To compare the MM algorithm with a widely used method, we choose the Gauss-Newton (GN) method with a line search [11, Algorithm 1], which solves
Ep,a(s) = ϵaJaT1480(s, s∗p) + 1
2∥fT1480(s, z∗pI) − gp∥2
by smoothing the regularization JaT1480 with
Ja,βT1480(s, s∗p) =∥Ca(s− da(s∗p))∥1,β,
where∥w∥1,β =∑
i
√wi2+ β for some smooth parameter β > 0. We set ϵ1 = 10−5and
ϵ2 = 0.5× 10−6 in this Chapter for performing reconstruction.
For MM algorithm, we compute
skp,a= MM-CP(s∗pI, k, 103, Ca, da(s∗p), gp, 10−6, ϵa) (6.14)
Then, we show the reconstructed conductivity distributions
RMMp,a = ς[T1480, s50p,a]
using the MM-CP algorithm in Fig.6.12 ((a + 1)-th column, p-th row).
For the GN method, we compute
tkp,a= GN(s∗pI, k, Ca, da(s∗p), gp, β, ϵa), (6.15)
which represents the k-th iteration of the Gauss-Newton method with smooth parameter β using the same initial guess s∗pI, simulated data gp, regularization parameter ϵa, and the same Ca, da(s∗p) used in (6.14).
We show the reconstructed conductivity distribution
RGN = ς[T , t50]
0.5 1 1.5
Figure 6.12: Comparison of the simulated and reconstructed conductivity distributions.
First column: simulated distributions. Second and third columns: Sparse and TV recon-structed distributions using the MM-CP algorithm, respectively. Fourth and fifth columns:
Sparse and TV reconstructed distributions using the GN method, respectively.
0 10 20 30 40 50
Figure 6.13: Comparison with the energy decrease of the MM algorithm and GN method.
Table 6.2: Error estimates of the reconstructed conductivities using the MM-CP algorithm and the Gauss-Newton method with β = 10−6.
p ∥Sp− RMMp,1 ∥△ ∥Sp− RMMp,2 ∥△ ∥Sp− RGNp,1 ∥△ ∥Sp − RGNp,2 ∥△
1 0.0031 0.0031 0.0036 0.0032
2 0.0031 0.0032 0.0039 0.0035
3 0.0036 0.0036 0.0045 0.0041
4 0.0040 0.0037 0.0042 0.0041
Table 6.3: Error estimates of the reconstructed conductivities using the GN method with smooth parameters β = 10−3, 10−9.
β = 10−3 β = 10−9
p ∥Sp− RGNp,1 ∥△ ∥Sp− RGNp,2 ∥△ ∥Sp− RGNp,1 ∥△ ∥Sp − RGNp,2 ∥△
1 0.0048 0.0035 0.0036 0.0037
2 0.0051 0.0039 0.0039 0.0040
3 0.0050 0.0045 0.0040 0.0041
4 0.0046 0.0043 0.0043 0.0042
using the Gauss-Newton method with β = 10−6in Fig.6.12 ((a + 3)-th column, p-th row).
Additionally, we show Ep,2(skp,2) and Ep,2(tkp,2) in Fig.6.13 to compare the energy decrease in MM algorithm and GN method, respectively.
To judge the reconstruction performance, given R ∈ ς[T1480,R1480], we define the measure
∥Sp− R∥△ = vu uu ut
∑
j=1,...,18230 cj is the centroid of△j
△j∈T18230
|Sp− R|2(cj)|△j|
to approximate∥Sp−R∥L2(Ω). We show in Table 6.2 that the error estimate∥Sp−RMMp,a ∥△
is less than∥Sp−RGNp,a ∥△, which means that the MM algorithm has better approximation to the simulated conductivitySp. Besides, we show the error estimates of the reconstructed conductivities using the GN method with β = 10−3, 10−9in Table 6.3, which shows worse approximations in comparison to the results with β = 10−6.
Chapter 7 Discussion
We now discuss the relevant numerical results. For the estimate of the pair of background conductivity and contact impedance without additional information, in Chapter 6.2 we found the optimal pair (s∗p, z∗p) by least-squares data fitting, which shows poor estimates of contact impedance in Table 6.1. However, from such pair (s∗p, z∗p), the numerical re-constructions using the MM-CP algorithm still show good estimates of the intensity and location of inclusions in Fig.6.7,6.8,6.9 and a fast decrease in objective energy in Fig.6.11.
The reconstruction insensitivity to the contact impedance was discussed in [18, §4.1.1] and attributed to the adjacent current pattern we adopted in Appendix 3. The least-squares data fitting in Chapter 6.2 need to be adjusted, e.g. in [27, (29)] an unknown constant is added in data fitting, and in [7, pg.920] a condition is proposed that s∗pz∗p = 0.24 cm.
In the test of MM-CP algorithm we find the smaller parameter δ shows faster energy decrease in MM algorithm (4.3) and faster convergent speed for CP algorithms (5.2), and we choose δ in Section 6.3 by Machine epsilon. For the choice of rescaled regularization parameter ϵ/δ defined in (5.10) we choose (ϵ1, ϵ2)/δ = (0.5, 10) in Section 6.3 empiri-cally. For other choice of regularization parameter in CEM-based EIT, a formula in [10, (21)] was proposed for TV reconstruction, where the formula was illustrated by L-curve [12] shown in [11, Fig. 6] for TV reconstruction, too.
We discuss the generality of Problem 1, which is capable of processing measured data produced from the trigonometric current pattern [18]. In addition to the adjacent current pattern we adopted, for the trigonometric current pattern, we need to change the voltage
measuring vector ϕi and current injecting vector ϕ′i. Other representations of the conduc-tivity distribution are allowed in Problem 1. The replacement of the conducconduc-tivity basis χ△j in Appendix 3 only changes Cj in Vs = ∑L
ℓ=1 1
zℓBℓ+∑n
j=1sjCj, whose form still remains.
Finally, we emphasize that the conductivity basis is independent of FEM basis. the confusion of dependence comes from the common use of FEM triangles for defining both conductivity and FEM basis; see (3.10) in Chapter 3 for detail. For example, It demon-strated in [19, (8)] the conductivity basis has no use of FEM triangles in the level set representation of conductivity distribution.
Chapter 8 Conclusion
In conclusion, based on monotonicity relation, we provide a globally convergent MM algorithm for CEM-based EIT. Furthermore, a general globally convergent MM algorithm for regularized least square problem is stated in Theorem 1, which provides a standard tool for engineers to develop customized MM algorithm. Besides, the numerical solver for proposed MM algorithm is flexible. For example, in addition to O(1/k2) CP solver used in this paper, a diagonally preconditioned CP algorithm cooperating with parallel GPU implementation for convex optimization was proposed in [21, §3.3]. For MM algorithm development in CEM-based EIT, we demonstrate a method to construct majorizers based on monotonicity relation, by which other kinds of majorizers with more efficient solver or improved convergence rate are expected.
Appendix A
Proof of Proposition 1
First, we prove following lemma.
Lemma 1. For all compact set U ⊂ (0, ∞)n, there exists C > 0 such that ∥Vs−1− Vt−1∥ ≤ C∥s − t∥ ∀ s, t ∈ U.
Proof. There exists a∈ (0, ∞)nsuch that aj ≤ sj for all s ∈ U, from which, in addition to the structure of Vs, we have∥Vs−1∥ ≤ ∥Va−1∥ ∀ s ∈ U and
Vs−1− Vt−1 = Vs−1(Vt− Vs)Vt−1
= Vs−1 ( n
∑
j=1
(tj− sj)Cj )
Vt−1. (A.1)
Then, we have∥Vs−1− Vt−1∥ ≤ ∥Va−1∥2∥s − t∥√∑n
j=1∥Cj∥2for all s, t∈ U and prove the lemma.
Let δj be the j-th unit coordinate. Since (A.1) implies Gφ(s + hδj)− Gφ(s)
h = 1
h
⟨ φ,
[ Vs+hδ−1
j − Vs−1] φ
⟩
=−⟨
φ, Vs+hδ−1
jCjVs−1φ
⟩
then (a) is proved by Lemma 1. To prove (b), it is straightforward to check boundedness of
Gφ,∇Gφ by Lemma 1 and the compactness of E. Then, the Lipschitz continuity of∇Gφ
follows from
|(∇Gφ(s)− ∇Gφ(t))j|
=|⟨
Vs−1φ, CjVs−1φ⟩
−⟨
Vt−1φ, CjVt−1φ⟩
|
=|⟨
(Vs−1+ Vt−1)φ, Cj(Vs−1− Vt−1)φ⟩
|
≤ ∥Vs−1+ Vt−1∥ ∥φ∥2∥Cj∥ ∥Vs−1− Vt−1∥.
To prove (c), by defining B =∑L
ℓ=1 1
zℓBℓ, we have
Gφ(s) = max
u ⟨u, 2φ − Bu⟩ −
∑n j=1
sj⟨u, Cju⟩ (A.2)
with u = Vs−1φ as the optimal solution. Then,∀ s, t ∈ (0, ∞)n
Gφ(s)≥⟨
Vt−1φ, 2φ− BVt−1φ⟩
−
∑n j=1
sj⟨
Vt−1φ, CjVt−1φ⟩
=⟨
Vt−1φ, 2φ− BVt−1φ⟩
+⟨∇Gφ(t), s⟩
with the equality holding when s = t, which implies⟨
Vt−1φ, 2φ− BVt−1φ⟩
= Gφ(t)−
⟨∇Gφ(t), t⟩, Then, (c) is proved. To prove (d),
Gφ(s) = min u, v1, . . . , vn Bu +
∑n j=1
Cjvj = φ
⟨u, Bu⟩ +
∑n j=1
⟨vj, Cjvj⟩ sj
with (u, vj) = (Vs−1φ, sjVs−1φ) being an optimal solution. Then, the same argument used in proving (c) yields (d).
Appendix B
Proof of Proposition 2
First, we give a definition that combines the concepts of majorization and CL1,1(Ω) men-tioned in the Preliminary.
Definition 1. We say{fω}ω∈Ω(uniformly) majorizes f in CL1,1(Ω) if (a) {fω}ω∈Ω(uniformly) majorizes f .
(b) f, fω ∈ CL1,1(Ω) for all ω∈ Ω.
By contrast, we say{fω}ω∈Ωminorizes f in CL1,1(Ω) if{−fω}ω∈Ωmajorizes−f in CL1,1(Ω).
Proposition 3. Consider{fω}ω∈Ωmajorizes f in CL1,1(Ω), G ∈ CM1,1(I) for I ⊂ R.
Then, for all x, y ∈ Ω
(a) |fy(x)− f(x)| ≤ 2L∥x − y∥.
(b) ∥∇fy(x)− ∇f(x)∥ ≤ 2L∥x − y∥.
(c) ∥∇G(fy(x))− ∇G(f(x))∥ ≤ 2M(L2+ L)∥x − y∥ if f(x), fy(x)∈ I.
Proof. We claim∇fy(y) =∇f(y) ∀y ∈ Ω, which follows from f(x) − f(y) ≤ fy(x)−
fy(y)∀x ∈ Rn, y ∈ Ω and
∂xjfy(y) = lim
h↗0
fy(y + hδj)− fy(y) h
≤ lim
h↗0
f (y + hδj)− f(y)
h = ∂xjf (y)
= lim
h↘0
f (y + hδj)− f(y) h
≤ lim
h↘0
fy(y + hδj)− fy(y)
h = ∂xjfy(y).
Then, (a) is proved by
|fy(x)− f(x)| ≤ |fy(x)− fy(y)| + |f(y) − f(x)|
≤ 2L∥x − y∥.
Moreover, (b) is proved by∇fy(y) = ∇f(y) and the same argument used in (a). Then, (c) is proved by
∥∇G(fy(x))− ∇G(f(x))∥
≤ |G′(fy(x))− G′(f (x)| ∥∇fy(x)∥+
|G′(f (x))| ∥∇fy(x)− ∇f(x)∥
≤ 2M(L2+ L)∥x − y∥.
Next, we prove Lemma 2 and Lemma 3, In Lemma 2, we use Definition 1 and semi-algebraic function [1, p.451], which is a function whose graph is a semisemi-algebraic set com-posed by the finite union of polynomial constraints. For example, polynomials and sets of polynomial constraints are trivially semialgebraic. According to [1, p.451], we summarize four properties for later use:
(S1) Indicator functions on semialgebraic sets are semialgebraic.
(S2) Finite sums and products of semialgebraic functions are semialgebraic.
(S3) f (x) = supy∈Cg(x, y)(or infy∈Cg(x, y)) is semialgebraic if C and g are
semialge-braic.
(S4) A semialgebraic function f satisfies the Kurdyka-Łojasiewicz property (KL prop-erty) [1, Def 3.1] on dom ∂f = {x : ∂f(x) ̸= ∅}.
Lemma 2. Underthe assumptions of Proposition 2, there exist αi, βi, γ(δ) > 0, i = 1, . . . , m such that
(a) IE + ϵJ +12∥f − g∥2is semialgebraic.
(b) {lti}t∈E minorizes fi− gi in Cα1,1
i (E).
(c) {hti}t∈E majorizes fi− giin Cβ1,1
i (E).
(d) {δDt}t∈E uniformly majorizes 0 in Cγ1,1(E).
Proof. By dom Dt = [s+,∞)n, and with the function
F (x, y) = 1
2(x− y x
2
)2 = 1
2(x− y)2(1 + y
x)2 ≥ 1
2(x− y)2
on [s, s]2 and F ∈ C2([s, s]2), we prove (d).
To prove (b) and (c), by Proposition 1(a) and dom hti = dom−lit= [s+,∞)n, we have hti,−lti ∈ Γ0(Rn). Then, majorization and minorization follow from (4.2). For the C1,1 regularity, by (4.1) and Proposition 1(b), there exists Li > 0 such that fi− gi ∈ CL1,1i(E).
From these results and the compactness of E, there exists Mi, Ni > 0 such that hti ∈ CM1,1
i(E),−lit∈ CN1,1i(E). Then, (b) and (c) are proved by choosing αi = max{Li, Ni} and βi = max{Li, Mi}.
To prove (a), denote (A.2) byGφ(s) = maxuGφ(s, u), from which and (4.1) we have
fi(s) = sup
u
[Idom fi(s) + Gψi(s, u)] + inf
u [Idom fi(s)− Gθi(s, u)].
Then, fiis semialgebraic by (S1), (S2), and (S3). Since J (s) =∥C(s−d)∥1 = max∥w∥∞≤1⟨w, C(s − d)⟩
is semialgebraic by (S3) andI is semialgebraic by (S1), by those and (S2), (a) is proved.
Lemma 3. Let G(r) = r22, and f1,2 ∈ Γ0(Rn)∩ CL1,1(Ω) with f = max{f1, f2} ≥ 0.
Then, G(f )∈ Γ0(Rn) and for x∈ Ω,
∂G(f (x))⊂ {c∇G(f1(x)) + (1− c)∇G(f2(x)) : c∈ [0, 1]}
Proof. Clearly, G(f )∈ Γ0(Rn). Then, for all y ∈ Rn, p∈ ∂G(f(x)), we have G(f(y)) ≥ G(f (x)) +⟨p, y − x⟩; see Preliminary. Therefore, x = argminyG(f (y))− ⟨p, y⟩. Then, by the definitions of G and f , we have
(x, f (x)) = argmin y, r f1(y)≤ r f2(y)≤ r
r 2
2− ⟨p, y⟩
with constraints that clearly satisfy Slater’s condition [5, (5.26)]. Then, (x, f (x)) satisfies the KKT condition [5, (5.49)]:
p = z1∇f1(x) + z2∇f2(x) (B.1)
f (x) = max{f1(x), f2(x)} = z1+ z2, z1,2 ≥ 0 , z1(f1(x)− f(x)) = z2(f2(x)− f(x)) = 0
which implies that for all c∈ [0, 1]
(z1, z2) =
(f1(x), 0) if f2(x) < f1(x) (0, f2(x)) if f2(x) > f1(x).
( cf1(x), (1− c)f2(x) ) if f2(x) = f1(x) By substituting (z1, z2) into (B.1), we prove the lemma.
Finally, Proposition 2 is proved and generalized by Lemma 2 and Theorem 1.
Theorem 1. Let Ω ⊂ Rn be compact and convex, and R ∈ Γ0(Rn) with |Ω ∩ dom R| > 0. If there exist L, Mi, Ni > 0 such that
(a) {uti}t∈Ωmajorizes vi in CM1,1
i(Ω), i = 1...m.
(b) {wit}t∈Ωminorizes vi in CN1,1
i(Ω), i = 1...m.
(c) {dt}t∈Ωuniformly minorizes 0 in CL1,1(Ω).
(d) F =IΩ+ R + 12∥v∥2 is semialgebraic.
Let rit= max{uti,−wit}. Then, {rit}t∈Ωmajorizes|vi|, and
sk+1 = argmin
s∈Ω dsk(s) + R(s) +1
2∥rsk(s)∥2
converges to a critical point of F for all s0 ∈ Ω.
Proof. Since (a) and (b) imply for i = 1, . . . , m,
|vi| = max{vi,−vi} ≤ max{uti,−wti} = rti ∈ Γ0(Rn),
we have that{rti}t∈Ωmajorizes|vi|.
The convergence is based on [2, Thm 2.9], which states a given sequence xkconverges to a critical point of a proper lower semicontinuous function Φ if∃ α, β > 0, gk ∈ ∂Φ(xk), and a convergent subsequence xkj → ¯x such that four conditions hold:
(C1) Φ satisfies the KL property at ¯x.
(C2) α∥xk+1− xk∥2+ Φ(xk+1)≤ Φ(xk) (C3)∥gk+1∥ ≤ β∥xk+1− xk∥
(C4) F (xkj)→ F (¯x).
By assumptions of Ω, R and v, the F defined in (d) is a proper lower semicontinuous function and there exists skj,bs ∈ Ω such that skj → bs. With these results and (d),(S4),
then (C1) holds.
Since{rti}t∈Ω majorizes|vi|, then (C2) holds by (c) because there exists κ > 0 such that
κ∥sk+1− sk∥2 + R(sk+1) + 1
2∥v(sk+1)∥2
≤ dsk(sk+1) + R(sk+1) + 1
2∥rsk(sk+1)∥2
≤ dsk(sk) + R(sk) + 1
2∥rsk(sk)∥2
= R(sk) + 1
2∥v(sk)∥2
To prove (C3), set G(x) = x2/2 and P = IΩ + R. We have sk+1 = argminsP (s) + dsk(s)+∑m
i=1G(rsik(s)) with P, dsk, G(rsik)∈ Γ0(Rn) and|dom P ∩dom dsk∩∩mi=1dom G(risk)| ≥
|Ω ∩ dom R| > 0. Then, by Fermat’s rule [3, Thm 16.3], the addition rule [3, Cor 16.50], and Lemma 3, there exists pk+1 ∈ ∂P (sk+1), qk+1 ∈ ∂∑m
i=1G(rsik(sk+1)) =
∑m
i=1∂G(rsik(sk+1)), and ci ∈ [0, 1] such that
pk+1+∇dsk(sk+1) + qk+1 = 0, qk+1 =
∑m i=1
ci∇G(usik(sk+1)) + (1− ci)∇G(−wsik(sk+1)).
From these results and∇G(vi(sk+1)) =∇G(−vi(sk+1)), we have
pk+1+
∑m i=1
∇G(vi(sk+1))
=−∇dsk(sk+1)+
∑m i=1
ci
[∇G(vi(sk+1))− ∇G(usik(sk+1)) ]
+
∑m i=1
(1− ci)
[∇G(−vi(sk+1))− ∇G(−wsik(sk+1)) ]
.
Since (a) and (b) imply−Ni ≤ wsik(sk+1)≤ vi(sk+1)≤ usik(sk+1)≤ Mifor all i, k, then
G∈ CK1,1i([−Ni, Mi]) for some Ki > 0. From these results and Proposition 3, we have
∥pk+1+
∑m i=1
∇G(vi(sk+1))∥
≤ 2 (
L +
∑m i=1
Ki(Mi2+ Mi+ Ni2+ Ni) )
∥sk+1− sk∥.
Then, (C3) holds since pk+1 +∑m
i=1∇G(vi(sk+1)) ∈ ∂P (sk+1) +∇(12∥v(sk+1)∥2) =
∂F (sk+1) [23, Exer 8.8(c)].
To prove (C4), it is sufficient to prove limj→∞P (skj) = P (bs). Since P ∈ Γ0(Rn), it is sufficient to prove lim supj→∞P (skj)≤ P (bs), which is proved by
P (bs) ≥ P (skj) +
⟨∇dskj −1(skj) + qkj, skj − bs⟩
and
∥∇dskj −1(skj) + qkj∥ ≤ L +
∑m i=1
Ki(Mi+ Ni).
Appendix C
Proximity Operator Formulas
To compute prox[σTδ] in Algorithm 1 line 10, by separation of variables, the problem reduces to solving
(ˆr1, ˆr2) = argmin
r1,2≥0 ω(r1+ r2)2+ (r1− a1)2+ (r2− a2)2
= argmin
r1,2≥0 ω(r1+ r2)2+ (r1− a+1)2+ (r2− a+2)2, where ω = σδ, a+i = max{ai, 0} for i = 1, 2. Then, we have
(ˆr1, ˆr2) =
(c1, c2) if c1,2 ≥ 0 (b1, 0) if c1 ≥ 0, c2 ≤ 0 (0, b2) if c1 ≤ 0, c2 ≥ 0
∅ if c1,2 < 0
,
where
b1,2 = a+1,2
ω + 1, c1,2 = (ω + 1)a+1,2− ωa+2,1
2ω + 1 .
From the relation b1,2− c1,2 = ω+1ω c2,1, we have
ˆ
ri = P[0,bi](ci), i = 1, 2.
To compute P in Algorithm 1 line 14, by separation of varibale, the problem reduces
to solving
(ˆu1, ˆu2) = argmin s ≤ u1 ≤ s u1u2 ≥ (ski)2
(u1− p1)2+ (u2− p2)2.
By the change of variables (u1, u2) = (skix1, skix2), the problem reduces to
(ˆx1, ˆx2) = argmin
x1 ∈ I, x ∈ C(x1− a1)2+ (x2− a2)2
for some a ∈ R2, compact interval I ⊂ (0, ∞), and the set C = {x ∈ R2 : x1x2 ≥ 1}.
Then, we have
ˆ
x1 = PI((PC(a))1) , xˆ2 = max{a2, 1/ˆx1}.
To compute PC(a), by whose symmetric property we can simply consider a∈ D = {a ∈ R2 : a /∈ C, a21 ≤ a22}, we have
PC(a)|a∈D = (x, 1/x),
where x is the unique positive solution of
x4− a1x3+ a2x− 1 = 0, a ∈ D.
To solve the quartic equation, we refer to [24] and derive a formula in Theorem 2.
Theorem 2. For a∈ D, let p = 4 − a1a2, q = a21− a22,
Then, we have
1. y≥ 0 and y3+ py + q = 0.
which implies
a2 =
g+h−+ g−h+ if a2 ≤ 0 g+h++ g−h− if a2 > 0 .
Then, for a2 ≤ 0, we have
0 = (x2 + g−x + h−)
= (x2 + g−x + h−)(x2+ g+x + h+)
= x4+ (g++ g−)x3+ (g+g−+ h++ h−)x2+ (g+h−+ g−h+)x + h+h−
= x4− a1x3+ a2x− 1.
For a2 > 0, we have the same result by the same argument.
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