DESARROLLO DEL

Numerical experiments were conducted to demonstrate the performance of each of the methods discussed above in terms of computational complexity. The experiments timed how long it took to evaluate a convex relaxation of one of the states δi at 36 p

values while varying the number of discretization points (size of the system) used to approximate the solution of the PDE. This simulates the computational complexity associated with evaluating the objective function of the convex lower-bounding prob- lem as a function of problem size. The results of the numerical experiments are shown in Figure 5-7. The results for the multiple-inclusion implementation were omitted for clarity.

It was observed that sufficiently tight relaxations of δ were obtained after NxNyNz

iterations of the relaxation technique discussed in Section 4.3.3. The unfortunate consequence of this result is the increased computational cost of calculating these relaxations. As discussed in the methods section above, the non-preconditioned and diagonally preconditioned implementations exhibit O(knz) complexity. For this ex- ample,

nz = 4NyNz+ 4(Nx− 2)Nz+ 4(Nx− 2)(Ny− 2) + 7(Nx− 2)(Ny− 2)(Nz− 2),

which is on the order of nx. Therefore, since k = nx, the complexity of calculating

relaxations of δ was hypothesized to be O(n2

x). As can be seen from Figure 5-7, the

(numerical) experimental data supports this hypothesis.

As discussed in the Methods section above, the Gauss elimination implementa- tions and the dense preconditioned implementation were hypothesized to haveO(n3_x)

Nodes O b je c t iv e F u nc. E v a lua t io n ( se c) y = 0.0002x2.1121 y = 2E-06x2.7778 y = 2E-06x2.8509 y = 1E-05x2.9917 y = 0.0001x2.1347 B C D F E 2 10 103 0 10 1 10 2 10 3 10 4 10 5 10

Figure 5-7: The scaling of the evaluation of convex relaxations for each method as a function of the problem size. Method B: Non-preconditioned impl. Method C: Direct method w/RCMK. Method D: Direct method. Method E: Diagonally preconditioned imp. Method F: Inverse-midpoint (dense) preconditioned imp.

complexity with the reordered implementation being slightly less expensive (due to minimizing fill-in). Figure 5-7 supports this hypothesis as well. Interestingly, the Gauss elimination implementations required less CPU seconds to calculate relax- ations for nx < 729 (corresponding to Nx = Ny = Nz = 9). This is likely due

to the overhead of calculating interval bounds using the (parametric) interval linear solver, which is equivalent to the interval Gauss-Seidel iteration [101]. Since the di- rect methods scale worse with system size than the iterative method, even with this overhead, the iterative methods perform better than the direct methods for nδ ≥ 729

(Nx = Ny = Nz ≥ 9). In order to adequately model the system, Nx = Ny = Nz ≥ 15

is required, and so for this model, the iterative methods perform more favorably.

5.4

Concluding Remarks

In this chapter, the results developed in Section 4.3.3 were explored further and applied to large sparse systems which commonly arise in engineering applications. A brief background on large sparse systems was given including various approaches for

solving them numerically as well as computational effort and storage requirements. However, the main contribution is a parameter-estimation case study motivated by an engineering design problem. The computational effort of the iterative relaxation method of Section 4.3.3 was compared against the direct relaxation method of [88]. The method of Section 4.3.3 performed favorably as it scaled with the square of the dimension of the problem as opposed to the cubic scaling of the direct method of [88]. These initial results demonstrate that the iterative approach of Section 4.3.3 for constructing relaxations may be quite effective for large sparse systems and with future research and the proper computer implementation, global optimization of large sparse systems can be solved quickly and efficiently.

Chapter 6

Relaxations of Implicit Functions

Revisited

In this chapter, the theoretical arguments behind the construction of convex and concave relaxations of implicit functions are revisited. In particular, in Chapter 4, the assumptions regarding uniqueness of an implicit function x on P may be too restrictive in some cases. The effects of relaxing the uniqueness assumptions are explored in this chapter.

6.1

Direct Relaxation of Fixed-Point Iterations

In Section 4.3.1, Assumption 4.3.1 was applied, which stated that for P ∈ IRnp_{, there} exists an implicit function x : P → Rnx _{such that it is a fixed-point of the function} ϕ (as defined in that section). Furthermore, it was assumed that an interval X was known such that x(P )⊂ X and x(p) is unique in X for all p ∈ P .

However, these assumptions are not entirely necessary for the main results of that section (Theorems 4.3.4, 4.3.5, and 4.3.6) to hold. The results of that section are generalized as follows.

Theorem 6.1.1. Let ϕ be defined as in Section 4.3.1. Suppose ϕ has n fixed-points xi(p), i = 1, . . . , n, for every p ∈ P such that xi(P ) ⊂ X for some X ∈ IRnx for

Proof. As long as xi(P ) ⊂ X, i = 1, . . . , n, the proof of Theorem 4.3.4 holds under

the relaxed hypotheses since xk,c₍· ) and xk,C₍· ) are relaxations of x

i(· ) = ϕ(xi(· ), · )

for i = 1, . . . , n on P .

Theorem 6.1.2. Suppose the hypotheses of Theorem 6.1.1 hold. Then, Theorem

4.3.5 holds.

Proof. The proof of Theorem 4.3.5 holds here with x replaced by xi, i = 1, . . . , n. Theorem 6.1.3. Suppose the hypotheses of Theorem 6.1.1 hold. Then, Theorem

4.3.6 holds.

Proof. The proof of Theorem 4.3.6 holds here without modification.

If more than one fixed point of ϕ exists in X, it is likely that the hypotheses of Theorem 4.3.6 will hold and therefore the relaxations calculated by applying Theorem 6.1.1 will not be improvements on some of the interval bounds.

6.2

Relaxations of Solutions of Parametric Linear

Systems

In Section 4.3.3, Assumption 4.3.9 was applied. Assumption 4.3.9-1 is essentially the same as Assumption 4.3.1 but stated with respect to parametric linear systems. In that section, it was already discussed how Assumption 4.3.9-2 can be relaxed by introducing a preconditioning matrix Y ∈ Rnx×nx_{. The results of Section 4.3.3 are} generalized as follows.

Theorem 6.2.1. Let A and b be as in Section 4.3.3 and suppose either Assumption

4.3.9-2 holds or that we have a proper preconditioning matrix Y. Suppose that there exist n implicit functions δi, i = 1, . . . , n, such that A(p)δi(p) = b(p), i = 1, . . . , n,

for all p ∈ P and δi(P ) ⊂ ∆, i = 1, . . . , n for ∆ ∈ IRnx. Then, Theorem 4.3.12

Proof. As long as δi(P ) ⊂ ∆, i = 1, . . . , n holds, the proof of Theorem 4.3.12 holds

under the relaxed hypotheses since δk,c and δk,C are relaxations of δi, i = 1, . . . , nx

on P .

Theorem 6.2.2. Suppose the hypotheses of Theorem 6.2.1 hold. Then, Theorem

4.3.14 holds.

Proof. The proof of Theorem 4.3.14 holds here with δ replaced with δ, i = 1, . . . , n.