A Higher Order Approach to Ontology Evolution in Physics Abstract A higher-order logical approach to ontology evolution is applied to examples in Physics. Based on this approach, a framework is proposed which includes a methodology for the formalisation of ontology evolution in higherorder logic and an implementation of this methodology in the theorem prover Isabelle. The proposed basic mechanisms for evolution are called ontology repair plans. These operate on ontologies formalised as contexts, i.e., as multiple logical theories. In such a setting, ontologies may contradict one another or introduce redundancies with respect to one another, without any of them containing logical contradictions or redundancies. When, though, an inconsistency or a redundancy between two or more ontologies becomes explicit, it may be resolved by the application of an ontology repair plan, as each plan compiles together a pattern for diagnosis and transformation rules for effecting a repair. The repair can combine the retraction and/or addition of axioms as well as the deeper modification of the language in which the ontology is represented. J. Lehmann: GALILEO project. Artificial Intelligence (AI) and, more generally, Computer Science (CS) are presently faced with the challenge of how autonomous software can manipulate its own knowledge. Such knowledge is typically represented in an ontology that conceptualises the entities of the software's application domain and allows the software to reason about such entities at a higher level of abstraction than simply the level of data or information. Just like any abstract model, ontologies are limited representations of the world, which is dynamic and inherently complex. If autonomous systems are to feature any kind of robustness with respect to the dynamics and complexity of changing environments and goals, of communication acts and of new information, they must be able to autonomously update their own ontologies.. The literature on the subject of updating an ontology in the face of new information often uses the phrase ontology evolution and usually concentrates on how Description Logic (dl) axiomatic theories for Semantic Web applications or First Order Logic (fol) theories need to retract axioms or modify entailments in order to maintain their coherence and consistency. Section 2 discusses how this type of work on ontology evolution is related to ours.. Based on this discussion, we present our approach to ontology evolution in higher-order logic (hol). This approach underlies a framework called galileo, for Guided Analysis of Logical Inconsistencies Leads to Evolved Ontologies. There are a few previous publications on galileo, e.g., [5, 7, 8] . Here we go beyond those papers by firmly positioning our proposal with respect to the wider literature on ontology evolution and by presenting a stable integration, standardisation, implementation and evaluation of the framework. 1 galileo includes a methodology for the formalisation of ontology evolution in higher-order logic (hol), as well as an implementation of this methodology in the automated theorem prover Isabelle. As discussed in more detail below, in galileo the term "ontology" is interpreted in an inclusive way, to encompass any representation of knowledge, usually as a logical theory. We have chosen a logic that is rich enough to represent, in a natural way, the object-level concepts and relationships of Physics as well as the meta-level concepts and relationships of ontology evolution processes. This argues for a logic such as hol, which is polymorphic, 2 typed, and higher-order.. In galileo, the basic evolution-mechanisms are called ontology repair plans (orps). Each orp compiles together a pattern for diagnosis of conflicts between ontologies and transformation rules for effecting a repair. For both development and testing, we rely on examples from Physics, as advances in this field may naturally be modelled as cases of ontology evolution and they are usually well documented. Physicists revise predictive theories when confronted with conflicting experimental evidence. Therefore, the orps typically assume there is one ontology representing a predictive theory and other ontologies representing an experimental or observational set-up for that theory. When the experimental ontology contains a theorem that contradicts or is redundant with respect to a theorem of the theoretical ontology, an orp is triggered and amends the two ontologies so that the conflict is resolved. The development methodology of galileo revolves around the selection of initial ontologies by the collection, analysis, formalisation, implementation and testing of appropriate case studies in the history of Physics and of the orps inspired by them.. The initial ontologies are formalised and implemented as contexts, i.e., as multiple logical theories, which are isolated from other theories and which may even use their own local language to describe the entities in the domain. In such a setting, if the ontologies share the same language, they may explicitly contradict one another or introduce redundancies with respect to one another, without any of them containing logical contradictions or redundancies. On the other hand, if the ontologies do not share the same language they may implicitly contradict one another or introduce redundancies with respect to one another without producing an explicit logical contradiction or a redundancy at the global level. When, 1 Further workshop reports and the implementation of galileo can be downloaded from http://dream.inf.ed.ac.uk/projects/ ontology_evolution. 2 Polymorphic means that some terms of the language can have more than one type. though, two or more such ontologies are bridged, i.e., mapped or aligned through a third one which aligns them and resolves their differences, they will be able to share axioms. This will allow the proof of contradictory facts or make some facts redundant, thus affecting the robustness of the system. In both these types of situation (i.e., shared or non shared language) we say that the ontologies are locally consistent or parsimonious but globally inconsistent or redundant. When the global inconsistency or redundancy becomes explicit the application of an orp may resolve the conflict between the two ontologies.. Shortly, we investigate in higher-order logic the problem of ontology evolution from the perspective of automating the mechanisms to repair locally consistent or parsimonious but globally inconsistent or redundant ontologies.. The empirical part of our methodology supports the longterm objectives of the definition of a theory of ontology evolution, as well as of an evolution calculus. In this article we test the following hypothesis:. Automated ontology evolution via orps is computationally feasible and can account for the kinds of ontology evolution that are observed in human problem solving in the Physics domain. We show that desirable properties, e.g., coverage, efficiency, maintainability, high quality of the repairs, can be achieved.. The empirical results provided by the evolutions proposed by galileo allow for the evaluation of all these desirable properties. In particular, the results allow us to conclude that the generality of a higher-order logical approach to ontology evolution is beneficial for the quality of the repairs, in terms of the meaningfulness of the proposed evolutions.. Section 2 discusses related work and makes the case for a higher order approach. Section 3 describes the adopted methodology. Sections 4, 5, 6 present three orps and applications for each them. These three sections are quite dense in formal content, they describe in detail the formal apparatus of our proposal as well as a number of models of case studies to which our orps apply. Section 7 illustrates the implementation of galileo. Section 8 discusses the evaluation. Section 9 discusses further work. Section 10 draws some conclusions. In this article we have presented a higher-order logical approach to ontology evolution in the domain of Physics. As argued in Sect. 2, the need for a higher order approach to the evolution of formal representations of concepts and their relationships (e.g., ontologies, knowledge bases, logical theories, etc.) arises from the inherent lack of generality of existing dl and fol approaches. In order to go beyond axiom retraction and/or addition and achieve subtler forms of automated diagnosis and repair, a meta-logic more expressive than dl or fol is required. We have shown that, especially because of the polymorphism of its symbols, hol can be such metalogic. hol's formulae can abstract over types, number of arguments, etc. thus expanding the logical search space and providing elements for the formulation of declarative theories of ontology evolution-as opposed to the algorithms usually found in the dl-or fol-based literature.. This higher-order approach under-pins a framework which includes both a methodology for the formalisation of ontology evolution in hol and an implementation of this methodology in the theorem prover Isabelle. The main elements of the galileo framework are:. Ontologies formalised as (sub-)contexts by using locales, i.e., as multiple logical theories. Bridges, i.e., contexts that are used to resolve possible linguistic differences between ontologies that do not share exactly the same language. Conflicts between ontologies that are each locally consistent or parsimonious but that globally contradict each other or contain redundancies with respect to one another. These conflicts are inspired by case studies in the history of Physics, in which clashes between theories, experiments, observations, etc. resulted in the field's evolution. Ontology Repair Plans, i.e., mechanisms for the diagnosis of conflicts between locally consistent or parsimonious but globally inconsistent or redundant ontologies as well as for their evolution by means of axiom retraction and/or addition and/or signature change.. The presentation, application and evaluation of three of galileo's ontology repair plans has provided empirical evidence of the value of a hol-based approach to ontology evolution. In Sects. 4 through 6 the orp Where's My Stuff? was applied to the discovery of latent heat and to the postulation of dark matter; Inconstancy was applied to Modified Newtonian Dynamics in the study of galaxies and to the observations that proved the speed of light to be finite; Unite was applied to the identification of the Morning and Evening Stars and to the assessment of the shape of the Earth. This formal apparatus set the stage for the discussion and evaluation of the implementation in Sects. 7 and 8. The implementation is prototyped in the higher-order theorem prover Isabelle, using Locales to represent ontologies and higher-order deduction and matching with heuristics to perform diagnosis and repair. The prototype has proven to be a valuable tool for exploring the space of solutions. On the one hand, the power of hol and hom is unleashed on the case studies, and this results in a number of matches that is often huge. On the other hand, the heuristics have proven effective in curbing this power, by bringing the solution space to a manageable size, yet always returning the historical solution alongside other physically meaningful ones. galileo is prototyped in the higher-order theorem prover Isabelle [29] . Ontologies and bridges are represented as Locales [1] , i.e., as independent proof contexts that can be extended. If a locale M is an extension of a locale L, then all symbols in the signature of L are in the signature of M and all theorems of L are theorems of M, as expected. With the ability to create extensions, merging locales is naturally possible, as merging two locales can be obtained by creating a new locale that extends from both of them. For our application, extensions essentially allow to modularise and express dependencies between the ontologies. Just like the corresponding orps illustrated in Figs. 3, 6 and 9, each of galileo's procedures supports two main reasoning tasks: the diagnosis of the conflict between the considered ontologies and their repair.

O O O O O B I O B O O O O O O B I O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Three-dimensional multiprocessor system-on-chip thermal optimization 3D stacked wafer integration has the potential to improve multiprocessor system-on-chip (MPSoC) integration density, performance, and power efficiency. However, the power density of 3D MPSoCs increases with the number of active layers, resulting in high chip temperatures. This can reduce system reliability, reduce performance, and increase cooling cost. Thermal optimization for 3D MPSoCs imposes numerous challenges. It is difficult to manage assignment and scheduling of heterogeneous workloads to maintain thermal safety. In addition, the thermal characteristics of 3D MPSoCs differ from those of 2D MPSoCs because each stacked layer has a different thermal resistance to the ambient and verticallyadjacent processors have strong temperature correlation.We propose a 3D MPSoC thermal optimization algorithm that conducts task assignment, scheduling, and voltage scaling. A power balancing algorithm is initially used to distribute tasks among cores and active layers. Detailed thermal analysis is used to guide a hotspot mitigation algorithm that incrementally reduces the peak MPSoC temperature by appropriately adjusting task execution times and voltage levels. The proposed algorithm considers leakage power consumption and adapts to inter-layer thermal heterogeneity. Performance evaluation on a set of multiprogrammed and multithreaded benchmarks indicates that the proposed techniques can optimize 3D MPSoC power consumption, power profile, and chip peak temperature. Multiprocessor system-on-chips (MPSoCs) are now widely used in application-specific systems and high-performance computing. They offer performance, design and implementation complexity, power consumption, and thermal benefits over massively superscalar uniprocessor architectures. Their use, and scales, will increase dramatically in the coming years. According to Tony Massimini, chief of technology at semiconductor research and consulting company Semico Research, 16-core processors will be common within the next four years [1] . Intel plans to deliver processors that have dozens or hundreds of cores during the next decade [2] .. Increasing functionality and performance requirements, com- bined with the increasing impact of global interconnect delay, will push designers toward more aggressive technologies in order to increase integration density and decrease communication delay. Using the mainstream two-dimensional (2D) planar CMOS fabrication process, on-chip interconnect shows poor scalability in both performance and power consumption. Three-dimensional (3D) fabrication technology can improve integration density and interconnect delay by using vertical interconnects [3, 4, 5, 6, 7] . Major vendors plan to start shipping 3D MPSoCs incorporating throughsilicon-vias within a year [8] .. Temperature control is one of the key challenges for 3D MPSoC design. Increasing chip power consumption and temperature affect circuit performance, reliability, cooling cost, packaging cost, and power consumption. 3D integration results in power density increasing linearly in the number of vertically-stacked active layers, complicating thermal-aware design relative to 2D MPSoCs. Thermal issues that can sometimes be safely ignored in 2D planar processes (e.g., self-heating, thermal runaway, and temperatureinduced performance degradation) become increasingly prominent in 3D MPSoCs. 3D integration holds promise for MPSoCs. However, before it is practical new solutions are needed for the thermal problems it brings.. Thermal optimization of 3D MPSoCs is complex. An MPSoC must support numerous concurrent tasks. Different tasks exhibit distinct power and thermal characteristics. Managing heterogeneous workloads to optimize performance and temperature is challenging. In addition, the thermal characteristics of 3D MPSoCs differ from those of 2D MPSoCs. In 3D MPSoCs, each active layer has a different thermal resistance to the ambient. Moreover, since the thickness of active layers in 3D MPSoCs is within the range of tens of microns, inter-layer temperature correlation is much higher than intra-layer temperature correlation, i.e., heat flows easily between vertically-adjacent processor cores while lateral core-to-core heat flow is limited. This heterogeneity complicates thermal optimization of 3D MPSoCs. High chip power density and temperature complicate 3D MPSoC design. In this paper, we have described 3D-Wave, a thermal optimization algorithm for 3D MPSoC design. 3D-Wave uses a two-stage optimization flow consisting of (1) a constructive power balancing and minimization algorithm and (2) a thermal analysis driven iterative hotspot mitigation algorithm. Experimental results indicate that 3D-Wave can reduces 3D MPSoC peak temperature below that of similar techniques that optimize power, alone. To the best of our knowledge, this is the first system-level thermal optimization algorithm for 3D MPSoCs. This section describes the 3D MPSoC chip package setup and the benchmarks used to evaluate 3D-Wave.. We consider a two-layer front-to-back eight-core 3D MPSoC architecture. Each silicon layer contains four Alpha 21264 microprocessor cores. Each processor core has a size of 4.56 mm×4.56 mm. The thickness of the top silicon layer is 50 µm. The thickness of the bottom silicon layer is 0.5 mm; this layer is thicker in order to provide mechanical support. There is a 10 µm polyimide glue layer between silicon layers. We model a forced-air cooling solution. The 3D MPSoC chip is attached to a copper heat sink through a 50 µm thermal grease interface layer. We use a detailed thermal analysis algorithm to evaluate heat flow through this stacked wafer thermal model [19] . These analysis results provide guidance to 3D-Wave during optimization.. The proposed 3D MPSoC thermal optimization algorithm is evaluated using testing traces generated from 22 multiprogrammed and multithreaded benchmarks. Table 1 shows the benchmarks we used from MediaBench, ALPBench, and SPEC2000 benchmark suits. The M5 multi-processor full-system simulation environment [28] with integrated dynamic and leakage power models is used to gather the execution and power consumption traces of these benchmarks and convert them into 22 task graphs. The traces are produced by dividing the execution of processes into short discrete time intervals corresponding to tasks and assigning each such task a power consumption based on processor power simulation. Using these 22 task graphs, ten benchmarks were constructed. Each of these is composed of 11 of the 22 task graphs.

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Evaluating the effectiveness of ICT to support globally distributed PBL teams In the 21st century it is becoming increasingly common to work and learn in teams that are globally distributed. Such teams rely heavily on ICT to facilitate communication. There are many communication tools and technologies to choose from. This study investigates the effectiveness of a rich set of tools and considers if synergy exists between them.This paper reports and analyses the use of communication tools by students in a distributed Problem-based Learning (dPBL) environment. The PBL teams consisted of four students from Singapore and four from the UK. Teams had a rich set of synchronous and asynchronous communication tools available to them, including ISDN videoconference, WebCam video/audio, Microsoft NetMeeting® Chat, Asynchronous drop box and threaded discussion Forum.This ethnographic study showed that semantic discussion threads included the use of all tools and enabled students to effectively co-construct meaning (and understanding). Students were able to deploy the use of different tools effectively to achieve the learning outcomes intended for various stages in the PBL cycle, with both asynchronous and synchronous media being considered of high importance. Globally distributed work teams are becoming more common and the range of tools and technologies to support them is also growing. This raises a number of issues for educators: Which tools are appropriate and how should they be used? How can students develop (employability) skills through experiencing such a distributed environment?. This paper reports on a project to explore and analyze these questions in the context of distributed Problem-based Learning (dPBL). In any ethnographic study the validity of the conclusions is heavily dependent on the context and we do not make any strong claims to be able to generalize these. However, we believe that our findings can help others in the design of collaborative environments.. The importance of the rich environment of communication tools became clear in this project, particularly given the global distribution of the teams. This rich set of tools enabled the students to overcome difficulties. For example students regarded synchronous discussion as essential for decision making. However, WebCam video and audio proved ineffective because of bandwidth, latency and language difficulties, though we may expect the technology to improve in the future [6] . Rapanotti [12] also reported benefits from the deployment of voice groupware in a small trial, despite reliability problems.. The students overcame these issues by using Microsoft Messenger® or NetMeeting® chat. They also demonstrated flexibility to arrange meetings to overcome the seven-hour time difference.. Asynchronous Dropbox and Forum facilities were widely used for complementary tasks, such as exchange of information and product delivery (Dropbox) and challenging contributions and constructing shared understanding (Forum).. Our analysis shows that the semantic threads incorporated all of these tools in a rational way, and this emphasizes the need for an integrated set of such tools to encourage the development of highperforming teams.. It is more difficult to draw conclusions regarding the ISDN Video conference. It provided good quality audio and video, but technical difficulties restricted its use to one conference at the end of the first phase. It was also the only tool that was not available 'on-demand'.. In the second phase (Sept-Oct 2003) three videoconferences were scheduled, at the start (for team orientation / introductions), middle and end (for group reflection) of the PBL case. The teams used the middle conference to resolve team conflict and organisational issues.. Differences in results between the two phases suggest that prior experience (and hence preparation) of students is significant in the way they select communication tools.. All the teams involved in this study achieved a high quality solution that incorporated significant input from both UK and Singapore parts of a team. On no occasion did any sub-team withdraw to produce their own solution. They successfully demonstrated interdependence and co-construction of knowledge. Students also achieved the learning outcomes at a high grade.. Cultural differences caused by international distribution of students were not investigated in any detail within this project, although it emerged as a factor in selecting communication tools.. Further to this investigation, the findings from the micro perspective of the individual and team also prompted the team to analyze the dPBL programme from a macro perspective, which is the study of the whole programme as three activity systems. The use of Activity Theory helped shed light on how the team functioned in a community and how it had learnt using the mediation tools at its disposal. The findings from this analysis will be presented at a later date.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


An Analysis of Visual Adaptation and Contrast Perception for Tone Mapping Abstract-Tone Mapping is the problem of compressing the range of a High-Dynamic Range image so that it can be displayed in a Low-Dynamic Range screen, without losing or introducing novel details: The final image should produce in the observer a sensation as close as possible to the perception produced by the real-world scene. We propose a tone mapping operator with two stages. The first stage is a global method that implements visual adaptation, based on experiments on human perception, in particular we point out the importance of cone saturation. The second stage performs local contrast enhancement, based on a variational model inspired by color vision phenomenology. We evaluate this method with a metric validated by psychophysical experiments and, in terms of this metric, our method compares very well with the state of the art. H IGH-DYNAMIC Range (HDR) images constitute a powerful tool to store real-world radiance values and have become more and more accessible during the last 15 years. However, radiances may span many orders of magnitude, while common displays and printers usually span up to two [1] . So, a further "compression step," called "Tone Mapping" (TM), is required to properly visualize the information stored in HDR images. Many tone mapping operators (TMOs) have been proposed in the literature; for a thorough review and analysis of the state of the art until 2005, we refer to the excellent book by Reinhard et al. [2] . Here we just want to give a very brief overview of the different schools of thought that have been proposed so far.. There is a category of perceptually-based TMOs which can be either spatially global or local. The former are, in general, very fast and do not introduce halos or artifacts, but their contrast rendition tends to be poor. The first global TMOs used Stevens' law [3] to achieve range compression and were proposed by Tumblin and Rushmeier [4] , Chiu et al. [5] , and Ward et al. [6] . Schlick [7] proposed a rational function very close to the Naka-Rushton formula (that will be discussed in detail in next section) showing improvements with respect to Stevens' law. The global Naka-Rushton formula has been exploited by Pattanaik et al. [8] , Reinhard and Devlin [9] , and more recently, by Kuang et al. [10] . More sophisticated vision models were also proposed, taking into account timeadaptation such as Ferwerda's model [11] or WeberFechner's law, e.g., the methods by Ward et al. [12] and Ashikhmin [13] . Reinhard et al. [14] and Tamburrino et al. [15] introduced local TMOs that improved detail but created halos and artifacts next to edges. A TMO based on a modified version of the Retinex model of color vision was proposed by Jobson et al. [16] , and the anchoring theory 1 has been used by Krawczyk et al. [17] after a suitable subdivision of the original image into layers of similar luminance.. There is another category, that of gradient-based TMOs, which rely on the idea of shrinking large intensity gradients while preserving small fluctuations, corresponding to fine details. Tumblin and Turk [18] used a hierarchical method based on a Partial Differential Equation (PDE) inspired by anisotropic diffusion. Durand and Dorsey [19] and Kuang et al. [10] obtained improved results by using techniques inspired by bilateral filtering. Fattal et al. [20] and Mantiuk et al. [21] proposed methods where spatially varying compression factors were used to implement suitable manipulations of the gradient field.. Finally, Pardo and Sapiro [22] and Lischinski et al. [23] proposed interactive TMOs in order to produce pleasant results for particular luminance zones selected by a user.. In this paper, we will follow the TM philosophy proposed by Ward et al. [12] , which is that the main purpose of TM is to emulate as much as possible the perception of contrast and color produced by the real-world scene. Lacking a perfect model of the Human Visual System (HVS), we concentrate instead on well-established facts of visual perception (experimental data on visual adaptation and contrast perception) to propose our TM operator.. Dunn et al. [24] refer to visual adaptation as a completely retinal process, while local contrast enhancement (and color constancy) also involves the visual cortex (see Hubel's book [25] ). This suggests that the processes involved in these two phases are quite different; therefore, we have decided to mimic them (or rather their outputs) in two independent stages of our algorithm.. The main contribution of this paper is to propose a TMO which produces good results by trying to replicate basic properties of the HVS. At the same time, we point out and make use of several aspects of the HVS, supported by psycho-physiological data, that we feel have been ignored in the TM literature and which are influential in the quality of the TM results. Some of these aspects are, for example, taking into account localized cone saturation or computing the semisaturation constant in the Naka-Rushton equation as an average of each chromatic channel, but not of the luminance. Let us start this section on the implementation details of our algorithm by saying that we deal with each of the three color channels separately (most TM methods consider the luminance channel and do not operate separately on each color channel, see [2] ). For the experiments we have used the original R, G, and B channels of the images. Hence, in what follows, when we talk about radiance values, averages, etc., we are referring to real-valued images corresponding to one color channel, not vector-valued images corresponding to three channels simultaneously.. As we mentioned in Section 2, several works in neuroscience point out that cones change their sensitivity in proportion to the light level they receive in their same waveband (see [29] , [30] ); that is why we have chosen to compute the background intensity and semisaturation constants independently for each channel, i.e., not based on the luminance of the image, as it is usually done.. For each channel, we compute the background intensity I b as the geometric average between the arithmetic mean and the median of the radiance values: I b ¼ median 0:5 mean 0:5 . We introduce one parameter, , that lets the user specify the overall brightness of the final output, so our estimated background intensity becomes: I b 10 À . The default value for is zero; > 0 increases the darkness of the output, < 0 increases its brightness. The effect of varying can be observed in Fig. 6 first row.. Valeton and van Norren [28] show how the NakaRushton curves shift to the right as I b increases, and how to compute the corresponding semisaturation constant I s from the background intensity. For the latter, we linearly interpolated their data obtaining the function 2 : log 10 I s ¼ log 10 ðI b Þ À 0:37ð4 þ log 10 ðI b Þ À Þ þ 1:9.. The semisaturation value is, by definition, the center value in the four order operative range of the cones, so the inflexion point I M in (6) where we switch from WeberFechner to Naka-Rushton as explained in Section 2 is computed as I M ¼ I s 10 2 . The values for the Weberfractions k in (6) are taken from experimental data in Wyszecki and Stiles [32] :k ¼ 100 1:85 for "red" and "green" cones,k ¼ 100 8:7 for "blue" cones. Following Valeton and van Norren [28] , we take n ¼ 0:74. The value for the constant s 0 is computed so that the curve c is continuous at I M . The value for m is chosen as m ¼ I s 10 À1:2 , following graphs in Wyszecki and Stiles [32] .. After we apply (6) to the input HDR image, we normalize the result (linearly scaling it so that its range becomes ½0; 1). Denoting by I 0 any of the color channels of this image, we now look for the imageĨ minimizing (8) through the gradient descent method by iterating the following procedure until convergence, as proposed by Palma-Amestoy et al. [39] : 2. The data in Valeton and van Norren [28] is in trolands, while calibrated HDR data is in cd m 2 . In log-scale this amounts to a shift that we introduced as the constant 4, which is added to log 10 ðI b Þ.. where Át ¼ 0:2, 0 is the mean of I 0 and R I n ðxÞ ¼ Z wðx; yÞsignðI n ðxÞ À I n ðyÞÞdy; ð10Þ. with w being a normalized 2D Gaussian kernel of effective radius . We choose to set as 30 percent of the number of rows or columns of the image (whichever is greater), and if we wanted to modify its value, we would see an effect on the local contrast properties of the output, as can be observed in Fig. 6 second row.. It is important to note that, since the values for and influence the overall brightness of the output and its local contrast properties, they could be optimized for any particular display: We could use a quality measure like that of Aydin et al. [42] (discussed in the following section) in order to find the values of and that, for a given display, produce less error.. The iteration process stops when the average difference between I n and I nÀ1 is less than 0.5 percent. Then we linearly scale I n so that its maximum is 1 and its minimum is 0, obtaining the final outputĨ. The rationale for this scaling is to make use of all of the available range in the LDR image, but we are aware that this potentially results in contrast stretching, independently in each color channel, possibly leading to color artifacts: We have not observed such problems in our experiments (see the next section) but in any case we want to remark that this last scaling is not at all an essential part of our procedure and, in fact, it may be omitted. Equation (9) has complexity of order OðN 2 Þ, where N is the total number of pixels, but it can be reduced to OðN log NÞ with the numerical approximation technique proposed by Bertalmío et al. [38] . Our implementation is written in C++ and runs under Linux on a computer with 3 GB RAM and two 2.8 GHz processors. For a 1;024 Â 768 image, the first stage of our algorithm takes 1.13 seconds and the second stage takes 10.8 seconds.

O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A filtration method for order-preserving matching The problem of order-preserving matching has gained attention lately. The text and the pattern consist of numbers. The task is to find all the substrings in the text which have the same length and relative order as the pattern. The problem has applications in analysis of time series. We present a new sublinear solution based on filtration. Any algorithm for exact string matching can be used as a filtering method. If the filtration algorithm is sublinear, the total method is sublinear on average. We show by practical experiments that the new solution is more efficient than earlier algorithms. String matching [1] is a widely known problem in Computer Science. Given a text T of length n and a pattern P of length m, both being strings over a finite alphabet , the task of string matching is to find all the occurrences of P in T . The problem of order-preserving matching [2] [3] [4] [5] [6] has gained attention lately. It considers strings of numbers. The task is to find all the substrings (also called factors) u in T which have the same relative order as P , and |u| = |P |. Suppose P = (10, 22, 15, 30, 20, 18, 27) and T = (22, 85, 79, 24, 42, 27, 62, 40, 32, 47, 69, 55, 25) , then the relative order of P matches the substring u = (24, 42, 27, 62, 40, 32, 47) of T , see Fig. 1 .. Several online [7, 5, 3, 4] and one offline solution [2] have been proposed for order-preserving matching. Kubica et al. [4] and Kim et al. [3] presented solutions based on the Knuth-Morris-Pratt algorithm (KMP) [8] . Later, Cho et al. [5, 6] gave a sublinear solution based on the bad character heuristic of the Boyer-Moore algorithm [9] . Al-most at the same time, Belazzougui et al. [7] derived an optimal sublinear solution. We will present a new practical solution based on filtration. We form a modified pattern and use an algorithm for exact string matching as a filtration method. Our approach is simpler and in practice more efficient than earlier solutions. We transform the original pattern P into a binary string P expressing increases (1), equalities (0), and decreases (0) between subsequent pattern positions. Then we search for P in the analogously transformed text T . For example, P = 101001 corresponds to P = (10, 22, 15, 30, 20, 18, 27) and T = 100101001100 to T above. Each occurrence is a match candidate which is verified following the numerical order of the positions of the original pattern P . Note that in this approach any algorithm for exact string matching can be used as a filtration method. If the filtration algorithm is sublinear and the text is transformed on line, the total method is sublinear on average.. We made experiments with two sublinear string matching algorithms and two linear string matching algorithms as the filtering method. Our approach with sublinear filters was considerably faster than the algorithm by Cho et al. [5] , which is the first sublinear solution of the problem. The paper is organized as follows. Section 2 describes the previous solutions for order-preserving matching, Section 3 presents our solution based on filtration, Section 4 analyses the new approach, Section 5 presents and discusses the results of practical experiments, and Section 6 concludes the article. We introduced a new practical solution based on filtration for order-preserving matching. Any exact string matching algorithm can be used as the filtration algorithm. In this paper, we utilized SBNDM2, SBNDM4, FSO, and KMP as the filtration method of our solution. The results of our practical experiments prove that the solutions based on SBNDM2 and SBNDM4 are faster than the earlier BMOPM algorithm. Moreover, the solution based on FSO is still faster for certain short pattern lengths. Research on filters and encodings for order-preserving matching is continuing. After submitting this paper, Chhabra et al. [13] improved filtering time with the SIMD technology. Chhabra et al. [14] applied the same encoding to approximate order-preserving matching. Cantone et al. [15] developed a more sophisticated encoding with a faster filter. We will prove that our approach is sublinear in the average case, if the filtration algorithm is sublinear. Sublinearity means that on average all the characters in the text are not examined.. Let us assume that the numbers in P and T are integers and they are statistically independent of each other and the distribution of numbers is discrete uniform. Let P and T be the transformed pattern and text. Let c be the count of the integer range (i.e. the alphabet size). The probability of one in a position of P or T (as a result of a comparison) is p = (c 2 /2 − c/2)/c 2 = (c − 1)/2c, because there are c 2 integer pairs and c equalities. So the probability of a character match q is. Because adjacent positions in P = b 1 b 2 · · · b m−1 and in T are not independent, let us consider matching of a relaxed We tested four string matching algorithms as filtration methods for order-preserving matching. Two of them, SBNDM2 and SBNDM4 [10] are based on the Backward Nondeterministic DAWG Matching (BNDM) algorithm [1] . In BNDM, each alignment window is processed from right to left like in the Boyer-Moore algorithm [9] by simulating the nondeterministic automaton of the reversed pattern with bitparallelism. SBNDMq starts the processing of each alignment window by reading a q-gram. The third algorithm is Fast Shift-Or (FSO) [11] . We utilized a version of FSO coded by B. Ď urian [10] . FSO was selected because it is fast on short binary patterns [10] . The fourth algorithm is the KMP algorithm [8] ; together with verification it was supposed to approximate the two earlier methods [3, 4] based on KMP. Of the algorithms, SBNDM2 and SBNDM4 are sublinear, whereas FSO and KMP are linear.. The tests were run on Intel 2.70 GHz i7 processor with 16 GB of memory running Ubuntu 12.10. All the algorithms were implemented in C in the 64-bit mode and run in the testing framework of Hume and Sunday [12] . Our solution based on filtration was compared with the BMH approach by Cho et al. [5] (the authors generously let us use their implementation). Because the BMH approach was clearly faster than the KMP-based algorithm [3] and slightly faster than the linear version of the BMH approach in the tests [6] , we tested only the first mentioned algorithm.. For testing we used three texts: a random text and two real texts, which were time series of the Dow Jones index and Helsinki temperatures. The random data contains 1,000,000 random integers between 0 and 2 30 . The Dow Jones data contains 15,248 integers pertaining to the daily values of the stock index in the years 1950-2011 and the Helsinki temperature data contains 6818 integers referring to the daily mean temperatures in Fahrenheit (multiplied by ten) in Helsinki in the years 1995-2005. From each text we picked randomly patterns of length 5, 8, 10, 15, 20, 30, and 50 . Each set contains 1000 patterns for the random text and 200 patterns for the real texts. Table 1 shows the average execution times per pattern of all the algorithms. The unit is 10 milliseconds for real data and one second for random data. In addition, a graph on times for the Dow Jones data is shown in Fig. 2 . The real texts were tested with 180 repeated runs and the random text was tested with 60 repeated runs. In Table 1 , S2OPM represents the algorithm based on SBNDM2 filtration, S4OPM represents the algorithm based on SBNDM4 filtration, BMOPM-q represents the BMH approach [5] for q = 3, 4, 5, KOPM represents the algorithm based on KMP filtration and FSO-OPM represents the algorithm based on Fast Shift-Or.. From Table 1 , it can be seen that in case of real data, S4OPM is a clear winner for most tested values of m, and FSO-OPM is the fastest for m = 5. With all the three data sets, S2OPM is mostly slower than S4OPM but its execution time approaches that of S4OPM as the value of m increases. For m = 50, the execution times of S2OPM and S4OPM are almost equal. Relatively, S2OPM and S4OPM perform better on the real data than on the random data. In the case of Helsinki daily temperatures, the execution times of S2OPM and S4OPM are comparable. In the case of random data, S2OPM and S4OPM are the best for m = 5 and FSO-OPM is the best for m = 8, 10.

O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems Abstract-This paper studies gradient-based schemes for image denoising and deblurring problems based on the discretized total variation (TV) minimization model with constraints. We derive a fast algorithm for the constrained TV-based image deburring problem. To achieve this task, we combine an acceleration of the well known dual approach to the denoising problem with a novel monotone version of a fast iterative shrinkage/thresholding algorithm (FISTA) we have recently introduced. The resulting gradient-based algorithm shares a remarkable simplicity together with a proven global rate of convergence which is significantly better than currently known gradient projections-based methods. Our results are applicable to both the anisotropic and isotropic discretized TV functionals. Initial numerical results demonstrate the viability and efficiency of the proposed algorithms on image deblurring problems with box constraints.

O O O O O B I B I O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O


ROUTE PLANNING AND OPTIMIZATION OF ROUTE USING SIMULATED ANT AGENT SYSTEM This research presents an optimization technique for route planning using simulated ant agents for dynamic online route planning and optimization of the route. It addresses the issues involved during route planning in dynamic and unknown environments cluttered with obstacles and objects. A simulated ant agent system (SAAS) is proposed using modi¯ed ant colony optimization algorithm for dealing with online route planning. It is compared with evolutionary technique on randomly generated environments, obstacle ratio, grid sizes, and complex environments. The evolutionary technique performs well in simple and less cluttered environments while its performance degrades with large and complex environments. The SAAS generates and optimizes routes in complex and large environments with constraints. The traditional route optimization techniques focus on good solutions only and do not exploit the solution space completely. The SAAS is shown to be an e±cient technique for providing safe, short, and feasible routes under dynamic constraints and its e±ciency has been tested in a mine¯eld simulation with di®erent environment con¯gurations and is capable of tracking the moving goal and performs equally well as compared to moving target search algorithm. A swarm is de¯ned as a set of nature-inspired mobile agents that collectively carry out a distributed problem solving. These mobile agents can achieve higher objectives when working in collaborative manner. They are able to communicate with each other directly or indirectly by acting on their local environment. Swarm intelligence *This paper was recommended by Regional Editor Majid Ahmadi. belong to this family. The planner is provided with the complete picture of the environment along with the starting and destination points. The destination point remains constant throughout the planning process. Static environments are those environments which, once discovered completely, do not change with respect to the environment components. The static environment can be known, partially known 9 or unknown and requires a di®erent approach for the each category.. On the other hand, an online planner generates a partial plan during the execution of the task. Online planners are mostly used for dynamic route planning. When the state space changes during the planning phase, the route planning is called dynamic online route planning. 3 The online planning approach is based on the assumption that the agents would be dealing with dynamic environment, and it would neither be feasible nor practical to re-plan the complete route. The online planners are mostly heuristic in nature as the problem instance reveals incrementally and is often prone to slow response time. They require a sophisticated algorithmic approach to¯nd the shortest route.. There are di®erent approaches for online route planning. Some of the online planners generate a route o²ine initially and then modify it during task execution. Some do planning gradually with task execution. While there are some planners that always hold a backup plan as prediction and when the environment changes or previous plan fails, they start executing the backup plan. The unknown and dynamic environments are most di±cult to deal. The tracking of the moving target by ACO can be considered as a dynamic online planning problem. There are a number of constraints to be considered during planning phase. The obstacle avoidance and nding the shortest route for each changing goal are the complex tasks to deal with. The simulated ant agent system has been used successfully for route planning and optimization of routes in static and dynamic environments. The SAAS has been tested with simple to complex environments and was found to be a robust, e±cient and scalable online line route planning system. The route optimizer repairs the generated routes while keeping the number of cells and distance same. The results for the static environment for each map size are also compared with the evolutionary technique and SAAS performed better then the genetic algorithm. For dynamic environments, the SAAS has been compared with the moving target search algorithm and proved to be an e±cient method for dealing with the dynamic environment. The target goals for the SAAS have been changed and SAAS e®ectively tracked the new goals. The distance of each new goal is compared with the A* algorithm and found to be comparable. This paper implements SAAS for mine detection and optimized online route planning for static as well as for the dynamic environment. The consistent experimental results with di®erent size maps have shown the scalability and robustness of the system for handling the dynamic environment. The SAAS can be further tested for other constraint satisfaction and multi-objective optimization problems in di®erent application areas. The experiments have been conducted separately for static goal and for the dynamic goal environment. The starting point and the goal point remains constant throughout each run. Each run consists of 50 iterations with a single ant. The maps have been generated randomly with di®erent obstacle ratios and placement of mines. The environments are from simple to complex. The SAAS has been used to¯nd the optimum route between the starting point and the goal point. The probability of selection for the next node depends on the values of alpha and beta. The alpha value represents the contribution of pheromone level and the value of beta represents the contribution of heuristic function. The higher the value of beta, the more is the in°uence of the distance in the selection of the next node. The¯gure below shows the e®ect of changing the values for alpha and beta on the convergence of SAAS. The paths converge to the optimum values with increasing beta value. The experiments have been conducted for 20 Â 20; 40 Â 40; 60 Â 60, and 80 Â 80 map sizes and the obtained results are consistent in all the maps. The performance of SAAS remains consistent with the increase in map size. The alpha value corresponds to the pheromone level and beta value corresponds to the heuristic, i.e., distance measure from the start point to the goal point. The results show the in°uence of beta on the cells traversed by guiding the ant agents towards the goal. The more the value of beta, the better the results. Each experiment has been conducted 30 times and the average value for each 30 runs is reported in the above¯gures. The starting point and the goal points have been selected as two extreme positions on the map. The maps are generated randomly with a di®erent number of mines, obstacle ratio and complexity. The color coding has been used to distinguish mine, obstacle and empty cell that can be traversed. The feasible routes are of the shortest distance while avoiding mines and obstacles.. The graphs in Figs. 2À5 show the cells traversed for each value of beta while keeping the value of alpha constant. It is observed in each of the graphs below that the route becomes shorter with an increasing value of beta. The results have been compared with ideal solutions using the Manhattan distance as well as the Euclidean distance for heuristic measure and the performance of the simulated ant agent system is comparable with ideal solutions generated by the A* algorithm. The SAAS has performed equally well as compared to the A* algorithm that is known to be the best algorithm to¯nd the shortest distance between two given points. The experiments show the scalability and robustness of the simulated ant agent system by using larger grid maps and its performance remains consistent throughout the simulation run.. The results for simple to complex environments have shown the scalability of the simulated ant agents. The convergence for each run depends on the values of alpha and beta. The table list has been used to restrict the ant agents to new routes and avoids loops. The SAAS without using a table list explores the same nodes again and again and gets trapped in cycles. The table list has been maintained to enhance the performance of SAAS by removing cycles and avoiding loops. The number of mines, obstacles and complexity of the grid maps play an important role in the convergence of the algorithm. The SAAS has been tested for static as well as dynamic goals and shows promising results.. The SAAS has been applied to the dynamic environment for presenting a moving target search. After the selection for the number of times to change the goal state, the SAAS runs for tracing the¯rst goal. After tracing the¯rst goal with the shortest route the goal changes randomly to a new random con¯guration. The next goal appears within 10% of the area of the previous goal. The ant agents continue to evaluate the probabilities of the new node selection and again run the roulette wheel. SAAS is capable of acquiring the moving goal with the shortest route. ACO is primarily an optimization technique and SAAS uses ACO for moving target search in the dynamic environment. The results showed the robustness and scalability of the SAAS in a dynamic environment. Table 5 . The intervals are¯xed to 40 for each run. The distance of the new goal along with the distance from the previous goal was taken into account for comparison. The path traversed by SAAS has been compared with both the heuristics, i.e., Manhattan distance as well as the Euclidean distance. Each experiment is conducted with di®erent obstacle ratios and map sizes.. The tracking of the moving target has been implemented using a simulated ant agent system with di®erent parameters and for di®erent grid sizes. Each experiment uses 5 di®erent goals generated at random locations with 200 iterations as one run. Table 5 shows the results for each goal and compares the route generated by SAAS as cells traversed with the Manhattan distance as well as with the Euclidean distance. The experimentation uses a di®erent obstacle ratio, grid sizes and heuristic measure. The simulation has been tested with di®erent options to generate new goals randomly. In option 1, we select the number of times to change the goal. In option 2, we provide the percentage of goal change during the complete run of iterations while the third option provides the exact gap between iterations to change the goal.. Finally, the results obtained by SAAS in a static environment are compared with the evolutionary technique using the genetic algorithm as shown in Table 6 .. The paths are represented as chromosomes and a population size of 20 has been used. The mutation and one point crossover have been used along with the¯tness function. The paths obtained have been compared with SAAS again with 20 Â 20; 40 Â 40; 60 Â 60, and 80 Â 80 maps and SAAS outperforms the evolutionary algorithm. The convergence rate of genetic algorithm is found to be slow as compared to the convergence rate of SAAS. The results shown in Table 6 clearly show the di®erence in cells traversed by both the techniques. The smallest values for each map have been used with the best values of alpha and beta for comparison. The number of cells traversed for SAAS and GA has been taken as the average of 30 runs.

B I O B O O O O O B O O O O O B O O B I O O O O O O O B I O B O O O O O O O O O B I O O O O O O O O O O O O O B O O O O O O B I I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Semi-supervised learning by search of optimal target vector We introduce a semi-supervised learning estimator which tends to the first kernel principal component as the number of labeled points vanishes. Our approach is based on the notion of optimal target vector, which is defined as follows. Given an input data-set of x values, the optimal target vector y is such that treating it as the target and using kernel ridge regression to model the dependency of y on x, the training error achieves its minimum value. For an unlabeled data set, the first kernel principal component is the optimal vector. In the case one is given a partially labeled data set, still one may look for the optimal target vector minimizing the training error. We use this new estimator in two directions. As a substitute of kernel principal component analysis, in the case one has some labeled data, to produce dimensionality reduction. Second, to develop a semisupervised regression and classification algorithm for transductive inference. We show application of the proposed method in both directions. The problem of effectively combining unlabeled data with labeled data, semi-supervised learning, is of central importance in machine learning; see, for example, [1, 2, 3] and references therein. Semi-supervised learning methods usually assume that adjacent points and/or points in the same structure (group, cluster) should have similar labels; one may assume that data are situated on a low dimensional manifold which can be approximated by a weighted discrete graph whose vertices are identified with the empirical (labeled and unlabeled) data points. This can be seen as a form of regularization [4] . A common feature of these methods, see also [5] , is that, as the number of labeled points vanishes, the solution tends to the constant vector. An interesting survey on semi-supervised learning literature may be found on the web [6] . Improving regression with unlabeled data is the problem considered in [7] , where co-training is achieved using k-NN regressors. A statistical physics approach, based on the Potts model, is described in [8] . An issue closely related to semi-supervised learning is active-learning: some attempts to combine active learning and semi-supervised learning has been made [9] .. The purpose of this work is to introduce a semi-supervised learning estimator which, as the number of labeled points vanishes, tends to the first kernel principal component [10] ; when a suitable number of labeled points is available, it may be used for transductive inference [11] . Our approach is based on the following fact. Given an unlabeled data set, its first kernel principal component is such that, treating it as target vector, supervised kernel ridge regression provides the minimum training error. Now, suppose that you are given a partially labeled data set: still one may look for the target vector minimizing the training error. This optimal target vector may be seen as the generalization of the first kernel principal component to the semi-supervised case.. The paper is organized as follows. In the next Section we describe our approach, while in Section 3 the experiments we performed are described. Some conclusions are drawn in Section 4.. The problem of effectively combining unlabeled data with labeled data, semi-supervised learning, is of central importance in machine learning; see, for example, [1, 2, 3] and references therein. Semi-supervised learning methods usually assume that adjacent points and/or points in the same structure (group, cluster) should have similar labels; one may assume that data are situated on a low dimensional manifold which can be approximated by a weighted discrete graph whose vertices are identified with the empirical (labeled and unlabeled) data points. This can be seen as a form of regularization [4] . A common feature of these methods, see also [5] , is that, as the number of labeled points vanishes, the solution tends to the constant vector. An interesting survey on semi-supervised learning literature may be found on the web [6] . Improving regression with unlabeled data is the problem considered in [7] , where co-training is achieved using k-NN regressors. A statistical physics approach, based on the Potts model, is described in [8] . An issue closely related to semi-supervised learning is active-learning: some attempts to combine active learning and semi-supervised learning has been made [9] .. The purpose of this work is to introduce a semi-supervised learning estimator which, as the number of labeled points vanishes, tends to the first kernel principal component [10] ; when a suitable number of labeled points is available, it may be used for transductive inference [11] . Our approach is based on the following fact. Given an unlabeled data set, its first kernel principal component is such that, treating it as target vector, supervised kernel ridge regression provides the minimum training error. Now, suppose that you are given a partially labeled data set: still one may look for the target vector minimizing the training error. This optimal target vector may be seen as the generalization of the first kernel principal component to the semi-supervised case.. The paper is organized as follows. In the next Section we describe our approach, while in Section 3 the experiments we performed are described. Some conclusions are drawn in Section 4. We have presented a new approach to semi-supervised learning based on the notion of optimal target vector, the target vector such that KRR provides the minimum training error over all the possible target vectors. The proposed (7) are depicted, for a typical instance of four labeled points in the IRIS data set. The star corresponds to the solution with µ < µ1, which has the smallest energy E . We introduce a semi-supervised learning estimator which tends to the first kernel principal component as the number of labelled points vanishes. Our approach is based on the notion of optimal target vector, which is defined as follows. Given an input data-set of x values, the optimal target vector y is such that treating it as the target and using kernel ridge regression to model the dependency of y on x, the training error achieves its minimum value. For an unlabeled data set, the first kernel principal component is the optimal vector. In the case one is given a partially labeled data set, still one may look for the optimal target vector minimizing the training error. We use this new estimator in two directions. As a substitute of kernel principal component analysis, in the case one has some labeled data, to produce dimensionality reduction. Second, to develop a semisupervised regression and classification algorithm for transductive inference. We show application of the proposed method in both directions.. We have presented a new approach to semi-supervised learning based on the notion of optimal target vector, the target vector such that KRR provides the minimum training error over all the possible target vectors. The proposed (7) are depicted, for a typical instance of four labeled points in the IRIS data set. The star corresponds to the solution with µ < µ1, which has the smallest energy E . Concerning IRIS data set and m = 4, we depict the distribution (over 10000 random selections of labeled points) of the linear correlation R between eigenvectors ofH and the true labels. From the left to the right and the top to the bottom, we refer to the first, the second, the third and the fourth eigenvector. Grey (black) histogram bars denote values of R lower (greater) than those of the corresponding fully unsupervised principal component.

B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Aircraft taxi time prediction: Comparisons and insights The predicted growth in air transportation and the ambitious goal of the European Commission to have on-time performance of flights within 1 minute makes efficient and predictable ground operations at airports indispensable. Accurately predicting taxi times of arrivals and departures serves as an important key task for runway sequencing, gate assignment and ground movement itself. This research tests different statistical regression approaches and also various regression methods which fall into the realm of soft computing, to more accurately predict taxi times. Historic data from two major European airports is utilised for cross-validation. Detailed comparisons show that a TSK fuzzy rule-based system outperformed the other approaches in terms of prediction accuracy. Insights from this approach are then presented, focusing on the analysis of taxi-in times, which is rarely discussed in literature. The aim of this research is to unleash the power of soft computing methods, in particular fuzzy rule-based systems, for taxi time prediction problems, and to show that, although these methods have only been recently applied to airport problems, they present promising and potential features for such problems. The latest vision for air transportation in Europe is predicting marked growth in this sector. The European Commission [1] assumes an increase in the global volume of air traffic from 2.5 billion passengers in 2011 to 16 billion passengers in 2050. Thus, the number of commercial flights in Europe per year is expected to increase from 9.4 million to 25 million during the same time period. Nevertheless, one of the formulated goals by 2050 is also that on-time performance of flights is within 1 minute.. Efficient ground movement operations are key to successful operations of air transportation networks [2] . Especially since airport ground movement operations link runway sequencing and gate assignment problems and accurate taxi time predictions support various different stakeholders at airports. For example, the benefits for take-off sequence of having accurate taxi times was shown in [3] and recent developments for Heathrow [4] require accurate taxi times both for take-off sequencing and for allocating pushback times to aircraft, at which they should leave the stands. A significant proportion of the actual travel time can be spent on airport's surfaces especially with short-haul flights. To achieve the stated on-time performance from the European Commission, it is crucial to more accurately predict taxi times at European airports. This will enable runway planning with less uncertainty and will provide important predictions for all of the parties which are involved in the handling of aircraft. Furthermore, it will also be increasingly important to have accurate control of the aircraft on the ground using an advanced decision support system [5] .. Idris et al. [6] published the first paper on taxi-out time estimation based on multiple linear regression, with data from Boston Logan International Airport (BOS). The importance of taking into account the current amount of traffic on the surface, when predicting taxi-out times, was highlighted for the first time. The take-off queue size was found to be an especially important explanatory variable for the problem. With the introduction of Collaborative Decision the taxiways, from the gate or runway exit to the runway entrance or back to the gate. Thereby, total taxi time is aimed to be reduced or, more broadly, it is aimed that the operations serve the other linked airport operations as smoothly as possible, while simultaneously reducing the environmental issues. Several operational constraints have to be considered. Aircraft have time windows when they can start taxiing and when they have to reach the target. For a departing aircraft this consists of an earliest pushback time and either a specific time or a time window at the runway where the aircraft has to take-off. An arriving aircraft has to leave the runway immediately after landing and taxi to the gate, which should be free when the aircraft is expected to be there. Depending upon the layout of an airport, not all aircraft can taxi on all taxiways due to limitations of size or weight. This means that the route may be aircraft type specific. In addition, aircraft cannot cross or overtake on a single taxiway, which makes the routing more complex.. This research supports the aforementioned problem by understanding and more accurately predicting taxi times. Such predictions can be used to make better overall decisions at airports and also to improve the quality of decision support systems for the ground movement problem at airports, by applying the findings and integrating the different aircraft speeds into such models. Airports commonly have many objectives and most will already be automating some of their operations or decision support, or be planning to do so in the future. Some of the aims are to improve predictability, improve on-time performance, reduce ground movement costs, enhance the use of ground handling resources, stands, gates and terminals and reduce apron and taxiway congestion [34] . Key to such improvements are better taxi time predictions, which can help many different decision support systems. This research should also help the development of new and more accurate decision support systems for the ground movement problem. A recent study [15] focused on finding the explanatory variables for taxi time prediction for both arrivals and departures, using multiple linear regression to highlight their statistical significance. This paper uses the same ex- The latter outperformed the other approaches on datasets from two European hub airports. TSK fuzzy rule-based systems use fuzzy membership functions to subdivide the input space in the premise part and a weighted sum of multiple linear regression approaches in the consequent part. As the different fuzzy rules work cooperatively, in contrast to approaches such as M5 model trees, the approach may potentially give more accurate estimates and can also model non-linear patterns in the data. Furthermore, this paper gave insights into the different rules found by the TSK fuzzy rule-based system and considered taxi-in times, which seems to be a less understood problem in this field. It would be interesting to also compare these regression approaches for other airports to see whether these findings can be extended to the busiest airports in the world, and particularly to US airports, where the airport operations are often managed differently or are operated under differing constraints. In addition, this research could be integrated into decision support systems which help controllers in the towers, followed by a fine-tuning phase of the models and the decision support systems to provide more valuable decision-making aids.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A minimum Sobolev norm technique for the numerical discretization of PDEs Partial differential equations (PDEs) are discretized into an under-determined system of equations and a minimum Sobolev norm solution is shown to be efficient to compute and converge under very generic conditions. Numerical results of a single code, that can handle PDEs in first-order form on complicated polygonal geometries, are shown for a variety of PDEs: variable coefficient div-curl, scalar elliptic PDEs, elasticity equation, stationary linearized Navier-Stokes, scalar fourth-order elliptic PDEs, telegrapher's equations, singular PDEs, etc. In this paper we describe a novel numerical technique for the discretization of partial differential equations (PDEs). The technique that we present in this paper shares a lot of ideas in common with non-conforming finite element methods (FEMs) [5] .. The FEM tries to set up discrete sets of square equations; that is, the number of equations exactly matches the number of unknowns. These are delicate to set up for each PDE type [22] and require sophisticated proofs of convergence [5, 15] . However, when successful, they can be very efficient. For example, Maxwell's equations which appear to be non-square in first-order form, can prove difficult to discretize correctly with FEM. Typically they are converted to second-order equations before being discretized [18] . The least-squares FEM [3, 15] , on the other hand, tries to set up a least-squares problem which can then be solved in a variety of ways. The normal equation approach can cause condition number problems and more refined least-squares approaches exist and these can handle first-order systems of PDEs provided the boundary conditions are of the Agmon-Douglis-Nirenberg type [3] .. The method proposed in this paper is a high-order method, that relies primarily on the linear independence of the PDE and its boundary conditions along with the assumption of the existence and uniqueness of a sufficiently smooth solution. The key ideas of the method are to discretize to an under-determined set of equations and then to pick the solution that minimizes a diagonal Sobolev norm. Unlike least-squares method which require determination of the right norm on the range space, the method proposed here requires the right norm on the domain space, and this seems to be an easier choice. 2 , and a set of convex quadrilaterals that cover the domain. The boundaries are shown with thicker lines. The interior of the domain has been shaded a light gray. Note that the curved internal boundary 2 is covered by three overlapping convex quadrilaterals; one of them has red edges, one has light blue edges, and the third has dark blue edges. The vertices are chosen to coincide with the location of the singularities of the solution. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article. ) This method has been observed to maintain high-order convergence even in the presence of curved boundaries. The method prefers that the PDE is presented in first-order system form [21] . In particular, Maxwell type equations and fourth-order equations seem to present no special problems. Some singular PDEs and over-posed PDEs have also been handled. These ideas are further clarified in the rest of the paper.. In Section 9 on numerical experiments we show a single (<400 lines) Octave code being used to solve a variety of PDEs (in two independent variables) on a variety of geometries. This is to be contrasted with existing techniques. All the code, including the geometry files and test files, is freely available from our web-site http :/ /scg .ece .ucsb .edu /octapde .html. We have presented a method for the numerical solution of partial differential equations. The main idea was to use more terms in the bases expansion of the solution than there were equations. The function that minimizes a diagonal Sobolev norm was chosen as the numerical solution. A convergence proof was presented for the Golomb-Weinberger solution. Based on this it was shown that truncating the bases expansions liberally would not loose convergence. Detailed discussion of the numerical solver that can overcome the high ill-conditioning associated with large Sobolev parameter values was presented. The numerical solver was also designed to minimize memory consumption. Several numerical experiments covering a gamut of PDEs and geometries showed the code in action.. In the future the code will be improved to handle inhomogeneous jump conditions and curved boundaries without loss of order of convergence. Applications to eigenvalue problems and non-linear PDEs is being investigated. A detailed theoretical study of the truncation parameter and order of convergence of the method is being carried out. Replacing collocation with Galerkin style functionals is also being investigated.

O B I I O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O