjective maps from the admissible parameter space tothe finite-dimensional representation, and that as the order of the FD representations goes toinfinity, it converges to the admissible parameter space. We do so for each of the two ‘types’ ofdomains we work with, L2 and L∞. We begin with L2.
Lemma 7. Let F := f ∈ L2 : ||f ||M ≤ α1, |f(x + h) − f(x)| ≤ hα2 ∀x ∈ D, for someα1, α2 > 0. Now, for the M parameter dimensions, K ∈ τ , and n the polynomial dimension forQ1, Q2, define
FM := f ∈ F : f∣∣K∈ Pn(K) k = 1, . . . ,M.
Then FM is compact in F , and there exists a surjective map V 1M : F → FM such that
ρ(V 1Mqn, qn)→ 0 as M →∞
for any convergent sequence qn ⊂ F .
Proof. To see that FM is compact, we note that F has been shown to be compact (Lemma 5),and FM is closed. As any closed subset of a compact set is compact, FM must be compact.
We now define the surjective operator. For simplicity, assume that the piecewise polynomialsare all of order 1, and that D ⊂ R. For higher-degree polynomials, the arguments remainsimilar. For D ⊂ Rm, m > 1, the arguments presented below are easily extended to each spatialdimension.
Let xk be the finite number of sample points. Assume that D is connected (as one can extendto each disconnected subset of D), and for f ∈ F , define
V 1M (f) =
f(xk)− f(xk−1)
xk − xk−1(x− xk) + f(xk) for xk−1 ≤ x < xk, for k = 1, . . . ,M − 1.
Clearly, as V 1M = Id for any f ∈ FM , and as FM ⊂ F , it follows that V 1
M is surjective. Now,note that the Lipschitz condition implies V 1
M (f) < α2(x − xk) + f(xk) for any f ∈ F and allxk−1 < x < xk. It then follows that
||V 1M (f)− f ||M ≤ α2(xk − xk−1).
Let qn ⊂ F be any convergent sequence. Then clearly, as M → ∞, (xk − xk−1) → 0, andthus the desired inequality holds.
We now show a similar result for L∞.
Lemma 8. Let G := g ∈ L∞ : ||g||L∞ ≤ γ, . Let GM := g ∈ G : g∣∣Dk∈ P0(Dk) k =
1, . . . ,M. Then GM is compact in G, and there is a surjective map V 2M : G → GM such that
ρ(V 2Mqn, qn)→ 0 as M,n→∞
for any convergent sequence qn ⊂ G.
Proof. As with Lemma 7, it has already been shown that G is compact, and it immediatelyfollows that GM is closed, and thus compact as a subset of G. Now, define the surjective mapsimilar to Lemma 7. For any g ∈ G,
V 2M (g) := g(xk)− g(xk−1) for xk−1 < x < xk, for k = 1, . . .M − 1.
Then V 2M = Id on GM , and as a subset of G, it is surjective. The inequality follows as
||V 2M (g)− g||M ≤ γmax
K∈τhK ,
with hk defined as in the proof for Lemma 7. By the definition of Dealuney triangulation, wehave M →∞, hK → 0 ∀K ∈ τ and thus the inequality holds.
Finally, we combine the ideas of the two above lemmas, applying them to the appropriatesubspaces of Q.
Proposition 4 (HQ4). There exists a sequence of finite dimensional compact sets QM ⊆ Qand surjective maps VM : Q → QM such that for any convergent (possible trivially convergent)sequence qn in Q we have ρ(VMqn, qn)→ 0 as n,M →∞.
Proof. For simplicity, defineQM = QM1 × . . .× QM5 .
Then from Lemma 7 Q1 and Q2 are compact. Similarly, from Lemma 8, Q3, Q4, Q5 are compact.Thus, as the product of compact spaces is compact, Q is compact.
Define VM to be the product of the projective mappings guaranteed by the two lemmas for eachof the spaces. Then it is immediate from these lemmas that the mappings are surjective, andthen finally, VM must satisfy the desired convergence as n,M go to ∞.
Now, with all of these propositions established, we turn to showing that the set is FSPEC.
4.3 Kinematic MHD Inverse Theorem
Following the work of Banks [3], we now define a set based on the above propositions that isFSPEC, showing that the inverse problem is well-posed. We do so by showing that the setsatisfies each of piece of the definition, FSPEC.i - FSPEC.v.
Theorem 6. The set HN , Un(q), C, QM is FSPEC.
Proof. From Theorems 3 and 4, it is immediate that U depends continuously on q, and thus sodoes JNM (q). Compactness of QM , as shown in HQ4, under the continuous map of U implies theexistence of solutions to IDN
M , as the image is compact and thus contains a minimum. Denotethis solution qNM . By definition of minimum, we have JNM (qNM ) ≤ JN (qM ) for all qM ∈ QM . Ofcourse, this implies that
JNM (qNM ) ≤ JNM (VM (q)) for all q ∈ Q. (25)
As VM is surjective by HQ4, choose qNM ∈ Q such that VM qNM = qNM . As Q is compact (see HQ1),
as N,M →∞, there exists some convergent subsequence of the above solutions, i.e. qNkMk ⊂ Q
such that qNkMk→ q∗ for some q∗ ∈ Q. Utilizing HQ4 again, along with triangle inequality, it
follows that
ρ(qNkMk, q∗) ≤ ρ(qNk
Mk, qNkMk
) + ρ(qNkMk, q∗)
= ρ(VMk(qNkMk
), qNkMk
) + ρ(qNkMk, q∗)→ 0.
implying that qNkMk→ q∗ in Q. This implies that both FSPEC.i and FSPEC.ii are satisfied.
For notational purposes, let qk = qNkMk
, and similarly for J . Then
||Jk(qk)− J(q∗)||L2 ≤∣∣∣∣(C(qk)UNk(·; qk)− d
)−(C(q∗)U(·; q∗)− d
)∣∣∣∣L2
=∣∣∣∣C(qk)UNk(·; qk)− C(q∗)U(·; q∗)
∣∣∣∣L2 .
This implies the FSPEC.iii holds. Now, from HQC and the lack of dependence of q on C, itimmediately follows that ||Jk(qk)− J(q∗)||L2 → 0 by the above. This implies that
Jk(qk)→ J(q∗). (26)
and thus FSPEC.iv holds. FSPEC.v holds by all of the above. The existence of one such subse-quence is guaranteed by applying Theorem 2 onto the subspaces Vh,Wh, with an appropriatelydefined parameter sequence.
We now turn to the numerical validation and subsequent demonstration of the numerical pa-rameter estimation scheme.
5 Foward Problem: Numerical Implementation
5.1 Introduction
With well-posedness of both the forward and inverse problem established, we now turn to thenumerical implementation of the governing equations. As noted previously, MHD has manyapplications, but we focus on a specific application, an MHD generator. As the above governing
equations apply only where there is an applied magnetic field, which exists only in one sectionof the MHD generator, the MHD channel, we model the MHD channel numerically.
There are many different electrode configurations for an MHD channel. Some geometries thathave been explored include Faraday, Hall, and disc [24]. We focus on the Faraday geometry,under the assumption that the Hall parameter is non-negligible, but also not of extreme magni-tude. In this configuration, the channel and flow are linear, with electrodes in the y-direction,on opposing sides of the channel.
For implementation simplicity, we make use of an engineering multi-physics finite-element sofware,COMSOL [1]. In the following section, we simulate two different Faraday models, continuousand segmented. We then validate these models and their implementation by comparing thepower within the channel to an ideal equation. Finally, qualitative and quantitative effects ofsome parameters are then discussed.
5.2 Faraday Generator Components
The MHD channel is composed of a channel, electrode, and resistor. Each of these componentsare governed by (10), with varying assumptions simplifying the system. These assumptions arebased on the underlying physics of each component. Note that for computational simplicity, allwalls are assumed to be infinitely thin.
Firstly, we introduce the channel. This component houses the plasma, and is thus where theelectric potential and induced current spontaneously arise. We prescribe a uniform fluid flowconsistent with other models, at a flow rate of 1800 m/s [13]. The applied magnetic field is givenan appropriate value of 6 T [13]. Lastly, the channel conductivity is given by a realistic valueof the conductivity of an ionized noble gas (e.g. Xenon) with seed (e.g. potassium) at 2000K,namely 60 S/m [10]. We choose the channel to have length 1.5 times the total electrode lengthto satisfy the ideal geometry restrictions presented in Rosa [20]. Note that within the channel,the entirety of the governing equations are in effect, and no physics-based simplifications aremade.
The next component included in the model are the electrodes. By definition, these are electricallyconductive materials that allow current to flow in a circuit. Thus, they have some assignedconductivity based on the material type. In our case, we use the conductivity of copper, 61S/m[10]. Being a solid material, electrodes have negligible ion and electron mobilities, by definition.Finally, u = 0, as there is no plasma flow within any component other than the channel. Thisallows for a physics-based reduction of the governing equations within the electrodes to Ohm’slaw, J = σ(E), Faraday’s law, ∇V = 0, and the divergence condition, ∇ · J = 0.
Lastly, the resistor mimics a load being placed on the channel. It is similar to the electrode,but the resistivity of the material, as opposed to being a material property, is varied to simulatedifferent loads being placed on the channel. All other assumptions made for the electrodes areequivalently placed on the resistor and thus the same set of governing equations apply.
The boundary conditions are made more complex to represent both generator configurationsand ideal conditions. Firstly, the resistor, in order to replicate a load, must connect two elec-trodes. Thus, a periodic condition is placed on the outer boundaries of the resistor and opposingelectrode. Similarly, the channel is assumed to be infinitely long, and thus periodic boundaryconditions are placed on the inlet and outlet to the computational domain. Elsewhere, the per-fectly electrically insulating boundary condition is assumed. Any boundary between componentsis assumed to be a continuity condition placed on each of the solutions. We now turn to thespecific geometries of the Faraday generators.
5.3 Geometries
We first examine the continuous Faraday channel geometry. A simple schematic of this geom-etry is presented in Figure 1. Each component described in Section 5.2 is labeled. Periodicboundary conditions are noted with the coordinated dashed lines, with red lines representingthe electrode-resistor periodic boundary conditions, and blue lines denoting the inlet-outletboundary conditions.
Similar to the continuous Faraday generator, the segmented Faraday geometry consists of elec-trodes, resistors, and a channel, the components described in section 5.2. In contrast with thecontinuous Faraday geometry, the electrodes are segmented and separated by portions of thechannel, denoted the inner-electrode space. The geometry is seen in Figure 2.
Again, the periodic conditions are marked with the corresponding colors, with each correspond-ing boundary having a periodic boundary condition. In this geometry, each pair of electrodesare loaded individually, and thus each resistor represents an individual load being placed on thechannel. The channel’s inlet and outlet periodic conditions again replicates the infinitely-longideal channel. One interesting aspect of the segmented Faraday geometry is that it will ideallynegate the effect of the Hall parameter. Further discussion of this concept can be seen in Section5.6.
5.4 Numerical Mesh and Solver
We now turn to the numerical implementation of each of the described geometries. As notedin the introduction, we utilize an engineering software called COMSOL [1], which is known forbeing able to couple complex multi-physics on complex geometries. The application is straight-forward with the system of equations rewritten in a mixed-Poisson form, i.e. the system (10),and with an explicit form of the conductivity tensor. Given that each component describedabove is expressed in such a form, implementation requires only defining the parameters for thesystem properly in each portion of the domain. Under the different Faraday geometries, twodistinct full 3-D models are implemented in COMSOL. Triangulation of each spatial domain isachieved using a flipping algorithm along with a psuedo-random mesh to generate a Delauneytriangulation.
Several examples of a properly produced Delauney trianguation mesh for a segmented Faradaygeometry are presented in Figure 3. Figure 3a is a physics-defined mesh, where comsol useda priori knowledge of electric currents and a chosen distribution level of ‘normal’ to generateand refine the mesh. Notice that at the boundaries and corners, a refinement of the otherwiseuniform Delauney mesh can be seen. Figure 3b uses same physics apriori knowledge, yet with a‘fine’ distribution. Lastly, Figure 3c represents a user-defined Delauney mesh, where COMSOLused the flip-algorithm to properly generate the Delauney mesh with given refinement levelsin volumes, edges, and corners of the domain. For example, some boundaries were chosen tohave a fixed number of elements. As well, various sub-sections of the domain were coarsenedand refined based on test runs, where high gradients of electric potential were seen, particularlycorners and on boundaries with periodic boundary conditions.
We use an iterative solver to converge to the solution of the resulting linear system of equations.The algorithm utilized for this purpose is functional generalized residual method, or FGMRES.This is a pre-conditioned method of GMRES. A further discussion of this algorithm can be seenin [16]. We now turn to a discussion of model validation.
Electrode
Channel
Electrode
Resistor
y
x
u
nothing
Electrode Channel Electrode
Res
isto
r
yz
B
nothing
Figure 1: Continuous Faraday Generator.
Figure 2: Segmented Faraday Geometry.
(a) (b) (c)
Figure 3: Three different COMSOL generated meshes. (a): physics-based, normal distribution,(b): physics-based, fine distribution, (c): user-defined.
5.5 Continuous Faraday: Ideal Power Equation
We define power within any MHD channel as
P = −J ·E.
Under the continuous Faraday geometry, Ex ≈ 0, as the continuous electrodes artificially negateits formation within the channel [20]. From a design stand-point, one can easily assume that Bis uni-directional and uniform, i.e., B = (0, 0,Bz) is the applied magnetic field. For ideal powerconditions to apply, one also must assume that u = (ux, 0, 0), which again is uniform. Underthese assumptions, u × B lies only in the y-direction, agreeing with the electrode placementpresented in Section 5.3. Again to satisfy the ideal power conditions, we assume that βe, βi, σare all spatially constant. Conventionally, one also defines a load-factor, representing the ratioof the average load placed on the channel over the maximum load allowed, as
K =Ey
uxBz.
Using (8a), it follows that
Jy =σ(1 + βi)
(1 + βi)2 + β2e
(Ey − uxBz
).
Noting that if Ex,Ez ≈ 0, the ideal continuous Faraday average power equation including ion-slipmust be
Pcf = J ·E ≈ JyEy = − σ(1 + βi)
(1 + βi)2 + β2e
K(1−K)u2xB
2z. (27)
Using these ideal power equations, we now validate the numerical model, using the average valuesof all the parameters within the channel. By varying the parameters to represent differentoperating conditions, we obtain different power within the channel. We then compare thenumerical model’s measured power to the ideal power equations. This is done with numericalintegration approximating the integral ∫
channel
−J ·E dV.
We do so for a realistic range of values for each parameter. The range of values tested can beseen in their related figures. Note that the load factor is varied by changing the resistivity ofthe resistor, as discussed in the components section. Figures 4a, 4b, 4c, ??, and 5 compare theideal and measured power in the top figure, with the lower figure giving the relative differencebetween the two. Clearly, for all parameters varied, good agreement is seen, with difference lessthan 1%.
5.6 Segmented Faraday: Ideal Power Equation
Unlike the continuous Faraday geometry, the segmented Faraday geometry no longer guaranteesthat Ex ≈ 0. However, this geometry does short-circuit the electric current in the x-direction,implying that Jx ≈ 0. Again using (8a) to solve for Ex yields that the ideal segmented Faradaypower is given by
Psf =σ
(1 + βi)K(1−K)u2
xB2z. (28)
Thus, by neglecting ion-slip, MHD generators could have power-losses on the order of 1/(1+βi).
To validate the model, a similar method to the continuous Faraday geometry is utilized. Weuse the same realistic ranges for B, σ,u and vary βe, βi,K by changing µe, µi, R respectively.
(a) Ideal power and Comsol power compared tovaried Bz (b) Ideal power and Comsol power compared to
varied σ
(c) Ideal power and Comsol power compared tovaried K
(d) Ideal power and Comsol power compared tovaried u
Figure 4
Figure 5: Ideal power (dashed) and computed power (solid) for varied µi, µe = 1/6, 5/6, 10/6
As with the continuous Faraday model validation, R is the resistivity of the resistor. Thedifferences between the (28) and the measured COMSOL power within the segmented Faradaymodel are seen in Figures 6a, 6b, 6c, ??, and 7. Again, as with the continuous Faraday, goodagreement between the ideal power equation and measured power is noted. However, pooreragreement than the continuous Faraday geometry can be attributed to geometrical errors fromthe more complex geometry, as well as numerical difficulties from this complex geometry forhigher electron mobilities.
6 Numerical Implementation of Parameter Estimation
We now turn to a numerical demonstration of the parameter estimation scheme presented above,using COMSOL and both Faraday geometries: continuous and segmented. For simplicity anddemonstration purposes, we only attempt to recover two parameters, βe, βi, by recovering theirrespective mobilities, µe and µi. To perform the parameter estimation, we utilize the livelinkcross software compatibility between Matlab and Comsol [2] for the scheme, using the optimiza-tion solver lsqnonlin [14] to recover the parameters
As is standard when other data is unavailable, we synthesize the data using our model. We thenperturb the data to represent instrument noise, and avoid an “inverse crime.” We demonstratethe recovery of the parameters under an additive Gaussian noise. Under the previous notation,let U(·; q) represent the solution to (10) for some q. The we define
DA(nL) := U(·; q) +N · nL,
as the additive-noise data. Here, nL represents the noise-level, N ∼(N (0, ||U ||)
)m, i.e. it is a
random-vector of dimension equal to U , with standard deviation given by the norm of U . Afteradding the noise as described above, we compare the recovered parameter to the true parameteras a function of noise-level, within the framework of the segmented Faraday geometry. For agiven noise-level, the parameter recovery scheme was implemented with initial guesses with anerror of 10% for 10 runs. The errors were then averaged to provide the results. The results areseen in Figure 8a and Figure 8b.
(a) Ideal power and Comsol power compared tovaried Bz
(b) Ideal power and Comsol power compared tovaried σ
(c) Ideal power and Comsol power compared tovaried K
(d) Ideal power and Comsol power compared tovaried u
Figure 6
Figure 7: Ideal power (dashed) and computed power (solid) for varied µi, µe = 1/6, 5/6, 10/6
(a)(b)
Figure 8: Mean relative error (solid) with asymmetrical error bars for recovery of µe (left) andµi (right). Asymmetrical error bars were generated by determining the standard deviation oferrors for over-estimates and under-estimates separately.
Comparing Figures 8a and 8b, we see that µi is more easily and accurately recovered that µe.For instance, at a noise level of 0.1, we have that the average relative error over 10 runs forµe is roughly 0.12, while the average relative error over 10 runs for µi is less than 0.025. Thevariance on the estimates of µi also grows slowly with respect to the noise level. However, theelectron mobility is not reliably recovered. Note here the lack of monotinicty and large variancesfor different noise levels. As well, note the asymmetric error bars on µe, implying the need fordistributional recovery to better describe the likely recovered parameter.
7 Conclusion
Within the framework of kinematic MHD, we have now established an efficient numerical modelfor the deterministic forward problem within COMSOL [1]. Furthermore, this model has beenvalidated, allowing for it’s use in future work. As well, the inclusion of ion-slip was noted bothanalytically, through the well-posedness arguments, and numerically, with an update of theideal power equations, and subsequent validation through the model. Finally, the parameterestimation scheme was also shown to be well-posed, with a numerical demonstration capturingthe convergence and subsequent difficulties of parameter recovery. Clearly, there is clearlydifficulty in obtaining accurate values of µe from noisy data. This points to the necessity ofrecovering a distribution of potential values, and the inclusion of uncertainty in the above work.If attempting to optimize an MHD generator in real-time, it is clear that uncertainty mustbe considered. Else, the recovered parameter for µe may poorly reflect the current operatingconditions of the generator, and thus result in sub-optimal power Future work will investigatethis further.
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