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Scientific Computing

Numerical simulation of real-world phenomena provides fertile ground for building interdisciplinary relationships. The SCI Institute has a long tradition of building these relationships in a win-win fashion – a win for the theoretical and algorithmic development of numerical modeling and simulation techniques and a win for the discipline-specific science of interest. High-order and adaptive methods, uncertainty quantification, complexity analysis, and parallelization are just some of the topics being investigated by SCI faculty. These areas of computing are being applied to a wide variety of engineering applications ranging from fluid mechanics and solid mechanics to bioelectricity.


martin

Martin Berzins

Parallel Computing
GPUs
mike

Mike Kirby

Finite Element Methods
Uncertainty Quantification
GPUs
pascucci

Valerio Pascucci

Scientific Data Management
chris

Chris Johnson

Problem Solving Environments
ross

Ross Whitaker

GPUs
chuck

Chuck Hansen

GPUs
   

Scientific Computing Project Sites:


Publications in Scientific Computing:


Fourier Series of Atomic Radial Distribution Functions: A Molecular Fingerprint for Machine Learning Models of Quantum Chemical Properties
O. A. von Lilienfeld, R. Ramakrishanan, M., A. Knoll. In International Journal of Quantum Chemistry, Wiley Online Library, 2015.

We introduce a fingerprint representation of molecules based on a Fourier series of atomic radial distribution functions. This fingerprint is unique (except for chirality), continuous, and differentiable with respect to atomic coordinates and nuclear charges. It is invariant with respect to translation, rotation, and nuclear permutation, and requires no pre-conceived knowledge about chemical bonding, topology, or electronic orbitals. As such it meets many important criteria for a good molecular representation, suggesting its usefulness for machine learning models of molecular properties trained across chemical compound space. To assess the performance of this new descriptor we have trained machine learning models of molecular enthalpies of atomization for training sets with up to 10 k organic molecules, drawn at random from a published set of 134 k organic molecules with an average atomization enthalpy of over 1770 kcal/mol. We validate the descriptor on all remaining molecules of the 134 k set. For a training set of 10k molecules the fingerprint descriptor achieves a mean absolute error of 8.0 kcal/mol, respectively. This is slightly worse than the performance attained using the Coulomb matrix, another popular alternative, reaching 6.2 kcal/mol for the same training and test sets.



Data Science: What Is It and How Is It Taught?,
H. De Sterck, C.R. Johnson. In SIAM News, SIAM, July, 2015.



A Scalable Algorithm for Radiative Heat Transfer Using Reverse Monte Carlo Ray Tracing,
A. Humphrey, T. Harman, M. Berzins, P. Smith. In High Performance Computing, Lecture Notes in Computer Science, Vol. 9137, Edited by Kunkel, Julian M. and Ludwig, Thomas, Springer International Publishing, pp. 212-230. 2015.
ISBN: 978-3-319-20118-4
DOI: 10.1007/978-3-319-20119-1_16

Radiative heat transfer is an important mechanism in a class of challenging engineering and research problems. A direct all-to-all treatment of these problems is prohibitively expensive on large core counts due to pervasive all-to-all MPI communication. The massive heat transfer problem arising from the next generation of clean coal boilers being modeled by the Uintah framework has radiation as a dominant heat transfer mode. Reverse Monte Carlo ray tracing (RMCRT) can be used to solve for the radiative-flux divergence while accounting for the effects of participating media. The ray tracing approach used here replicates the geometry of the boiler on a multi-core node and then uses an all-to-all communication phase to distribute the results globally. The cost of this all-to-all is reduced by using an adaptive mesh approach in which a fine mesh is only used locally, and a coarse mesh is used elsewhere. A model for communication and computation complexity is used to predict performance of this new method. We show this model is consistent with observed results and demonstrate excellent strong scaling to 262K cores on the DOE Titan system on problem sizes that were previously computationally intractable.

Keywords: Uintah; Radiation modeling; Parallel; Scalability; Adaptive mesh refinement; Simulation science; Titan



TOD-Tree: Task-Overlapped Direct send Tree Image Compositing for Hybrid MPI Parallelism
A. V. P. Grosset, M. Prasad, C. Christensen, A. Knoll, C. Hansen. In Eurographics Symposium on Parallel Graphics and Visualization (2015), Edited by C. Dachsbacher, P. Navrátil, 2015.

Modern supercomputers have very powerful multi-core CPUs. The programming model on these supercomputer is switching from pure MPI to MPI for inter-node communication, and shared memory and threads for intra-node communication. Consequently the bottleneck in most systems is no longer computation but communication between nodes. In this paper, we present a new compositing algorithm for hybrid MPI parallelism that focuses on communication avoidance and overlapping communication with computation at the expense of evenly balancing the workload. The algorithm has three stages: a direct send stage where nodes are arranged in groups and exchange regions of an image, followed by a tree compositing stage and a gather stage. We compare our algorithm with radix-k and binary-swap from the IceT library in a hybrid OpenMP/MPI setting, show strong scaling results and explain how we generally achieve better performance than these two algorithms.



An inter-model comparison of three computation fluid dynamics techniques for step-up and step-down street canyon flows
R. Stoll, E. Pardyjak, J.J. Kim, T. Harman, A.N. Hayati. In ASME FEDSM/ICNMM symposium on urban fluid mechanics, August, 2014.



Computational Determination of the Modified Vortex Shedding Frequency for a Rigid, Truncated, Wall-Mounted Cylinder in Cross Flow
A. Faucett, T. Harman, T. Ameel. In Volume 10: Micro- and Nano-Systems Engineering and Packaging, Montreal, ASME International Mechanical Engineering Congress and Exposition (IMECE), International Conference on Computational Science, November, 2014.
DOI: 10.1115/imece2014-39064



Data-Parallel Halo Finding with Variable Linking Lengths
W. Widanagamaachchi, P.-T. Bremer, C. Sewell, L.-T. Lo; J. Ahrens, V. Pascucci. In Proceedings of the 2014 IEEE 4th Symposium on Large Data Analysis and Visualization (LDAV), pp. 27--34. November, 2014.

State-of-the-art cosmological simulations regularly contain billions of particles, providing scientists the opportunity to study the evolution of the Universe in great detail. However, the rate at which these simulations generate data severely taxes existing analysis techniques. Therefore, developing new scalable alternatives is essential for continued scientific progress. Here, we present a dataparallel, friends-of-friends halo finding algorithm that provides unprecedented flexibility in the analysis by extracting multiple linking lengths. Even for a single linking length, it is as fast as the existing techniques, and is portable to multi-threaded many-core systems as well as co-processing resources. Our system is implemented using PISTON and is coupled to an interactive analysis environment used to study halos at different linking lengths and track their evolution over time.



In-situ feature extraction of large scale combustion simulations using segmented merge trees
A.G. Landge, V. Pascucci, A. Gyulassy, J.C. Bennett, H. Kolla, J. Chen, P.-T. Bremer. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC 2014), New Orleans, Louisana, IEEE Press, Piscataway, NJ, USA pp. 1020--1031. 2014.
ISBN: 978-1-4799-5500-8
DOI: 10.1109/SC.2014.88

The ever increasing amount of data generated by scientific simulations coupled with system I/O constraints are fueling a need for in-situ analysis techniques. Of particular interest are approaches that produce reduced data representations while maintaining the ability to redefine, extract, and study features in a post-process to obtain scientific insights.

This paper presents two variants of in-situ feature extraction techniques using segmented merge trees, which encode a wide range of threshold based features. The first approach is a fast, low communication cost technique that generates an exact solution but has limited scalability. The second is a scalable, local approximation that nevertheless is guaranteed to correctly extract all features up to a predefined size. We demonstrate both variants using some of the largest combustion simulations available on leadership class supercomputers. Our approach allows state-of-the-art, feature-based analysis to be performed in-situ at significantly higher frequency than currently possible and with negligible impact on the overall simulation runtime.



Efficient I/O and storage of adaptive-resolution data
S. Kumar, J. Edwards, P.-T. Bremer, A. Knoll, C. Christensen, V. Vishwanath, P. Carns, J.A. Schmidt, V. Pascucci. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, IEEE Press, pp. 413--423. 2014.
DOI: 10.1109/SC.2014.39

We present an efficient, flexible, adaptive-resolution I/O framework that is suitable for both uniform and Adaptive Mesh Refinement (AMR) simulations. In an AMR setting, current solutions typically represent each resolution level as an independent grid which often results in inefficient storage and performance. Our technique coalesces domain data into a unified, multiresolution representation with fast, spatially aggregated I/O. Furthermore, our framework easily extends to importance-driven storage of uniform grids, for example, by storing regions of interest at full resolution and nonessential regions at lower resolution for visualization or analysis. Our framework, which is an extension of the PIDX framework, achieves state of the art disk usage and I/O performance regardless of resolution of the data, regions of interest, and the number of processes that generated the data. We demonstrate the scalability and efficiency of our framework using the Uintah and S3D large-scale combustion codes on the Mira and Edison supercomputers.



Visual Exploration of High-Dimensional Data: Subspace Analysis through Dynamic Projections
SCI Technical Report, Shusen Liu, Bei Wang, J.J. Thiagarajan, P.-T. Bremer, V. Pascucci. No. UUSCI-2014-003, SCI Institute, University of Utah, 2014.

Understanding high-dimensional data is rapidly becoming a central challenge in many areas of science and engineering. Most current techniques either rely on manifold learning based techniques which typically create a single embedding of the data or on subspace selection to find subsets of the original attributes that highlight the structure. However, the former creates a single, difficult-to-interpret view and assumes the data to be drawn from a single manifold, while the latter is limited to axis-aligned projections with restrictive viewing angles. Instead, we introduce ideas based on subspace clustering that can faithfully represent more complex data than the axis-aligned projections, yet do not assume the data to lie on a single manifold. In particular, subspace clustering assumes that the data can be represented by a union of low-dimensional subspaces, which can subsequently be used for analysis and visualization. In this paper, we introduce new techniques to reliably estimate both the intrinsic dimension and the linear basis of a mixture of subspaces extracted through subspace clustering. We show that the resulting bases represent the high-dimensional structures more reliably than traditional approaches. Subsequently, we use the bases to define different “viewpoints”, i.e., different projections onto pairs of basis vectors, from which to visualize the data. While more intuitive than non-linear projections, interpreting linear subspaces in terms of the original dimensions can still be challenging. To address this problem, we present new, animated transitions between different views to help the user navigate and explore the high-dimensional space. More specifically, we introduce the view transition graph which contains nodes for each subspace viewpoint and edges for potential transition between views. The transition graph enables users to explore both the structure within a subspace and the relations between different subspaces, for better understanding of the data. Using a number of case studies on well-know reference datasets, we demonstrate that the interactive exploration through such dynamic projections provides additional insights not readily available from existing tools.

Keywords: High-dimensional data, Subspace, Dynamic projection



Systematic Debugging of Concurrent Systems Using Coalesced Stack Trace Graphs
D.C.B. de Oliveira, A. Humphrey, Q. Meng, Z. Rakamaric, M. Berzins, G. Gopalakrishnan. In Proceedings of the 27th International Workshop on Languages and Compilers for Parallel Computing (LCPC), September, 2014.

A central need during software development of large-scale parallel systems is tools that help help to identify the root causes of bugs quickly. Given the massive scale of these systems, tools that highlight changes--say introduced across software versions or their operating conditions (e.g., inputs, schedules)--can prove to be highly effective in practice. Conventional debuggers, while good at presenting details at the problem-site (e.g., crash), often omit contextual information to identify the root causes of the bug. We present a new approach to collect and coalesce stack traces, leading to an efficient summary display of salient system control flow differences in a graphical form called Coalesced Stack Trace Graphs (CSTG). CSTGs have helped us understand and debug situations within a computational framework called Uintah that has been deployed at large scale, and undergoes frequent version updates. In this paper, we detail CSTGs through case studies in the context of Uintah where unexpected behaviors caused by different vesions of software or occurring across different time-steps of a system (e.g., due to non-determinism) are debugged. We show that CSTG also gives conventional debuggers a far more productive and guided role to play.



Distortion-Guided Structure-Driven Interactive Exploration of High-Dimensional Data
S. Liu, Bei Wang, P.-T. Bremer, V. Pascucci. In Computer Graphics Forum, Vol. 33, No. 3, Wiley-Blackwell, pp. 101--110. June, 2014.

Dimension reduction techniques are essential for feature selection and feature extraction of complex high-dimensional data. These techniques, which construct low-dimensional representations of data, are typically geometrically motivated, computationally efficient and approximately preserve certain structural properties of the data. However, they are often used as black box solutions in data exploration and their results can be difficult to interpret. To assess the quality of these results, quality measures, such as co-ranking [ LV09 ], have been proposed to quantify structural distortions that occur between high-dimensional and low-dimensional data representations. Such measures could be evaluated and visualized point-wise to further highlight erroneous regions [ MLGH13 ]. In this work, we provide an interactive visualization framework for exploring high-dimensional data via its two-dimensional embeddings obtained from dimension reduction, using a rich set of user interactions. We ask the following question: what new insights do we obtain regarding the structure of the data, with interactive manipulations of its embeddings in the visual space? We augment the two-dimensional embeddings with structural abstrac- tions obtained from hierarchical clusterings, to help users navigate and manipulate subsets of the data. We use point-wise distortion measures to highlight interesting regions in the domain, and further to guide our selection of the appropriate level of clusterings that are aligned with the regions of interest. Under the static setting, point-wise distortions indicate the level of structural uncertainty within the embeddings. Under the dynamic setting, on-the-fly updates of point-wise distortions due to data movement and data deletion reflect structural relations among different parts of the data, which may lead to new and valuable insights.



Analyzing Simulation-Based PRA Data Through Clustering: a BWR Station Blackout Case Study
D. Maljovec, S. Liu, Bei Wang, V. Pascucci, P.-T. Bremer, D. Mandelli, C. Smith. In Proceedings of the Probabilistic Safety Assessment & Management conference (PSAM), 2014.

Dynamic probabilistic risk assessment (DPRA) methodologies couple system simulator codes (e.g., RELAP, MELCOR) with simulation controller codes (e.g., RAVEN, ADAPT). Whereas system simulator codes accurately model system dynamics deterministically, simulation controller codes introduce both deterministic (e.g., system control logic, operating procedures) and stochastic (e.g., component failures, parameter uncertainties) elements into the simulation. Typically, a DPRA is performed by 1) sampling values of a set of parameters from the uncertainty space of interest (using the simulation controller codes), and 2) simulating the system behavior for that specific set of parameter values (using the system simulator codes). For complex systems, one of the major challenges in using DPRA methodologies is to analyze the large amount of information (i.e., large number of scenarios ) generated, where clustering techniques are typically employed to allow users to better organize and interpret the data. In this paper, we focus on the analysis of a nuclear simulation dataset that is part of the risk-informed safety margin characterization (RISMC) boiling water reactor (BWR) station blackout (SBO) case study. We apply a software tool that provides the domain experts with an interactive analysis and visualization environment for understanding the structures of such high-dimensional nuclear simulation datasets. Our tool encodes traditional and topology-based clustering techniques, where the latter partitions the data points into clusters based on their uniform gradient flow behavior. We demonstrate through our case study that both types of clustering techniques complement each other in bringing enhanced structural understanding of the data.

Keywords: PRA, computational topology, clustering, high-dimensional analysis



Systematic Debugging Methods for Large-Scale HPC Computational Frameworks
A. Humphrey, Q. Meng, M. Berzins, D. Caminha B.de Oliveira, Z. Rakamaric, G. Gopalakrishnan. In Computing in Science Engineering, Vol. 16, No. 3, pp. 48--56. May, 2014.
ISSN: 1521-9615
DOI: 10.1109/MCSE.2014.11

Parallel computational frameworks for high performance computing (HPC) are central to the advancement of simulation based studies in science and engineering. Unfortunately, finding and fixing bugs in these frameworks can be extremely time consuming. Left unchecked, these bugs can drastically diminish the amount of new science that can be performed. This paper presents our systematic study of the Uintah Computational Framework, and our approaches to debug it more incisively. Our key insight is to leverage the modular structure of Uintah which lends itself to systematic debugging. In particular, we have developed a new approach based on Coalesced Stack Trace Graphs (CSTGs) that summarize the system behavior in terms of key control flows manifested through function invocation chains. We illustrate several scenarios how CSTGs could help efficiently localize bugs, and present a case study of how we found and fixed a real Uintah bug using CSTGs.

Keywords: Computational Modeling and Frameworks, Parallel Programming, Reliability, Debugging Aids



ASCAC Workforce Subcommittee Letter
B. Chapman, H. Calandra, S. Crivelli, J. Dongarra, J. Hittinger, C.R. Johnson, S.A. Lathrop, V. Sarkar, E. Stahlberg, J.S. Vetter, D. Williams. Note: Office of Scientific and Technical Information, DOE ASCAC Committee Report, July, 2014.
DOI: 10.2172/1222711

Simulation and computing are essential to much of the research conducted at the DOE national laboratories. Experts in the ASCR-relevant Computing Sciences, which encompass a range of disciplines including Computer Science, Applied Mathematics, Statistics and domain sciences, are an essential element of the workforce in nearly all of the DOE national laboratories. This report seeks to identify the gaps and challenges facing DOE with respect to this workforce.

The DOE laboratories provided the committee with information on disciplines in which they experienced workforce gaps. For the larger laboratories, the majority of the cited workforce gaps were in the Computing Sciences. Since this category spans multiple disciplines, it was difficult to obtain comprehensive information on workforce gaps in the available timeframe. Nevertheless, five multi-purpose laboratories provided additional relevant data on recent hiring and retention.

Data on academic coursework was reviewed. Studies on multidisciplinary education in Computational Science and Engineering (CS&E) revealed that, while the number of CS&E courses offered is growing, the overall availability is low and the coursework fails to provide skills for applying CS&E to real-world applications. The number of graduates in different fields within Computer Science (CS) and Computer Engineering (CE) was also reviewed, which confirmed that specialization in DOE areas of interest is less common than in many other areas.

Projections of industry needs and employment figures (mostly for CS and CE) were examined. They indicate a high and increasing demand for graduates in all areas of computing, with little unemployment. This situation will be exacerbated by large numbers of retirees in the coming decade. Further, relatively few US students study toward higher degrees in the Computing Sciences, and those who do are predominantly white and male. As a result of this demographic imbalance, foreign nationals are an increasing fraction of the graduate population and we fail to benefit from including women and underrepresented minorities.

There is already a program that supports graduate education that is tailored to the needs of the DOE laboratories. The Computational Science Graduate Fellowship (CSGF) enables graduates to pursue a multidisciplinary program of education that is coupled with practical experience at the laboratories. It has been demonstrated to be highly effective in both its educational goals and in its ability to supply talent to the laboratories. However, its current size and scope are too limited to solve the workforce problems identified. The committee felt strongly that this proven program should be extended to increase its ability to support the DOE mission.

Since no single program can eliminate the workforce gap, existing recruitment efforts by the laboratories were examined. It was found that the laboratories already make considerable effort to recruit in this area. Although some challenges, such as the inability to match industry compensation, cannot be directly addressed, DOE could develop a roadmap to increase the impact of individual laboratory efforts, to enhance the suitability of existing educational opportunities, to increase the attractiveness of the laboratories, and to attract and sustain a full spectrum of human talent, which includes women and underrepresented minorities.



A survey of high level frameworks in block-structured adaptive mesh refinement packages
A. Dubey, A. Almgren, John Bell, M. Berzins, S. Brandt, G. Bryan, P. Colella, D. Graves, M. Lijewski, F. Löffler, B. O’Shea, E. Schnetter, B. Van Straalen, K. Weide. In Journal of Parallel and Distributed Computing, 2014.
DOI: 10.1016/j.jpdc.2014.07.001

Over the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, providing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available. The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, their design trade-offs and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah.

Keywords: SAMR, BoxLib, Chombo, FLASH, Cactus, Enzo, Uintah



Parallel Breadth First Search on GPU Clusters
SCI Technical Report, Z. Fu, H.K. Dasari, M. Berzins, B. Thompson. No. UUSCI-2014-002, SCI Institute, University of Utah, 2014.

Fast, scalable, low-cost, and low-power execution of parallel graph algorithms is important for a wide variety of commercial and public sector applications. Breadth First Search (BFS) imposes an extreme burden on memory bandwidth and network communications and has been proposed as a benchmark that may be used to evaluate current and future parallel computers. Hardware trends and manufacturing limits strongly imply that many core devices, such as NVIDIA® GPUs and the Intel® Xeon Phi®, will become central components of such future systems. GPUs are well known to deliver the highest FLOPS/watt and enjoy a very significant memory bandwidth advantage over CPU architectures. Recent work has demonstrated that GPUs can deliver high performance for parallel graph algorithms and, further, that it is possible to encapsulate that capability in a manner that hides the low level details of the GPU architecture and the CUDA language but preserves the high throughput of the GPU. We extend previous research on GPUs and on scalable graph processing on super-computers and demonstrate that a high-performance parallel graph machine can be created using commodity GPUs and networking hardware.

Keywords: GPU cluster, MPI, BFS, graph, parallel graph algorithm



Fast Multi-Resolution Reads of Massive Simulation Datasets
S. Kumar, C. Christensen, P.-T. Bremer, E. Brugger, V. Pascucci, J. Schmidt, M. Berzins, H. Kolla, J. Chen, V. Vishwanath, P. Carns, R. Grout. In Proceedings of the International Supercomputing Conference ISC'14, Leipzig, Germany, June, 2014.

Today's massively parallel simulation code can produce output ranging up to many terabytes of data. Utilizing this data to support scientific inquiry requires analysis and visualization, yet the sheer size of the data makes it cumbersome or impossible to read without computational resources similar to the original simulation. We identify two broad classes of problems for reading data and present effective solutions for both. The first class of data reads depends on user requirements and available resources. Tasks such as visualization and user-guided analysis may be accomplished using only a subset of variables with restricted spatial extents at a reduced resolution. The other class of reads require full resolution multi-variate data to be loaded, for example to restart a simulation. We show that utilizing the hierarchical multi-resolution IDX data format enables scalable and efficient serial and parallel read access on a variety of hardware from supercomputers down to portable devices. We demonstrate interactive view-dependent visualization and analysis of massive scientific datasets using low-power commodity hardware, and we compare read performance with other parallel file formats for both full and partial resolution data.



Scalable large-scale fluid-structure interaction solvers in the Uintah framework via hybrid task-based parallelism algorithms
Q. Meng, M. Berzins. In Concurrency and Computation: Practice and Experience, Vol. 26, No. 7, pp. 1388--1407. May, 2014.
DOI: 10.1002/cpe

Uintah is a software framework that provides an environment for solving fluid–structure interaction problems on structured adaptive grids for large-scale science and engineering problems involving the solution of partial differential equations. Uintah uses a combination of fluid flow solvers and particle-based methods for solids, together with adaptive meshing and a novel asynchronous task-based approach with fully automated load balancing. When applying Uintah to fluid–structure interaction problems, the combination of adaptive mesh- ing and the movement of structures through space present a formidable challenge in terms of achieving scalability on large-scale parallel computers. The Uintah approach to the growth of the number of core counts per socket together with the prospect of less memory per core is to adopt a model that uses MPI to communicate between nodes and a shared memory model on-node so as to achieve scalability on large-scale systems. For this approach to be successful, it is necessary to design data structures that large numbers of cores can simultaneously access without contention. This scalability challenge is addressed here for Uintah, by the development of new hybrid runtime and scheduling algorithms combined with novel lock-free data structures, making it possible for Uintah to achieve excellent scalability for a challenging fluid–structure problem with mesh refinement on as many as 260K cores.

Keywords: MPI, threads, Uintah, many core, lock free, fluid-structure interaction, c-safe



An Alternative Formulation of Lyapunov Exponents for Computing Lagrangian Coherent Structures
A.R. Sanderson. In Proceedings of the 2014 IEEE Pacific Visualization Symposium (PacificVis), Yokahama Japan, 2014.

Lagrangian coherent structures are time-evolving surfaces that highlight areas in flow fields where neighboring advected particles diverge or converge. The detection and understanding of such structures is an important part of many applications such as in oceanography where there is a need to predict the dispersion of oil and other materials in the ocean. One of the most widely used tools for revealing Lagrangian coherent structures has been to calculate the finite-time Lyapunov exponents, whose maximal values appear as ridgelines to reveal Lagrangian coherent structures. In this paper we explore an alternative formulation of Lyapunov exponents for computing Lagrangian coherent structures.