Christopher Bailey-KelloggDartmouth College

6211 Sudikoff LaboratoryHanover, NH 03755

cbk@cs .dartmouth .edu

Abstract

Recent rapid advances in MEMS and information pro-cessing technology have enabled a new generation of Alrobotic systems - so-called Smart Matter systems --that are sensor rich and physically embedded . Thesesystems range from decentralized control systems thatregulate building temperature (smart buildings) to ve-hicle on-board diagnostic and control systems that in-terrogate large amounts of sensor data . One of thecore tasks in the construction and operation of theseSmart Matter systems is to synthesize optimal controlpolicies using data rich models for the systems and en-vironment . Unfortunately, these models may containthousands of coupled real-valued variables and are pro-hibitively expensive to reason about using traditionaloptimization techniques such as neural nets and geneticalgorithms . This paper introduces a general mech-anism for automatically decomposing a large modelinto smaller subparts so that these subparts can beseparately optimized and then combined . The mech-anism decomposes a model using an influence graphthat records the coupling strengths among constituentsof the model . This paper demonstrates the inecha-nism in an application of decentralized optimization fora temperature regulation problem . Performance datahas shown that the approach is much more efficientthan the standard discrete optimization algorithms andachieves comparable accuracy .

IntroductionThe new-generation sensor-rich AI robotic systemspresent a number of challenges . First, these systemsmust reason about large amounts of sensor data inreal time . Second, they must construct and reasonabout large models of - themselves and the environ-ment in order to rapidly determine optimal control re-sponse (Williams & Nayak 1996) . This paper describesan efficient computational mechanism to automate oneof the major tasks in reasoning about distributed phys-ical systems : decomposition of large models for thesesYsten-is into a set of submodels that can be separatelyoptimized and then combined .

This paper also appears in the Proceedings of AAA11999 . Copyright 1999 . American Association for ArtificialIntelligence (www.aaai.org) . All rights reserved .

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Influence-Based Model Decomposition

Feng ZhaoXerox Palo Alto Research Center

3333 Coyote Hill RoadPalo Alto . CA 94304zhao0parc .xerox .com

This paper makes important contributions to qual-itative and model-based reasoning in two ways . (1)The paper introduces a novel graph formalization fora, model decomposition problem upon which powerfulgraph partitioning algorithms can be brought to bear .The graph formalism is applicable to a large numberof problems where the dependency information in amodel of a distributed svstern can be either derivedfrom numerical trial data or reconstructed from analyticdescriptions commonly used in science and engineer-ing . (2) The paper develops two efficient partitioningalgorithms to decompose a large model into submod-els . The first algorithm employs spectral partitioning tomaximize intea-component dependencies (called the in-fluences) while minimizing inter-component dependen-cies . The second algorithm determines weakly coupledgroups of model components by systematically and ef-ficiently examining the structural similarities exhibitedby trial partitions . To illustrate the utility of these algo-ritlims, this paper applies the decomposition algorithmsto a distributed thermal control problem . Performancedata has confirmed that the model decomposition al-gorithms have yielded an efficient control optimizationalgorithm that outperforms standard optimization al-gorithms such as genetic algorithms and simulated an-nealing . Our optimization algorithm exploits the local-ity in the decomposition to attain efficiency and is ableto generate solutions that are interpretable in terms ofproblem structures . These contributions significantlyextend our previous work (Bailey-Kellogg & Zhao 1998)on qualitative models and parametric optimization forlarge distributed physical systems .Other researchers in qualitative reasoning . Bayesian

nets . and image processing have also investigated theproblem of using decomposition to efficiently modelcomplex physical systems . Williams and Millar de-veloped a decomposition algorithm for parameter es-timation that, determines for each unknown variablein a model a minimally overdetermined subset of con-straints (Williams & Millar 1996)'. The algorithms ofthis paper identify similar dependencies among nodes ofa net either from a constraint net or directly from nu-inerical data, and then partition the dependency graphinto nearly decoupled subsets . Clancy introduced an

Wafer

Rings of Lamps(,-d from me wafer luoking up)

Figure 1 : Rapid thermal processing for semiconductormanufacturing maintains a uniform temperature profileby independent control to separate rings of heat lamps .

algorithm for generating an envisionment of a modelexpressed as a qualitative differential equation, once apartition of the model is given by the modeler (Clancy1997) . Our influence-based decomposition algorithmscan produce the model partitions required by Clancy'salgorithm . Recent work in image segmentation has in-troduced measures of dissimilarity to decompose im-ages, based on pixel intensity differences (Shi & Malik1997 ; Felzenszwalb & Huttenlocher 1998) . Friedman etal . in probabilistic reasoning have introduced a methodto decompose a large Bavesian belief net into weakly-interacting components by examining the dependencystructure in the net (Friedman, Koller, & Pfeffer 1998) .Many scientific and engineering applications have ex-

ploited similar insights in order to divide and conquerlarge computational problems . In the well-studied N-body problem, the interactions among particles are clas-sified into near and far field so that they can be decom-posed into a hierarchy of local interactions to achieve alinear-time speed-up (Zhao 1987) . In engineering com-putation, domain decomposition techniques (Chan &Mathew 1994) have been developed to separately simu-late submodels of large models, based on connectivity inthe models . This paper utilizes a similar insight to for-malize the task of model decomposition based on influ-ences in the models . Furthermore, our approach explic-itly represents the physical knowledge and structuresthat it exploits, so that higher-level reasoning mecha-nisms have an explainable basis for their decisions .

Problem DescriptionWe develop the influence-based decomposition mecha-nism for large models typically arising from distributedsensing and control problems . For example, considera distributed thermal regulation system for rapid ther-mal processing in semiconductor curing, where a uni-form temperature profile must be maintained to avoiddefects (Figure 1) . The control strategy is decentral-ized, providing separate power zones for three ringsof heat lamps (Kailath & others 1996) . As a similarexample, rapid prototyping in thermal fabrication canemploy moving plasma-arc heat sources to control thetemperature of parts to be joined (Doumanidis 1997) .

Abstracting these real-world applications, this paperadopts as a running example the generic problem of

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decentralized temperature regulation for a piece of ma-terial . The temperature distribution over the materialmust be regulated to some desired profile by a set of in-dividually controlled point heat sources . Many sensor-rich systems employ such decentralized control that ac-complishes global objectives through local actions inorder to ensure adaptivity, robustness, and scalability .For example, a smart building regulates its tempera-ture using a network of sensors and actuators ; decen-tralized control allows the network to overcome failuresin individual control elements and to scale up withoutincurring exponential complexity. Rapid synthesis ofoptimal control policies for these distributed systemsrequires efficient methods for reasoning about large cou-pled models .

In particular, we focus on the control placement de-sign task : to determine the number and location of heatsources, subject to a variety of structural constraints(e.g . geometry, material properties, and boundary con-ditions) and performance constraints (e.g . maximumoutput and maximum allowed error) . While we onlyconsider the placement design here, we have also stud-ied the parametric optimization (in this example, theactual heat output) and reported it elsewhere (Bailey-Kellogg & Zhao 1998) . The engineering community hasapplied various discrete optimization techniques (e .g .genetic algorithms in (Dhingra & Lee 1994) and simu-lated annealing in (Chen . Bruno, & Salama 1995)) tothe control placement design problem . In contrast tothese techniques, we seek to use domain knowledge toextract and exploit qualitative structural descriptions ofphysical phenomena in the design process . This yieldsa principled method for reasoning about designs anddesign trade-offs, based on an encapsulation of deepknowledge in structures uncovered for a particular prob-lem . This in turn supports reasoning about and expla-nation of the design decisions .

The control placement design task will be used as aspecific example illustrating our general mechanism forpartitioning distributed models ; the discussion sectionfurther discusses the generality of our approach . Thegoal here will be to design a placement that aids para-metric optimization, by placing controls so that theyminimally interfere with each other . This is particu-larly appropriate for applications where the placementdesign is performed once, and the parametric designis performed repeatedly (e.g . for various desired tem-perature profiles) . The design approach taken here isto decompose a problem domain into a set of decou-pled, atomic subregions, and then independently designcontrols for the separate subregions . Regions are con-sidered decoupled if the exact control design in one re-gion is fairly independent of the exact control designin another . A region is considered atomic if it needsno further decomposition - control design for the re-gion yields adequate control of the region . Influence-based model decomposition provides a powerful high-level mechanism for achieving such designs .

1 2

Figure 2 : Thermal hill for a heat source .

Influence GraphIn order to design decentralized controls for a physi-cal field, it is necessary to reason about the effects ofthe controls on the field . We previously introduced theinfluence graph (Bailey-Kellogg & Zhao 1998) to repre-sent such dependencies . Figure 2 shows an example ofinfluences in a temperature domain - a "thermal hill,"with temperatures decaying away from the location ofa heat source . When multiple sources affect a thermalfield, their influences interact, yielding multiple peaksand valleys .The influences in this example obey the locality prin-

ciple : a heat source strongly affects nearby field nodesand only weakly affects further away field nodes, de-pending on the conduction properties of the mate-rial . In addition, despite nonlinearities in the spatialvariables (e.g . non-uniform conduction characteristicsor irregular geometry), influences from multiple heatsources can be linearly superposed to find joint influ-ences . These properties are characteristic of a varietyof physical phenomena . In order to take advantage ofthese and other properties, influence graphs serve asan abstract, domain-independent representation of thisknowledge . The definition of influence graph assumes adiscretized model, as in (Bailey-Kellogg & Zhao 1998),and as is common to standard engineering methods .Definition 1 (Influence Graph) An

influencegraph is a tuple (F, C, E ; w) where

" F is a set of field nodes ." C is a set of control nodes." E = C x F is a set of edges from control nodes to

field nodes." w : E -4 TZ is an edge weight function with w((c, f))

the field value at f given a unit control value at c.

Hence, the graph edges record a normalized influencefrom each control node to each field node . A thermalhill (e.g . Figure 2) is a pictorial representations of theedge weights for an influence graph from one heat sourceto the nodes of a temperature field .An influence graph is constructed by placing a control

with unit value at each control location of interest, oneat a time, and evaluating the field at field node loca-tions of interest . The method of evaluation is problem-specific . For example, it could be found by numericalsimulation, experimental data, or even explicit inver-sion of a capacitance matrix . An influence graph then

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serves as a high-level interface caching the dependencyinformation . The following two sections use this depen-dency information in order to decompose models basedon influences between their parts .

Graph DecompositionAs discussed in the introduction, many applications re-quire decomposing models into smaller pieces, in orderto reason more tractably with the components or todivide-and-conquer a problem. Many models can beformalized in terms of graphs describing the structuraldependencies of components . Decomposing a modelis then equivalent to partitioning the correspondinggraph. Given a graph, a decomposition identifies sub-sets of vertices so that a metric such as the number ofedges or total edge weight between vertices in differentsubsets is minimized . In particular, the decompositionof an influence graph partitions a model so that theconnected components maximize internal influence andminimize external influence . For the running exampleof decentralized heat control, this decomposes a ther-mal field so that controls in one part are maximallyindependent from those in other parts .The numerical analysis and computational geometry

communities have developed a number of methods forpartitioning graphs ; these methods have varying costsand varying effectiveness . In particular, we concentratehere on spectral partitioning (Simon 1991), which ex-amines the structure of a graph's Laplacian matrix en-coding the connectivity between points . Specifically,entry (i, j) in the Laplacian matrix has value -1 if andonly if there is an edge from node i to node j in thegraph ; entry (i, i) has value equal to the total numberof edges from node i . It turns out that a .good approx-imation to the optimal partitioning (minimum numberof cut edges) can be achieved by separating nodes ac-cording to the corresponding values in the eigenvectorfor the first non-trivial eigenvalue of this matrix (theFiedler vector) . Intuitively, in a one-dimensional do-main, this is similar to partitioning the domain by look-ing at the sign of a sine wave stretched over it . Thistechnique can be extended to minimize the total weightof edges cut, and normalization of edge weights allowssimultaneous optimization of both inter-partition dis-similarity and intea-partition similarity. Shi and Malikshowed, in the context of image segmentation, that thisapproach yields a good estimate of the optimal decom-position (Shi & Malik 1997) .

This novel formalization of control placement de-sign in terms of influence graph partitioning providesa graph-based framework in which to develop designalgorithms . The spectral partitioning algorithm servesas one instantiation of this framework, based on an all-to-all influence graph . The next section introduces analternative approach that uses a less detailed model,decomposing a model using an influence graph from aset of representative nodes to all the other nodes .

Probes

Probe classes

Figure 3 : Overview of influence-based decentralizedcontrol design for thermal regulation .

Equivalence Class PartitioningThe equivalence class partitioning mechanism decom-poses a model based on structural similarities exhibitedin a field in response to the actions of sample controlprobes . Figure 3 overviews the mechanism . In the ex-ample shown, the geometric constraint imposed by thenarrow channel in the dumbbell-shaped piece of mate-rial results in similar field responses to the two probesin the left half of the dumbbell and similar responses tothe two probes in the right half of the dumbbell . Basedon the resulting classes, the field is decomposed into re-gions to be separately controlled . In this case, the lefthalf of the dumbbell is decomposed from the right half.The following subsections detail the components of

this mechanism .

Control Probes

For a temperature field to exhibit structures, heatsources must be applied ; then an influence graph canbe constructed . For example, Figure 4 shows theiso-influences resulting from two different heat sourceplacements ; in both cases, the structure of the contoursindicates the constraint on heat flow due to the nar-row channel . The control placement design algorithmis based on the response of temperature fields to suchcontrol probes . The number and placement of controlprobes affects the structures uncovered in a tempera-ture field, and thus the quality of the resulting con-trol design . Possible probe placement strategies includerandom, evenly-spaced, or dynamically-placed (e.g . ininadequately explored regions or to disambiguate incon-sistent interpretations) . Experimental results presentedlater in this paper illustrate the trade-off between num-ber of probes and result quality ; using the simple ran-dom probe placement strategy.

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Controls

Figure 4 : Temperature fields exhibit structures in re-sponse to heat source probes .

Probes serve as representatives for the effects ofarbitrarily-placed controls . In particular, the probesthat most strongly affect a location (e.g . influencegreater than average or a fixed threshold) serve to ap-proximate the effects that would be produced by a con-trol placed at that location . The quality of the approx-imation of controls at arbitrary locations by represen-tative probes depends on the effects of geometry andmaterial properties . Since the influence graph encapsu-lates these effects, it provides a natural mechanism forreasoning about approximation quality .

Using control probes as representatives of controlplacement effects supports reformulation of the decom-position problem into that of partitioning probes intoequivalence classes . Each equivalence class of probesserves as a representatives for a region of strongly-affected potential control locations, as discussed above .A good decomposition produces probe classes with re-gions decoupled from the regions of other classes, andwhich have no acceptable subclasses .

Evaluating Control DecouplingThe first criterion for evaluating a decomposition is thateach region be decoupled from other regions ; that is,that controls in one region have little effect on nodesin the other, and vice-versa . In terms of control probeequivalence classes, decoupling will be evaluated by con-sidering independence of both control placement andcontrol parameters .To evaluate independence of control placement, con-

sider the influence gradient vectors induced by a setof probes ; Figure 5 shows a simple example for twoprobes .' While the flows are different near the loca-tions of the two probes, they are quite similar far awayfrom the probe locations . This similarity is due to con-straints imposed by geometry and material properties ;in this case, the narrow channel of the material effec-tively decouples the left and right halves . A numericalmeasure for the similarity is implemented, for exam-ple, by averaging the angular difference between gradi-ent vectors produced by different probes . This measureevaluates the indistinguishability of control placementwithin the set of probe locations, and thus is correlatedwith a good decomposition into decoupled regions .To evaluate independence of control parameters, we

distinguish between nodes strongly and weakly influ-enced by a control (the near field and the far field, re-

'Recall that an influence graph is constructed from fieldvalues for unit controls . By influence gradient vectors, wemean vectors in the gradient field for a control - rate anddirection of steepest change in field value .

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Figure 6 : An influence hill partitions a field into nearand far fields relative to a control .

spectively), due to locality in the domain (Figure 6) .Possible near/far distinctions include a fixed influencethreshold, a threshold proportional to the peak of thehill, or a threshold based on the "knee" of the hill . Aprobe only weakly affects its far field, and thus can beeffectively decomposed from it . Alternatively, probesthat have significant overlap in their near fields can bereasonably grouped together .

In addition to independence from any single control,a well-decoupled region must be independent from thecombined effects of other controls . That is, for a set ofprobes, the total influence on its controlled region fromcontrols for other probe sets must be sufficiently small .

Evaluating Region AtomicityThe second criterion for evaluating a decomposition isthat each region be decomposed far enough . A regionis considered atomic if none of its subregions are ad-equately decoupled from each other . For example, inFigure 7 a partition {{A, B, C, D}, {E, F, G}} achievesgood decoupling, since the probes in the first class arerelatively independent, from those in the second class .However, it is not atomic, since {A, B, C, D} can befurther decomposed into {{A, B}, {C, D}} .One approach to ensuring atomicity of the classes of

a decomposition is to recursively test subsets of probesto see if they result in. valid decompositions . For exam-ple, by testing partitions of the class {A, B, C, D} forindependence, the partition {{A, B}, {C, D}} would be

Figure 7 : Control probe placement with potential mon-atomic decomposition .

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1-Near field

ar field

Given probes P = {p1,p2, . . .,pn} .

Form classes C = ffPl}, {P2} ; . . . , {p,,}}.

Repeat until stableFor each neighboring ci, cj E C

If Forall k with k i4 i and k 54 jdecoupling of ci and ck anddecoupling of cj and ck are better thandecoupling of c i and cj

And ci U c j is atomicThen replace ci and cj in C with ci U cj

Table 1 : Probe clustering algorithm .

uncovered . The test can use heuristics to avoid testingall possible subclasses . For example, just by examin-ing overlap in influences in the class {A, B, C, D}, thepartition {{A, B}, {C, D}} can be generated as a coun-terexample to the atomicity of {A, B, C, D} . If a classis already small, out-of-class probes can be used in sucha test, and, if necessary, new probes can be introduced .For example, in an atomicity test for {A, B}, checkingindependence of {A, C} from {B, D} would show that{A, B} is indeed atomic . An inexpensive and empiri-cally effective method is to allow grouping of pairs ofprobes only if their near fields sufficiently overlap .

Probe Clustering Algorithm

Based on these criteria, control probes can be clusteredinto decoupled, atomic equivalence classes . One effec-tive clustering method is greedily merging neighboringprobe classes based on similarity. Start with each probein its own class, and form a graph of classes basedon proximity (e.g . Delaunay triangulation or nearnessneighborhood) . Then greedily merge neighboring pairsof classes that are most similar, as long as a regionis strongly influenced by other regions, and until amerger would result in a non-atomic class . Table 1provides pseudocode for this algorithm . Figure 8 il-lustrates a sample probe neighborhood graph . Figure 9depicts some influence gradients for sample probes, andFigure 10 shows the controlled regions for equivalenceclasses of probes after the merging process . While moresophisticated clustering mechanisms could be applied,the results in the next section show this algorithm tobe empirically effective on a set of example problems .

Implementation ResultsThe influence-based decomposition algorithms haveproved effective in designing control placements for de-centralized thermal regulation . . The performance hasbeen measured in two ways : quality of the decompo-sition, and ability of the resulting control design toachieve an objective .Data for three sample problems are given here : a

Figure 8 : Probe clustering example : probe neighbor-hood graph .

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Figure 10 : Probe clustering example : region decompo-sition after merging .

plus-shaped piece of material, a P-shaped piece of ma-terial, and an anisotropic (non-uniform conduction co-efficient) bar . These problems illustrate different ge-ometries, topologies (the P-shaped material has a hole),and material properties . Other problems have also beentested ; the results are similar .The decomposition algorithm forms groups of con-

trol probes with similar effects on the field . This re-quires that probes be dense enough, relative to condi-tions imposed by geometry and material properties, sothat groups of probes with. .similar effects can be uncov-ered . Otherwise, each probe ends up in its own class,and the decomposition is too dependent on probe place-ment . To study the impact of the number of controlprobes on the effectiveness of the resulting design, dif-ferent numbers of probes (4, 8, 16, and 32) were placedat random in a given domain, and results were aver-aged over a number of trial runs . While smarter probeplacement techniques might yield more consistently ef-fective designs, this approach provides a baseline andillustrates the trade-off between number of probes anderror/variance . The probe clustering algorithm used aDelaunay triangulation probe neighborhood graph, anear field based on influence of at least 10 percent ofpeak, and a similarity measure based on flow vector di-rection . For comparison, merging was performed untilfour classes remained .

Decomposition QualityThe goal of a decomposition algorithm is to partitiona domain into regions such that source placement andparametric optimization in each region is relatively in-dependent of that in other regions (decomposed) andhas no internally independent regions (atomic) . Theestimate of the quality of a decomposition used here isbased on a corresponding formalization for image seg-mentation by Shi and Malik (Shi & Malik 1997) : com-pare the total influence from each control location on lo-cations in other regions (decomposed), and the amountof influence from that location on locations in its ownregion (atomic) . To be more specific, define the decom-position quality q (0 < q < 1) for a partition P of a setof nodes S as follows(i is the influence) :

_

ErE R i (c, r)q-~~

GrSES 2(C, S)

For each control node, divide its influence on nodesin its own region by its total influence . Summing thatover each region yields an estimate of the fraction ofcontrol output of any control location in the region thatis used to control the other locations in that region . Thequality measure is combined over all regions by takingthe product of each region's quality .

Figure 11 compares the performance of the equiv-alence class partitioning mechanism with that of thespectral partitioning mechanism . It provides the aver-age error and standard deviation in error over a numberof trial runs, with respect to different numbers of control

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probes . For all three problems, the average quality forthe equivalence class partitioning mechanism naturallydecreases as the number of probes decreases . (Thereis a slight taper in the performance for the plus shape,due to statistical sampling .) Furthermore, the standarddeviation of quality tends to increase as the number ofprobes decreases, since the partition is more sensitive tospecific probe placements . The curve indicates a trade-off between amount of computation and resulting de-composition quality. With enough probes, the qualityis commensurate with that of spectral partitioning .

Control Placement QualityThe ultimate measure of the control design algorithm ishow well a design based on a decomposition can achievea control objective . This section evaluates the abil-ity of decomposition-based control designs to achievea uniform temperature profile. This profile is betterthan other, non-uniform profiles at indicating the per-formance of a decomposition, since it does not dependas much on local placement adjustment and paramet-ric optimization . Intuitively, if a decomposition clumpstogether sources, then some other region will not getenough heat and thus- will have a large error .

Simulated annealing (Metropolis et al . 1953) servesas a baseline comparison for error ; an optimizer was runfor 100 steps . The decomposition-based control designused a simple approach : for each region of a decomposi-tion, place controls in the "center of influence" (like thecenter of mass, but weighted with total influence fromthe probes, rather than mass, at each point) . In bothcases, only the global control placement was designed ;local adjustment could somewhat reduce the error .

Figure 12 illustrates average error and standard de-

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1,3

1 .2P Shape

Figure 12 : Performance data indicate that equivalenceclass-based control placement design supports tradingcontrol quality for computation . Results are relative tosimulated annealing .

viation in error over a set of trial runs . The equiva-lence class algorithm was tested with different numbersof control probes ; as shown above, the spectral parti-tioning algorithm would yield similar results . The er-ror here is the sum of squared difference between actualtemperature profile and desired temperature profile . Aswith decomposition quality, the average and standarddeviation of control quality tend to improve with thenumber of probes ; enough probes yields quality com-mensurate with that of simulated annealing .

Run-Time Performance

There are two implementation strategies to consider interms of run-time performance . The centralized ap-proach explicitly inverts the capacitance matrix describ-ing the system (e.g . from a finite-element mesh), yield-ing influences from every potential control node to everyfield node . The matrix inversion constitutes the bulkof the run time . Spectral partitioning then performs aneigenvalue computation . Simulated annealing examinessome number of configurations (in the tests above, 100)with respect to the effects encoded in this matrix . Ourapproach does simple comparisons between fields dueto different probe locations .The decentralized approach treats the system as a

black box function to be computed for each configu-ration ; the run time primarily depends on the numberof function evaluations . The spectral method is not di-rectly amenable to this approach . Simulated annealingrequires the function to be evaluated for each configu-ration (in the tests above, 100) . Our apprach requiresthe function to be evaluated for each probe (in the testsabove, 4, 8, 16, or 32 times) .

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DiscussionThis paper has developed mechanisms for automaticallydecomposing large models, based on structural repre-sentations of influences . This formalization in terms ofinfluences differs from the formalisms in related engi-neering work, which typically use topology and geome-try, and related AI work, which typically use constraintnets . Redescribing a model decomposition problem asan influence graph partitioning problem allows applica-tion of powerful, general algorithms .We first introduced the influence graph mechanism

in (Bailey-Kellogg & Zhao 1998) as a generic basis forrepresenting and manipulating dependency informationbetween control nodes and field nodes . The propertiesof physical fields that it encapsulates (locality, lineardependence, etc .) are exhibited by a variety of physicalphenomena, from electrostatics to incompressible fluidflow . Note that these phenomena are all governed bydiffusion processes ; it remains future work to developsimilar mechanisms for wave processes . Since they arebased on influence graphs, the model decomposition for-malism and algorithms developed here are applicable tothis wide variety of models.The influence graph provides a common interface to

the model of a distributed physical system, whether itis derived from partial differential equations, simulationdata, or actual physical measurements . In the case ofphysical measurements, sensor outliers can be detectedby comparing the data with the expected value basedon nearby points . Additionally, since the algorithmsreason about the qualitative structures of influences,they are less sensitive to individual sensor errors .The decomposition-based control design algorithms

search a design space in a much different manner fromthat of other combinatorial optimization algorithms,such as genetic algorithms (Holland 1975) and simu-lated annealing (Metropolis et al . 1953) . Rather than(perhaps implicitly) searching the space of all possiblecombinations of source locations, the influence-baseddecomposition approach divides and conquers a prob-lem, breaking a model into submodels based on in-fluences . This approach explicitly forms equivalenceclasses and structures in the domain, rather than im-plicitly representing them in terms of, for example, in-creased membership of highly-fit members in a popula-tion . Since design decisions are based on the influencestructure of the field, this approach supports higher-level reasoning about and explanation of its results ; forexample, a design decision could be explained in termsof constrained influence flows through a field .

ConclusionThis paper has developed efficient influence-basedmodel decomposition algorithms for optimization prob-lems for large distributed models . Model decompositionis formalized as a graph partitioning problem for an in-fluence graph representing node dependencies . The firstalgorithm applies spectral partitioning to an influence

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graph . The second algorithm decomposes a graph us-ing structural similarities among representative controlprobes . Both algorithms reason about the structure ofa problem using influences derived from either a con-straint net or numerical data . Computational experi-ments show that the algorithms compare favorably withmore exhaustive methods such as simulated annealingin both solution quality and computational cost .

AcknowledgmentsThe work is supported iA part by ONR YI grantN00014-97-1-0599, NSF NYI grant CCR-9457802, anda Xerox grant to the Ohio State University .

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