In this paper, we study the propagation of data uncertainty through the marching cubes algorithm for isosurface visualization for correlated uncertain data. Consideration of correlation has been shown paramount for avoiding errors in uncertainty quantification and visualization in multiple prior studies. Although the problem of isosurface uncertainty with spatial data correlation has been previously addressed, there are two major limitations to prior treatments. First, there are no analytical formulations for uncertainty quantification of isosurfaces when the data uncertainty is characterized by a Gaussian distribution with spatial correlation. Second, as a consequence of the lack of analytical formulations, existing techniques resort to a Monte Carlo sampling approach, which is expensive and difficult to integrate into visualization tools. To address these limitations, we present a closed-form framework to efficiently derive uncertainty in marching cubes level-sets for Gaussian uncertain data with spatial correlation (MAGIC). To derive closed-form solutions, we leverage the Hinkley's derivation on the ratio of Gaussian distributions. With our analytical framework, we achieve a significant speed-up and enhanced accuracy of uncertainty quantification over classical Monte Carlo methods. We further accelerate our analytical solutions using many-core processors to achieve speed-ups up to $\text{585} \times$585× and integrability with production visualization tools for broader impact. We demonstrate the effectiveness of our correlation-aware uncertainty framework through experiments on meteorology, urban flow, and astrophysics simulation datasets.

Sparse observations and coarse-resolution climate models limit effective regional decision-making, underscoring the need for robust downscaling. However, existing AI methods struggle with generalization across variables and geographies and are constrained by the quadratic complexity of Vision Transformer (ViT) self-attention. We introduce ORBIT-2, a scalable foundation model for global, hyper-resolution climate downscaling. ORBIT-2 incorporates two key innovations: (1) Residual Slim ViT (Reslim), a lightweight architecture with residual learning and Bayesian regularization for efficient, robust prediction; and (2) TILES, a tile-wise sequence scaling algorithm that reduces self-attention complexity from quadratic to linear, enabling long-sequence processing and massive parallelism. ORBIT-2 scales to 10 billion parameters across 65,536 GPUs, achieving up to 4.1 ExaFLOPS sustained throughput and 74–98% strong scaling efficiency. It supports downscaling to 0.9 km global resolution and processes sequences up to 4.2 billion tokens. On 7 km resolution benchmarks, ORBIT-2 achieves high accuracy with R2 scores in range of 0.98–0.99 against observation data.

Abstract Particle advection is a foundational algorithm for analyzing a flow field. The commonly used Parallelization‐Over‐Data (POD) strategy for particle advection can become slow and inefficient when there are unbalanced workloads, which are particularly prevalent in in situ workflows. In this work, we present an in situ workflow containing workload estimation for block assignment and duplication in a parallelization‐over‐data algorithm. With tightly coupled workload estimation and load‐balanced block assignment strategy, our workflow offers a considerable improvement over the traditional round‐robin block assignment strategy. Our experiments demonstrate that particle advection is up to 3X faster and associated workflow saves approximately 30% of execution time after adopting strategies presented in this work.

This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.

Particle advection is one of the foundational algorithms for visualization and analysis and is central to understanding vector fields common to scientific simulations. Achieving efficient performance with large data in a distributed memory setting is notoriously difficult. Because of its simplicity and minimized movement of large vector field data, the Parallelize over Data (POD) algorithm has become a de facto standard. Despite its simplicity and ubiquitous usage, the scaling issues with the POD algorithm are known and have been described throughout the literature. In this paper, we describe a set of in-depth analyses of the POD algorithm that shed new light on the underlying causes for the poor performance of this algorithm. We designed a series of representative workloads to study the performance of the POD algorithm and executed them on a supercomputer while collecting timing and statistical data for analysis. we then performed two different types of analysis. In the first analysis, we introduce two novel metrics for measuring algorithmic efficiency over the course of a workload run. The second analysis was from the perspective of the particles being advected. Using particle-centric analysis, we identify that the overheads associated with particle movement between processes (not the communication itself) have a dramatic impact on the overall execution time. These overheads become particularly costly when flow features span multiple blocks, resulting in repeated particle circulation (which we term "ping pong particles") between blocks. Our findings shed important light on the underlying causes of poor performance and offer directions for future research to address these limitations.

Understanding the time-varying magnetic field within a fusion device is critical for the successful design and construction of clean-burning fusion power plants. Poincaré analysis provides a powerful method for the analysis and visualization of magnetic field lines in fusion devices. Current state-of-the-art relies on manually and iteratively generating Poincaré plots from simulation data. Using Poincaré plots in deep analysis is very time consuming because Poincaré plots can be very computationally expensive, especially for a time-varying simulation with thousands of time steps. Further, the visualization results are so complex that only expert users know how to explore, interpret, and control. In addition, collaboration is hampered due to the high barrier to entry. To this end, we contribute Fugent, a reinforcement learning-based agent capable of recommending and evaluating the importance of exploration regions based on training data captured from historic expert user usage. Using Fugent, we show that important regions can be identified and recommended for further exploration. Fugent is open source.

The rapid advancement in scientific simulations and experimental facilities has resulted in the generation of vast amounts of data at unprecedented scales. The analysis and visualization of large amounts of data is a challenge in and of itself, but the requirements for timeliness significantly magnify these difficulties. Near real-time visualization is critical to monitor and analyze the data produced by these large facilities, but current production tools are not well-suited to these requirements. In this position paper, we share our perspective on some of the challenges, and thus, opportunities for research that stand in the way of near-real-time visualization of large scientific data.

Color maps are a commonly used visualization technique in which data are mapped to optical properties, e.g., color or opacity. Color maps, however, do not explicitly convey structures (e.g., positions and scale of features) within data. Topology-based visualizations reveal and explicitly communicate structures underlying data. Although our understanding of what types of features are captured by topological visualizations is good, our understanding of people's perception of those features is not. This paper evaluates the sensitivity of topology-based isocontour, Reeb graph, and persistence diagram visualizations compared to a reference color map visualization for synthetically generated scalar fields on 2-manifold triangular meshes embedded in 3D. In particular, we built and ran a human-subject study that evaluated the perception of data features characterized by Gaussian signals and measured how effectively each visualization technique portrays variations of data features arising from the position and amplitude variation of a mixture of Gaussians. For positional feature variations, the results showed that only the Reeb graph visualization had high sensitivity. For amplitude feature variations, persistence diagrams and color maps demonstrated the highest sensitivity, whereas isocontours showed only weak sensitivity. These results take an important step toward understanding which topology-based tools are best for various data and task scenarios and their effectiveness in conveying topological variations as compared to conventional color mapping.

Abstract The computational work to perform particle advection‐based flow visualization techniques varies based on many factors, including number of particles, duration, and mesh type. In many cases, the total work is significant, and total execution time (“performance”) is a critical issue. This state‐of‐the‐art report considers existing optimizations for particle advection, using two high‐level categories: algorithmic optimizations and hardware efficiency. The sub‐categories for algorithmic optimizations include solvers, cell locators, I/O efficiency, and precomputation, while the sub‐categories for hardware efficiency all involve parallelism: shared‐memory, distributed‐memory, and hybrid. Finally, this STAR concludes by identifying current gaps in our understanding of particle advection performance and its optimizations.

Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of level-set visualization for univariate data to multivariate data. In this paper, we present a statistical framework to quantify positional probabilities of fibers extracted from uncertain bivariate fields. Specifically, we extend the state-of-the-art Gaussian models of uncertainty for bivariate data to other parametric distributions (e.g., uniform and Epanechnikov) and more general nonparametric probability distributions (e.g., histograms and kernel density estimation) and derive corresponding spatial probabilities of fibers. In our proposed framework, we leverage Green's theorem for closed-form computation of fiber probabilities when bivariate data are assumed to have independent parametric and nonparametric noise. Additionally, we present a nonparametric approach combined with numerical integration to study the positional probability of fibers when bivariate data are assumed to have correlated noise. For uncertainty analysis, we visualize the derived probability volumes for fibers via volume rendering and extracting level sets based on probability thresholds. We present the utility of our proposed techniques via experiments on synthetic and simulation datasets.

The current practice for fusion scientists running first principle simulations on high performance computing plat-forms is to either run their simulations and output their data for post-hoc analysis, or to place in situ analytics into their code. In this paper we examine a complex workflow using XGC fusions simulation run on the Oak Ridge Leadership Computing Facility's supercomputer Summit, which also involve three anal-yses as part of the results necessary for scientific discovery. We discuss the challenges faced when implementing these algorithms and present an original hybrid staging technique to help enable the physicists to make discoveries during the execution of the simulation. By creating this infrastructure, we can examine complicated physics results, which may not have been possible without the infrastructure. For example, our work enables the online visualization of turbulent homoclinic tangle around the magnetic X-point, breaking the last confinement surface. This visualization could help fusion scientists to better understand and improve the turbulence spread of plasma exhaust heat, which is crucial toward realizing plasmas beyond the currently accessible physics regimes of present-day tokamak reactors. The physics of turbulent homoclinic tangle will be reported in a future physics publication, by utilizing the original online analysis/visualization framework presented in this paper.

Nowadays, data reduction is becoming increasingly important in dealing with the large amounts of scientific data. Existing multilevel compression algorithms offer a promising way to manage scientific data at scale, but may suffer from relatively low performance and reduction quality. In this paper, we propose MGARD+, a multilevel data reduction and refactoring framework drawing on previous multilevel methods, to achieve high-performance data decomposition and high-quality error-bounded lossy compression. Our contributions are four-fold: 1) We propose to leverage a level-wise coefficient quantization method, which uses different error tolerances to quantize the multilevel coefficients. 2) We propose an adaptive decomposition method which treats the multilevel decomposition as a preconditioner and terminates the decomposition process at an appropriate level. 3) We leverage a set of algorithmic optimization strategies to significantly improve the performance of multilevel decomposition/recomposition. 4) We evaluate our proposed method using four real-world scientific datasets and compare with several state-of-the-art lossy compressors. Experiments demonstrate that our optimizations improve the decomposition/recomposition performance of the existing multilevel method by up to$70 \times$, and the proposed compression method can improve compression ratio by up to$2 \times$compared with other state-of-the-art error-bounded lossy compressors under the same level of data distortion.

The rate of data generated by cutting-edge experimental science facilities and large-scale simulations enabled by current high-performance computing (HPC) systems has continued to grow at a far greater pace than the development of the network and storage capabilities on which these systems rely. To cope with this challenge, scientist are moving toward the creation of autonomous experiments and HPC simulations using machine learning. However, efficiently moving, storing, and processing large amounts of data away from the point of origin presents an incredible challenge. In-memory computing, in situ analysis, data staging, and data streaming are recognized viable alternatives to traditional file-based methods for transferring data between coupled workflows. However, the performance trade-offs and limitations for these methods are not fully understood when used in HPC applications. This article presents a comprehensive performance assessment of the current solutions for data staging when applied to applications that are not necessary I/O intensive which makes them not ideal candidates for these methods. Our study is based on experiments running at scale on Oak Ridge National Laboratory's Summit supercomputer using applications and simulations that cover typical computational motifs and patterns. We investigated the usability and cost/benefit trade-offs of staging algorithms for HPC applications under different scenarios and highlight opportunities for optimizing the dataflow between coupled simulation workflows.

Rapid growth in scientific data and a widening gap between computational speed and I/O bandwidth make it increasingly infeasible to store and share all data produced by scientific simulations. Instead, we need methods for reducing data volumes: ideally, methods that can scale data volumes adaptively so as to enable negotiation of performance and fidelity tradeoffs in different situations. Multigrid-based hierarchical data representations hold promise as a solution to this problem, allowing for flexible conversion between different fidelities so that, for example, data can be created at high fidelity and then transferred or stored at lower fidelity via logically simple and mathematically sound operations. However, the effective use of such representations has been hindered until now by the relatively high costs of creating, accessing, reducing, and otherwise operating on such representations. We describe here highly optimized data refactoring kernels for GPU accelerators that enable efficient creation and manipulation of data in multigrid-based hierarchical forms. We demonstrate that our optimized design can achieve up to 250 TB/s aggregated data refactoring throughput-83% of theoretical peak-on 1024 nodes of the Summit supercomputer. We showcase our optimized design by applying it to a large-scale scientific visualization workflow and the MGARD lossy compression software.

Extreme-scale simulations and high-resolution instruments have been generating an increasing amount of data, which poses significant challenges to not only data storage during the run, but also post-processing where data will be repeatedly retrieved and analyzed for a long period of time. The challenges in satisfying a wide range of post-hoc analysis needs while minimizing the I/O overhead caused by inappropriate and/or excessive data retrieval should never be left unmanaged. In this paper, we propose a data refactoring, compressing, and retrieval framework capable of 1) fine-grained data refactoring with regard to precision; 2) incrementally retrieving and recomposing the data in terms of various error bounds; and 3) adaptively retrieving data in multi-precision and multi-resolution with respect to different analysis. With the progressive data re-composition and the adaptable retrieval algorithms, our framework significantly reduces the amount of data retrieved when multiple incremental precision are requested and/or the downstream analysis time when coarse resolution is used. Experiments show that the amount of data retrieved under the same progressively requested error bound using our framework is 64% less than that using state-of-the-art single-error-bounded approaches. Parallel experiments with up to 1, 024 cores and ~ 600 GB data in total show that our approach yields 1.36× and 2.52× performance over existing approaches in writing to and reading from persistent storage systems, respectively.

There are multiple algorithms for parallelizing particle advection for scientific visualization workloads. While many previous studies have contributed to the understanding of individual algorithms, our study aims to provide a holistic understanding of how algorithms perform relative to each other on various workloads. To accomplish this, we consider four popular parallelization algorithms and run a “bake-off” study (i.e., an empirical study) to identify the best matches for each. The study includes 216 tests, going to a concurrency of up to 8192 cores and considering data sets as large as 34 billion cells with 300 million particles. Overall, our study informs three important research questions: (1) which parallelization algorithms perform best for a given workload?, (2) why?, and (3) what are the unsolved problems in parallel particle advection? In terms of findings, we find that the seeding box is the most important factor in choosing the best algorithm, and also that there is a significant opportunity for improvement in execution time, scalability, and efficiency.

With the growing computational complexity of science and the complexity of new and emerging hardware, it is time to re-evaluate the traditional monolithic design of computational codes. One new paradigm is constructing larger scientific computational experiments from the coupling of multiple individual scientific applications, each targeting their own physics, characteristic lengths, and/or scales. We present a framework constructed by leveraging capabilities such as in-memory communications, workflow scheduling on HPC resources, and continuous performance monitoring. This code coupling capability is demonstrated by a fusion science scenario, where differences between the plasma at the edges and at the core of a device have different physical descriptions. This infrastructure not only enables the coupling of the physics components, but it also connects in situ or online analysis, compression, and visualization that accelerate the time between a run and the analysis of the science content. Results from runs on Titan and Cori are presented as a demonstration.

One of the core issues across computer and computational science today is adapting to, managing, and learning from the influx of "Big Data". In the commercial space, this problem has led to a huge investment in new technologies and capabilities that are well adapted to dealing with the sorts of human-generated logs, videos, texts, and other large-data artifacts that are processed and resulted in an explosion of useful platforms and languages (Hadoop, Spark, Pandas, etc.). However, translating this work from the enterprise space to the computational science and HPC community has proven somewhat difficult, in part because of some of the fundamental differences in type and scale of data and timescales surrounding its generation and use. We describe a forward-looking research and development plan which centers around the concept of making Input/Output (I/O) intelligent for users in the scientific community, whether they are accessing scalable storage or performing in situ workflow tasks. Much of our work is based on our experience with the Adaptable I/O System (ADIOS 1.X), and our next generation version of the software ADIOS 2.X [1].

Scientific simulations on high performance computing (HPC) platforms generate large quantities of data. To bridge the widening gap between compute and I/O, and enable data to be more efficiently stored and analyzed, simulation outputs need to be refactored, reduced, and appropriately mapped to storage tiers. However, a systematic solution to support these steps has been lacking on the current HPC software ecosystem. To that end, this paper develops Canopus, a progressive JPEGlike data management scheme for storing and analyzing big scientific data. It co-designs the data decimation, compression and data storage, taking the hardware characteristics of each storage tier into considerations. With reasonably low overhead, our approach refactors simulation data into a much smaller, reduced-accuracy base dataset, and a series of deltas that is used to augment the accuracy if needed. The base dataset and deltas are compressed and written to multiple storage tiers. Data saved on different tiers can then be selectively retrieved to restore the level of accuracy that satisfies data analytics. Thus, Canopus provides a paradigm shift towards elastic data analytics and enables end users to make trade-offs between analysis speed and accuracy on-the-fly. We evaluate the impact of Canopus on unstructured triangular meshes, a pervasive data model used by scientific modeling and simulations. In particular, we demonstrate the progressive data exploration of Canopus using the “blob detection” use case on the fusion simulation data.

As we continue toward exascale, scientific data volume is continuing to scale and becoming more burdensome to manage. In this paper, we lay out opportunities to enhance state of the art data management techniques. We emphasize well-principled data compression, and using it to achieve progressive refinement. This can both accelerate I/O and afford the user increased flexibility when she interacts with the data. The formulation naturally maps onto enabling partitioning of the progressively improving-quality representations of a data quantity into different media-type destinations, to keep the highest priority information as close as possible to the computation, and take advantage of deepening memory/storage hierarchies in ways not previously possible. Careful monitoring is requisite to our vision, not only to verify that compression has not eliminated salient features in the data, but also to better understand the performance of massively parallel scientific applications. Increased mathematical rigor would be ideal,to help bring compression on a better-understood theoretical footing, closer to the relevant scientific theory, more aware of constraints imposed by the science, and more tightly error-controlled. Throughout, we highlight pathfinding research we have begun exploring related these topics, and comment toward future work that will be needed.

With the push to exascale, in situ visualization and analysis will continue to play an important role in high performance computing. Tightly coupling in situ visualization with simulations constrains resources for both, and these constraints force a complex balance of trade-offs. A performance model that provides an a priori answer for the cost of using an in situ approach for a given task would assist in managing the trade-offs between simulation and visualization resources. In this work, we present new statistical performance models, based on algorithmic complexity, that accurately predict the run-time cost of a set of representative rendering algorithms, an essential in situ visualization task. To train and validate the models, we conduct a performance study of an MPI+X rendering infrastructure used in situ with three HPC simulation applications. We then explore feasibility issues using the model for selected in situ rendering questions.

Scientific communities have benefitted from a significant increase of available computing and storage resources in the last few decades. For science projects that have access to leadership scale computing resources, the capacity to produce data has been growing exponentially. Teams working on such projects must now include, in addition to the traditional application scientists, experts in various disciplines including applied mathematicians for development of algorithms, visualization specialists for large data, and I/O specialists. Sharing of knowledge and data is becoming a requirement for scientific discovery, providing useful mechanisms to facilitate this sharing is a key challenge for e-Science. Our hypothesis is that in order to decrease the time to solution for application scientists we need to lower the barrier of entry into related computing fields. We aim at improving users' experience when interacting with a vast software ecosystem and/or huge amount of data, while maintaining focus on their primary research field. In this context we present our approach to bridge the gap between the application scientists and the visualization experts through a visualization schema as a first step and proof of concept for a new way to look at interdisciplinary collaboration among scientists dealing with big data. The key to our approach is recognizing that our users are scientists who mostly work as islands. They tend to work in very specialized environment but occasionally have to collaborate with other researchers in order to take full advantage of computing innovations and get insight from big data. We present an example of identifying the connecting elements between one of such relationships and offer a liaison schema to facilitate their collaboration.

Predictive pre-fetcher, which predicts future data access events and loads the data before users requests, has been widely studied, especially in file systems or web contents servers, to reduce data load latency. Especially in scientific data visualization, pre-fetching can reduce the IO waiting time. In order to increase the accuracy, we apply a data mining technique to extract hidden information. More specifically, we apply a data mining technique for discovering the hidden contexts in data access patterns and make prediction based on the inferred context to boost the accuracy. In particular, we performed Probabilistic Latent Semantic Analysis (PLSA), a mixture model based algorithm popular in the text mining area, to mine hidden contexts from the collected user access patterns and, then, we run a predictor within the discovered context. We further improve PLSA by applying the Deterministic Annealing (DA) method to overcome the local optimum problem. In this paper we demonstrate how we can apply PLSA and DA optimization to mine hidden contexts from users data access patterns and improve predictive pre-fetcher performance.

Streamline computation in a very large vector field data set represents a significant challenge due to the nonlocal and data-dependent nature of streamline integration. In this paper, we conduct a study of the performance characteristics of hybrid parallel programming and execution as applied to streamline integration on a large, multicore platform. With multicore processors now prevalent in clusters and supercomputers, there is a need to understand the impact of these hybrid systems in order to make the best implementation choice. We use two MPI-based distribution approaches based on established parallelization paradigms, parallelize over seeds and parallelize over blocks, and present a novel MPI-hybrid algorithm for each approach to compute streamlines. Our findings indicate that the work sharing between cores in the proposed MPI-hybrid parallel implementation results in much improved performance and consumes less communication and I/O bandwidth than a traditional, nonhybrid distributed implementation.

In the development of magnetic confinement fusion which will potentially be a future source for low cost power, physicists must be able to analyze the magnetic field that confines the burning plasma. While the magnetic field can be described as a vector field, traditional techniques for analyzing the field's topology cannot be used because of its Hamiltonian nature. In this paper we describe a technique developed as a collaboration between physicists and computer scientists that determines the topology of a toroidal magnetic field using fieldlines with near minimal lengths. More specifically, we analyze the Poincaré map of the sampled fieldlines in a Poincaré section including identifying critical points and other topological features of interest to physicists. The technique has been deployed into an interactive parallel visualization tool which physicists are using to gain new insight into simulations of magnetically confined burning plasmas.

Understanding vector fields resulting from large scientific simulations is an important and often difficult task. Streamlines, curves that are tangential to a vector field at each point, are a powerful visualization method in this context. Application of streamline-based visualization to very large vector field data represents a significant challenge due to the non-local and data-dependent nature of streamline computation, and requires careful balancing of computational demands placed on I/O, memory, communication, and processors. In this paper we review two parallelization approaches based on established parallelization paradigms (static decomposition and on-demand loading) and present a novel hybrid algorithm for computing streamlines. Our algorithm is aimed at good scalability and performance across the widely varying computational characteristics of streamline-based problems. We perform performance and scalability studies of all three algorithms on a number of prototypical application problems and demonstrate that our hybrid scheme is able to perform well in different settings.

This paper describes the first use of a Network Processing Unit (NPU) to perform hardware-based image composition in a distributed rendering system. The image composition step is a notorious bottleneck in a clustered rendering system. Furthermore, image compositing algorithms do not necessarily scale as data size and number of nodes increase. Previous researchers have addressed the composition problem via software and/or custom-built hardware. We used the heterogeneous multicore computation architecture of the Intel IXP28XX NPU, a fully programmable commercial off-the-shelf (COTS) technology, to perform the image composition step. With this design, we have attained a nearly four-times performance increase over traditional software-based compositing methods, achieving sustained compositing rates of 22-28 fps on a 1,024 x 1,024 image. This system is fully scalable with a negligible penalty in frame rate, is entirely COTS, and is flexible with regard to operating system, rendering software, graphics cards, and node architecture. The NPU-based compositor has the additional advantage of being a modular compositing component that is eminently suitable for integration into existing distributed software visualization packages.

This paper exploresthe use of visual operators for solids modeling. We focus on designing interfaces for free-form operators such as blends, sweeps, and deformations, because these operators have a large number of interacting parameters whose effects are often determined by an underlying parameterization. In this type of interactive modeling good solutions to the design problem have aesthetic as well as engineering components. Traditionally, interaction with the parameters of these operators has been through text editors, curve editors, or trial-and-error with a slider bar. Parametric values have been estimated from data, but not interactively. These parametersare usually one- or two-dimensional,but the operatorsthemselves are intrinsically three-dimensional in that they are used to model surfaces visualized in 3D. The traditional textual style of interaction is tedious and interposes a level of abstraction between the parameters and the resulting surface. A 3D visual interface has the...