3616

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 11, NOVEMBER 2005

Application of a Parallel Particle Swarm Optimization Scheme to the Design of Electromagnetic Absorbers Suomin Cui, Senior Member, IEEE, and Daniel S. Weile, Senior Member, IEEE

Abstract—The application of parallel (synchronous) particle swarm optimization (PSO) to electromagnetic absorber design is described. Synchronous PSO is a version of PSO that allows for parallelization and hence faster optimization. The velocity updating rule is empirically determined, and a simple approach is presented to improve its robustness. Absorbers designed with the PSO approach are compared with absorbers designed by genetic algorithms (GAs). Numerical results demonstrate that the PSO is able to deliver absorbers with comparable performance to GA-designed absorbers, with much less computational cost. Index Terms—Electromagnetic absorbers, genetic algorithms, optimization, parallel processing, particle swarm optimization.

I. INTRODUCTION

I

N 1995, Kennedy and Eberhart introduced particle swarm optimization (PSO) [1]. PSO is a population-based stochastic optimization technique based on the movement of swarms and inspired by social behavior of bird flocking or fish schooling. In the past several years, PSO has been successfully applied in many different application areas due to its robustness and simplicity [2]–[5]. In comparison with other stochastic optimization techniques like genetic algorithms (GAs) [6]–[8], PSO has fewer complicated operations and fewer defining parameters, and can be coded in just a few lines. Because of these advantages, the PSO has received increasing attention in electromagnetic community in recent years [9]–[13]. PSO is applied to the design of electromagnetic absorbers in this work. Most previous studies of electromagnetic applications of PSO focus on the standard asynchronous PSO scheme in which the global optimum value is updated after each objective function evaluation. Because this rule ensures that the best information at hand is always immediately used, it would be expected to lead to faster convergence. The drawback of the asynchronous approach is that the behavior of each particle depends on all of the particles evaluated previously; thus, the inherent parallelism of population-based optimization techniques is destroyed. This disadvantage severely restricts the practical applications of PSO in electromagnetic engineering optimization, because the problems in this area are often expensive to analyze, and require parallel computing to be successfully optimized. Therefore, it is of importance to develop easily implemented parallel PSO.

Manuscript received January 31, 2005; revised July 21, 2005. This work was supported by the U.S. Department of the Army through Applied EM, Inc., Hampton, VA, under Grant A02-155. The authors are with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2005.858866

In contrast to the asynchronous scheme, the synchronous scheme updates the global optimum only after the fitness values of all particles have been evaluated. Thus, velocity updates can be carried out in parallel, and the inherent parallelism of population-based optimizers like GAs is preserved. This paper will investigate the performance of the parallel PSO by comparing it with GAs in electromagnetic absorber designs, and thus clarify its place in the pantheon of available automated design methods. The rest of this paper will proceed as follows. Section II presents the synchronous PSO algorithm. Sections III and IV apply the new algorithm to design planar and polygonal absorbers, and confirm some advantages of the new technique. Conclusions are given in Section V. II. DESCRIPTION OF SYNCHRONOUS PSO As mentioned above, PSO was inspired by flocks of birds. Suppose that a group of birds is randomly searching for food in an area where only one piece of bread exists. Although none of the birds knows the exact location of the bread, if they each have some idea of their proximity to the bread, they can search the area cooperatively by communicating their positions and proximity to one another. The birds may fly to search the area either asynchronously or synchronously. In the asynchronous scheme, the first bird flies a certain distance and in a certain direction based on his own experience and the known best position found so far. When he is done flying, he communicates this information to the second bird. The second bird then uses this information to update his speed and position, and so on. On the other hand, in the synchronous scheme, all birds stop after flying in the same time interval. Once stopped, they communicate each other to get the current global best position. They then update their directions and speeds based on their own history and the globally best information available, and fly again. Like the analogous birds’ flying schemes, PSO can be classified into asynchronous PSO and synchronous PSO [10]–[12]. Asynchronous PSO can be found in many works [1], [11], [12], [14], [15]. The synchronous PSO, which can be easily parallelized, is described in pseudocode below. Let us consider an -dimensional minimization problem. The vector variable to be optimized will be denoted by , where . In the description of the algorithm, it will be useful to instead refer to a , where , and the position variable are related to the via the formula

0018-926X/$20.00 © 2005 IEEE

(1)

CUI AND WEILE: PARALLEL PARTICLE SWARM OPTIMIZATION SCHEME

3617

The objective function itself will be written as , as it will be simpler to discuss the algorithm ignoring the (trivial) decoding of the decision variables. Before beginning the optimization, a population size (i.e., and a maximum number of iterations number of particles) are chosen. Synchronous PSO can be accomplished by the following algorithm. a) Randomly initialize the population: Select the (normaland velociized) particle positions from unities form distributions with , and . b) Evaluate the objective function values . (Of course, this may require mapping back to the parameter space.) Note that objective function evaluation is the only step dependent on the objective function itself, and that it can be accomplished in parallel. c) Assign the global and local best positions: Set the local , best position for each particle using the formula and compare the evaluated fitness values and find the global , for some , such that best position , for . d) Search for the minimum value of . i) Update the particle velocities according to a predefined velocity calculation formula (this will be discussed in more detail below.). ii) Update all positions using the formula ; check all positions to ensure that . If any of the components of the position vectors go out of bounds, they can be clipped. , . iii) Evaluate then iv) Update the local best positions: if . v) Update the global best position , by letting , , such that , for for some . e) Repeat d) until a goal is reached or the number of iterations is surpassed. The above description is complete, except for the description and the velocity update. We describe of the clipping of the each of these issues in turn. There are several approaches to clip if it is less than zero or larger than one [11]. Different approaches may affect the convergence rate. We use a direct approach: is set to be a random number between zero and one if it is out of the range [0, 1]. The most important factor for success in PSO is the selection of the velocity-updating rule [14], [15]. While there exist many formulas for accomplishing this, all are fairly close to the two rules below. The first rule presented here is the standard PSO update rule, given by rand rand

(2)

indicates a random number chosen from a uniwhere rand and are two accelerform distribution between and , ation constants (often chosen to be two from past experience),

and is the inertia weight factor. Previous studies [14] have shown that a time-dependent weight factor often outperforms a fixed factor. The most common functional form for this weight factor is linear, and changes with time step as follows: (3) is the maximum number of iterations defined above where and and are often selected to be 0.9 and 0.4, respectively. The second rule used here comes from [15] and may be written as rand rand

(4)

where and are subject to and (the values of and are often set to be 2.05). Finally, no matter which of these two rules is chosen, the velocity calculation is ultimately subject to a limiting process in which the components of the velocity are restricted to a value less than some predetermined value. Specifically, the velocity limit rule used in this paper is as follows: sign

(5)

is a positive number between zero and one. where We found after a number of numerical experiments with simple problems that parallel PSO can be made more robust with a very slight change to the algorithm given above. Specially, the parallel PSO described above has the tendency to or , gravitate toward the boundaries, namely, even when these points are suboptimal. To overcome this problem, the component of the vector on the boundary is artificially reset to either a random number or the corresponding component of the current global optimum. While this approach could potentially decrease the efficiency of the search when the optimum is actually located on the boundary, such cases are likely rare in practice. It is worth noting that this phenomenon has not been reported in the literature for asynchronous PSO. We used several toy problems in the literature to check the performance of synchronous PSO and found a few general rules of thumb. 1) PSO with the first velocity updating rule is more robust than that with the second one. 2) PSO incorporating a velocity limit is a slightly better than the PSO without it, and reasonable limits are around 0.4. 3) PSO is competitive with GAs in finding the optimal solution (i.e., PSOs accuracy is comparable to that of GA’s); 4) PSO can be more efficient than GAs; that is, PSO often finds the solution with fewer objective function evaluations than are required by GAs. To demonstrate how these conclusions were reached, results for the optimization of the Griewank function are presented. (These results are meant to be representative of results too extensive to be reported here.) The Griewank function is given by (6)

3618

Fig. 1. (c)

N

Fig. 2.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 11, NOVEMBER 2005

Comparison of the performance of the PSO, BGA, and RGA for different population sizes for Griewank function. (a) = 40, and (d) different population sizes.

Planar four-layer coating absorber.

Fig. 1 displays the convergence of the mean best objective function values for 1000 trial runs of PSO, binary GA (BGA), and real-coded GA (RGA) for different population sizes. From Fig. 1(a) to (c), the PSO converges faster than the GAs when the

N

= 5, (b)

N

= 15,

and . Fig. 1(d) demonstrates population size that PSO running with a small population size can get better results than either BGA or RGA, even when these algorithms are run with larger population sizes . In short, for some problems, the PSO appears as effective and more efficient than the GAs. In the following sections, the PSO is applied to design different types of absorbers, and similar results are obtained. Both the BGA and the RGA used for the above example are standard versions. The BGA used here is the GA170.f coded by Carroll.1 In this BGA, the tournament selection and uniform crossover are used. The niching technique and elitism procedure are applied to enhance the performance of the GA. The RGA used here is the Pikaia.f coded by Charbonneau and Knapp.2 In this RGA, the roulette wheel selection, either one-point or two-point crossover with equal probabilities, and elitism procedure are used. To make the code more consistent with current 1D.

L. Carroll, http://cuaerospace.com/carroll/ga/. Charbonneau and B. Knapp, http://download.hao.ucar.edu/archive/ pikaia/. 2P.

CUI AND WEILE: PARALLEL PARTICLE SWARM OPTIMIZATION SCHEME

3619

Fig. 3. Comparison of the performance of the PSO, BGA, and RGA for different population sizes for the four-layer coating absorber. (a) = 40, and (d) = 80.

10, (c) N

N

TABLE I COMPARISON OF THE PERFORMANCE OF THE THREE DIFFERENT ALGORITHMS ON THE FOUR-LAYER ABSORBER DESIGN PROBLEM

N

= 5, (b) N

=

which also match parameters widely recommended in the literature. The only alteration made to these codes is that the random number generator included in the codes was changed to the random number generator in the Fortran 90 compiler, which is also used in the PSO. III. COATING ABSORBER DESIGNS

practice and with our BGA implementation, the RGA roulette wheel selection was replaced with tournament selection. The parameters (i.e., crossover rate, mutation rate, etc.) used for the GA runs were those recommended by the authors of the codes,

In this section, the parallel PSO described above is applied to design a four-layer coating absorber backed by a perfect electric conductor (PEC) as shown in Fig. 2. Each layer of the coating absorber is occupied by a homogenous material whose thickness, permittivity, and permeability are denoted by , , and ( ), respectively, and these 20 design variables make up the design space. The optimization goal is the minimization of the maximum the reflected power over a given frequency band and incident angle range for both polarizations. The design variables are limited as

3620

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 11, NOVEMBER 2005

Fig. 4. Two polygonal geometrical absorber models. (a) Model 1 and (b) model 2.

follows: cm, , , , . The PSO, BGA, and RGA are used to optiand mize this problem. For the PSO and both GAs, the population size and the maximum number of generations are set to 80 and 300, respectively. The results obtained by these optimizers are shown in Table I. From this table, we can see all these optimizers return almost the same results. Fig. 3 shows the average best objective function value profile for 20 trials for the PSO, BGA, and RGA with different population sizes for the frequency band 20–40 GHz and incident and angle range 0–10 . For the population sizes shown in Fig. 3(a)–(c), the PSO clearly converges faster than the BGA and RGA. In these cases, the PSO converges by iteration 50 to a result better than that achieved by the GAs at generation 150. For a large population size, the convergence rates of both algorithms are almost the same, and the results are very . This example close as shown in Fig. 3(d), which set shows that PSO can get good optimum values by using a very small population size, and thus it confirms the high efficiency are not too of the PSO. Indeed, the PSO results for from an engineering point of different from those for view.

IV. POLYGONAL ABSORBER DESIGNS In this section, PSO is applied to optimize the polygonal geometrical absorbers proposed in [16] and [17]. The absorbers are divided into two regions: a substrate region that is backed by PEC and a shaped region. The absorber is symmetric with respect to the axis. The substrate is occupied by a homogenous nonmagnetic material whose permittivity is given by (material 1 of the material database Table I in [16]). The thickness of the substrate is denoted as . The shaped region is also occupied by another homogenous nonmagnetic ma(material 2 in [16]). terial of permittivity (The fact that these numbers were fixed in PSO when they were

Fig. 5. Four absorbers designed by the PSO and BGA (model 1).

chosen from a database in [16] points to perhaps the most profound drawback of PSO—its difficulty to combine continuous and discrete variables. In cases where database design is a necessity, this advantage of GAs relative to PSO may be much more important than the efficiency difference.) The period is centered on the axis and divided into 2 segments of equal length, and the endpoints of these segments are located at the . points Two different absorber models are optimized here. For abare sorber model 1, the coordinates of the endpoints given by (7) for for

(8)

CUI AND WEILE: PARALLEL PARTICLE SWARM OPTIMIZATION SCHEME

3621

Fig. 6. Frequency response of the four absorbers for both polarizations. (a) TM polarization and (b) TE polarization.

Fig. 7. Comparison of the convergence tendency of three methods for the model 1 absorber. Comparison of (a) PSO with BGA and (b) PSO with RGA.

For absorber model 2, the coordinate is given by

s (the cone heights) are less than 5 cm. The maximum height for both models is thus 5.5 cm. For model 1, the number of the particles was set to 40 and the number of the iterations was set to 150. (The computation carried out using a cluster with 50 slave processors with 1.4 GHz CPU.) We ran the PSO twice and found worst case reflection coefficients of 33.50 and 33.84 dB, respectively. As a basis for comparison, this problem was also investigated by BGA (the population size and the generation number were selected to be 500 to ensure that a good optimum was located) and the results returned were 33.36 and 33.59 dB for two runs [16]. (As will be seen below, a population size of 500 is overkill for this problem. This was only done to ensure a reasonable result.) These values confirm that the PSO achieves reliable design results and is competitive with BGA in terms of finding the optimum values, but show that PSO may be more efficient. The four geometries designed by both methods and the corresponding frequency responses are displayed in Figs. 5 and 6,

for for

is given by (7), and

(9)

are shown in Fig. 4(a) These geometrical models with and (b), respectively. The finite-element boundary integral method [18], [19] is applied to compute the reflection coefficient from the absorber as defined in [16] and [17]. The optimization goal is to minimize the maximum reflection coefficient over a given frequency band (19–37 GHz) at normal incidence for both polarizations. The op. We set timized variables are and cm for both models. For model 1, each of the s is restricted to be less then 0.5 cm. For model 2, (the substrate thickness) is also forced to be less than 0.5 cm, while all other

3622

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 11, NOVEMBER 2005

Fig. 9.

Two selected absorbers designed by PSO and RGA (model 2).

Fig. 8. Comparison of the convergence tendency for the model 2 absorber: PSO versus BGA and RGA.

respectively. As can be seen from Fig. 5, every run of the different optimizers returns a different geometry, but all of the geometries give roughly the same absorptive performance as seen from Fig. 6. This implies that the problem is multimodal, but in such a way that several optima have the same performance. (Of course, the only way we could prove this contention is with an exhaustive search.) This fact provides designers more freedom to construct absorbers according to different requirements, or enforce other constraints not explicitly included in the optimization. To further study the convergence properties of the various algorithms, ten trials of PSO, BGA, and RGA were investigated for different population sizes. Fig. 7(a) compares the convergence properties of the PSO and BGA. This figure demonstrates that when population size is ten, PSO achieves a better objective function value at iteration 50 than BGA at generation 150. Indeed, the value found by PSO at generation 50 is 33.50 dB, which is very close to the value ( 33.58 dB) obtained by BGA with a population size of 40. In this case, PSO is at least four times faster than the GA. When the population size is 40, the convergence rate of PSO is almost identical to that of BGA. Fig. 7(b) compares the convergence properties of PSO and RGA. It is clear that PSO with a population size of ten converges faster than RGA with population sizes of ten and 40; the result by PSO with a population size of ten is better by approximately 0.3 dB than RGA with a population size of 40. For model 2, we ran PSO, BGA, and RGA ten times with different population sizes, and the mean best objective function values are displayed in Fig. 8. From Fig. 8, we can see the following. 1) PSO with a population size of 20 is much better than BGA and RGA with a population of 20. 2) PSO with a population size of 20 obtains a result as good as BGA with a population of 40. 3) RGA with a population size of 40 can get a slightly better result than the other schemes. (The difference between RGA

Fig. 10. Frequency response of the two designed absorbers (model 2) for both polarizations. (a) TM polarization and (b) TE polarization.

with a population size of 40 and PSO with a population size of 20 is 0.3 dB.)

CUI AND WEILE: PARALLEL PARTICLE SWARM OPTIMIZATION SCHEME

4) Finally, for a given population size, RGA is better than BGA. Nonetheless, this example confirms again that PSO with small population sizes can get results as good as the BGA and RGA with large populations. For this example, each run of the PSO, BGA, and RGA returns different geometries. This again demonstrates the multimodal nature of the problem. Fig. 9 shows the selected absorber profiles designed by using PSO and RGA, and Fig. 10 shows the frequency response of the designed absorbers.

V. CONCLUSION In this paper, we investigated the performance of the synchronous PSO, which is inherently parallel. The parallelism of the synchronous PSO makes practical the design of complex absorbers on clusters. It is found that parallel PSO must incorporate a special operator not found in PSO to keep the population from converging suboptimally to the boundary. This synchronous PSO was applied to optimize multilayer coatings and polygonal absorbers for wide-band frequency and/or wide incident range. The apparent advantages of the PSO are summarized below. 1) Simplicity: PSO is simpler to code than GAs. 2) Effectiveness: Numerical simulations for both toy and practical problems in this paper show that PSO is competitive with other relatively complicated algorithms. 3) Efficiency: Numerical results presented in this paper show that, for at least some classes of problems, PSO may be more efficient than GAs. Nonetheless, PSO does seem to have a few drawbacks with respect to GAs, including the difficulty of coding discrete parameters; thus, further extensions of this paper will concentrate on overcoming these foibles of PSO. Moreover, it is not clear that the good results can be returned quickly for broader classes of problems. In such cases, PSO may converge prematurely but could still be useful for a quick, first-cut optimization. It remains to be seen if an algorithm with the flexibility of GAs and the efficiency of PSO can be designed.

REFERENCES [1] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in Proc. IEEE Int. Conf. Neural Networks, Piscataway, NJ, 1995, pp. 1942–1948. [2] S. Naka, T. Genji, T. Yura, and Y. Fukuyama, “A hybrid particle swarm optimization for distribution state estimation,” IEEE Trans. Power Syst., vol. 18, pp. 60–68, 2003. [3] Z. Gaing, “A particle swarm optimization approach for optimum design of PID controller in AVR system,” IEEE Trans. Energy Conv., vol. 19, pp. 384–391, 2004. [4] C. Juan, “A hybrid of genetic algorithm and particle swarm optimization for recurrent network design,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, pp. 997–1006, 2004. [5] M. P. Wachowiak, R. Smolikova, Y. Zheng, J. M. Zurada, and A. S. Elmaghraby, “An approach to multimodal biomedical image registration utilizing particle swarm optimization,” IEEE Trans. Evol. Comput., vol. 8, pp. 289–301, 2004.

3623

[6] D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Reading, MA: Addison-Wesley, 1989. [7] D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans. Antennas Propag., vol. 45, pp. 343–353, Mar. 1997. [8] Y. Rahmat-Samii and E. Michielssen, Eds., Electromagnetic Optimization by Genetic Algorithms. New York: Wiley, 1999. [9] G. Ciuprina, D. Ioan, and I. Munteanu, “Use of intelligent-particle swarm optimization in electromagnetics,” IEEE Trans. Magn., vol. 38, no. 2, p. 10 370, 2002. [10] D. Gies and Y. Rahmat-Samii, “Reconfigurable array designed using parallel particle swarm,” in IEEE Antennas Propagation Soc. Dig., vol. 1, Columbus, OH, Jun. 22–27, 2003, pp. 177–180. [11] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, 2004. [12] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, 2004. [13] U. Baumgartner, C. Magele, and W. Renhard, “Pareto optimality and particle swarm optimization,” IEEE Trans. Magn., vol. 40, no. 2, pp. 1172–1175, 2004. [14] Y. Shi and R. C. Eberhart, “Parameter selection in particle swarm optimization,” in Evolutionary Programming VII: Proc. EP98, New York, 1998, pp. 591–600. [15] M. Clerc and J. Kennedy, “The particle swarm—explosion, stability, and convergence in a multidimensional complex space,” IEEE Trans. Evol. Comput., vol. 6, pp. 58–73, Feb. 2002. [16] S. Cui, A. Mohan, and D. S. Weile, “Pareto optimal design of absorbers using a parallel elitist nondominated sorting genetic algorithm and the finite element-boundary integral method,” IEEE Trans. Antennas Propag., vol. 53, pp. 2099–2107, Jun. 2005. [17] S. Cui and D. S. Weile, “Robust design of absorbers using genetic algorithms and the finite element-boundary integral method,” IEEE Trans. Antennas Propag, vol. 51, pp. 3249–3258, Dec. 2003. [18] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993. [19] J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics. New York: IEEE Press, 1998.

Suomin Cui (M’01–SM’03) received the B.S. degree in physics from Shaanxi Normal University, in 1989, and the M.S. and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1992 and 1995, respectively. In 1995, he joined the Department of Electrical Engineering, Nanjing University of Science and Technology, Nanjing, China, as a Postdoctoral Fellow. In 1997, he became an Associate Professor at the same university. He was a Research Fellow with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo, Japan, from September 1997 to August 2000. He is currently with the Department of Electrical and Computer Engineering, University of Delaware, Newark, as a Postdoctoral Fellow. His primary fields of interest include high-frequency diffraction analysis, computational electromagnetics, RCS computation of complex objects, and optimization methods. Dr. Cui received postdoctoral fellowships from the China Postdoctoral Council in 1995 and the Japan Society for the Promotion of Science in 1997. He received a research fellowship from the International Communication Foundation for Foreign Researchers in Japan in 2000. He received the second and first-class scientific and technical achievement awards from the Ministry of Electronic Industry of China in 1996 and 1997 and the third-class national scientific and technical achievement award in 1998.

3624

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 11, NOVEMBER 2005

Daniel S. Weile (S’92–M’99–SM’05) received the B.S.E.E. and the B.S. degrees in mathematics from the University of Maryland at College Park, in 1994 and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign, in 1995 and 1999, respectively. In 1994, he was with the Institute for Plasma Research developing interactive software for the design of depressed collectors for gyrotron beams. As a Research Assistant and Visiting Assistant Professor at the University of Illinois at Urbana-Champaign, he worked on the efficient design of electromagnetic devices using stochastic optimization techniques, and fast time-domain integral equation methods for the solution of scattering problems. Currently, he is an Assistant Professor of electrical engineering at the University of Delaware, Newark. His current research interests include computational electromagnetics (especially time-domain integral equations), periodic structures, and the use of evolutionary optimization in electromagnetic design. Dr. Weile is a Member of Eta Kappa Nu, Tau Beta Pi, Phi Beta Kappa, and the International Union for Radio Science (URSI) Commission B. He received a 2004 CAREER award from the National Science Foundation and a 2004 Young Investigator award from the Office of Naval Research.