Header logo is

22 results (View BibTeX file of all listed publications)

2016

Thumb xl nonlinear approximate vs exact
A New Perspective and Extension of the Gaussian Filter

Wüthrich, M., Trimpe, S., Garcia Cifuentes, C., Kappler, D., Schaal, S.

The International Journal of Robotics Research, 35(14):1731-1749, December 2016 (article)

Abstract
The Gaussian Filter (GF) is one of the most widely used filtering algorithms; instances are the Extended Kalman Filter, the Unscented Kalman Filter and the Divided Difference Filter. The GF represents the belief of the current state by a Gaussian distribution, whose mean is an affine function of the measurement. We show that this representation can be too restrictive to accurately capture the dependences in systems with nonlinear observation models, and we investigate how the GF can be generalized to alleviate this problem. To this end, we view the GF as the solution to a constrained optimization problem. From this new perspective, the GF is seen as a special case of a much broader class of filters, obtained by relaxing the constraint on the form of the approximate posterior. On this basis, we outline some conditions which potential generalizations have to satisfy in order to maintain the computational efficiency of the GF. We propose one concrete generalization which corresponds to the standard GF using a pseudo measurement instead of the actual measurement. Extending an existing GF implementation in this manner is trivial. Nevertheless, we show that this small change can have a major impact on the estimation accuracy.

am ics

PDF DOI Project Page [BibTex]

2016

am ics Wüthrich, M., Trimpe, S., Garcia Cifuentes, C., Kappler, D., Schaal, S. A New Perspective and Extension of the Gaussian Filter The International Journal of Robotics Research, 35(14):1731-1749, December 2016 (article)

PDF DOI Project Page [BibTex]

Thumb xl psychscience
Creating body shapes from verbal descriptions by linking similarity spaces

Hill, M. Q., Streuber, S., Hahn, C. A., Black, M. J., O’Toole, A. J.

Psychological Science, 27(11):1486-1497, November 2016, (article)

Abstract
Brief verbal descriptions of bodies (e.g. curvy, long-legged) can elicit vivid mental images. The ease with which we create these mental images belies the complexity of three-dimensional body shapes. We explored the relationship between body shapes and body descriptions and show that a small number of words can be used to generate categorically accurate representations of three-dimensional bodies. The dimensions of body shape variation that emerged in a language-based similarity space were related to major dimensions of variation computed directly from three-dimensional laser scans of 2094 bodies. This allowed us to generate three-dimensional models of people in the shape space using only their coordinates on analogous dimensions in the language-based description space. Human descriptions of photographed bodies and their corresponding models matched closely. The natural mapping between the spaces illustrates the role of language as a concise code for body shape, capturing perceptually salient global and local body features.

ps

pdf [BibTex]

ps Hill, M. Q., Streuber, S., Hahn, C. A., Black, M. J., O’Toole, A. J. Creating body shapes from verbal descriptions by linking similarity spaces Psychological Science, 27(11):1486-1497, November 2016, (article)

pdf [BibTex]

Thumb xl webteaser
Body Talk: Crowdshaping Realistic 3D Avatars with Words

Streuber, S., Quiros-Ramirez, M. A., Hill, M. Q., Hahn, C. A., Zuffi, S., O’Toole, A., Black, M. J.

ACM Trans. Graph. (Proc. SIGGRAPH), 35(4):54:1-54:14, July 2016 (article)

Abstract
Realistic, metrically accurate, 3D human avatars are useful for games, shopping, virtual reality, and health applications. Such avatars are not in wide use because solutions for creating them from high-end scanners, low-cost range cameras, and tailoring measurements all have limitations. Here we propose a simple solution and show that it is surprisingly accurate. We use crowdsourcing to generate attribute ratings of 3D body shapes corresponding to standard linguistic descriptions of 3D shape. We then learn a linear function relating these ratings to 3D human shape parameters. Given an image of a new body, we again turn to the crowd for ratings of the body shape. The collection of linguistic ratings of a photograph provides remarkably strong constraints on the metric 3D shape. We call the process crowdshaping and show that our Body Talk system produces shapes that are perceptually indistinguishable from bodies created from high-resolution scans and that the metric accuracy is sufficient for many tasks. This makes body “scanning” practical without a scanner, opening up new applications including database search, visualization, and extracting avatars from books.

ps

pdf web tool video talk (ppt) [BibTex]

ps Streuber, S., Quiros-Ramirez, M. A., Hill, M. Q., Hahn, C. A., Zuffi, S., O’Toole, A., Black, M. J. Body Talk: Crowdshaping Realistic 3D Avatars with Words ACM Trans. Graph. (Proc. SIGGRAPH), 35(4):54:1-54:14, July 2016 (article)

pdf web tool video talk (ppt) [BibTex]

Thumb xl ijcv tumb
Capturing Hands in Action using Discriminative Salient Points and Physics Simulation

Tzionas, D., Ballan, L., Srikantha, A., Aponte, P., Pollefeys, M., Gall, J.

International Journal of Computer Vision (IJCV), 118(2):172-193, June 2016 (article)

Abstract
Hand motion capture is a popular research field, recently gaining more attention due to the ubiquity of RGB-D sensors. However, even most recent approaches focus on the case of a single isolated hand. In this work, we focus on hands that interact with other hands or objects and present a framework that successfully captures motion in such interaction scenarios for both rigid and articulated objects. Our framework combines a generative model with discriminatively trained salient points to achieve a low tracking error and with collision detection and physics simulation to achieve physically plausible estimates even in case of occlusions and missing visual data. Since all components are unified in a single objective function which is almost everywhere differentiable, it can be optimized with standard optimization techniques. Our approach works for monocular RGB-D sequences as well as setups with multiple synchronized RGB cameras. For a qualitative and quantitative evaluation, we captured 29 sequences with a large variety of interactions and up to 150 degrees of freedom.

ps

Website pdf link (url) DOI Project Page [BibTex]

ps Tzionas, D., Ballan, L., Srikantha, A., Aponte, P., Pollefeys, M., Gall, J. Capturing Hands in Action using Discriminative Salient Points and Physics Simulation International Journal of Computer Vision (IJCV), 118(2):172-193, June 2016 (article)

Website pdf link (url) DOI Project Page [BibTex]

Thumb xl teaser web
Human Pose Estimation from Video and IMUs

Marcard, T. V., Pons-Moll, G., Rosenhahn, B.

Transactions on Pattern Analysis and Machine Intelligence PAMI, 38(8):1533-1547, January 2016 (article)

ps

data pdf dataset_documentation [BibTex]

ps Marcard, T. V., Pons-Moll, G., Rosenhahn, B. Human Pose Estimation from Video and IMUs Transactions on Pattern Analysis and Machine Intelligence PAMI, 38(8):1533-1547, January 2016 (article)

data pdf dataset_documentation [BibTex]

no image
Supplemental material for ’Communication Rate Analysis for Event-based State Estimation’

Ebner, S., Trimpe, S.

Max Planck Institute for Intelligent Systems, January 2016 (techreport)

am ics

PDF [BibTex]

am ics Ebner, S., Trimpe, S. Supplemental material for ’Communication Rate Analysis for Event-based State Estimation’ Max Planck Institute for Intelligent Systems, January 2016 (techreport)

PDF [BibTex]

Thumb xl cloud tracking
Gaussian Process-Based Predictive Control for Periodic Error Correction

Klenske, E. D., Zeilinger, M., Schölkopf, B., Hennig, P.

IEEE Transactions on Control Systems Technology , 24(1):110-121, 2016 (article)

ei pn

PDF DOI [BibTex]

ei pn Klenske, E. D., Zeilinger, M., Schölkopf, B., Hennig, P. Gaussian Process-Based Predictive Control for Periodic Error Correction IEEE Transactions on Control Systems Technology , 24(1):110-121, 2016 (article)

PDF DOI [BibTex]

Thumb xl dual control sampled b
Dual Control for Approximate Bayesian Reinforcement Learning

Klenske, E. D., Hennig, P.

Journal of Machine Learning Research, 17(127):1-30, 2016 (article)

ei pn

PDF link (url) [BibTex]

ei pn Klenske, E. D., Hennig, P. Dual Control for Approximate Bayesian Reinforcement Learning Journal of Machine Learning Research, 17(127):1-30, 2016 (article)

PDF link (url) [BibTex]

Thumb xl siyong
Shape estimation of subcutaneous adipose tissue using an articulated statistical shape model

Yeo, S. Y., Romero, J., Loper, M., Machann, J., Black, M.

Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 0(0):1-8, 2016 (article)

ps

publisher website preprint pdf link (url) DOI Project Page [BibTex]

ps Yeo, S. Y., Romero, J., Loper, M., Machann, J., Black, M. Shape estimation of subcutaneous adipose tissue using an articulated statistical shape model Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 0(0):1-8, 2016 (article)

publisher website preprint pdf link (url) DOI Project Page [BibTex]

no image
Event-based Sampling for Reducing Communication Load in Realtime Human Motion Analysis by Wireless Inertial Sensor Networks

Laidig, D., Trimpe, S., Seel, T.

Current Directions in Biomedical Engineering, 2(1):711-714, De Gruyter, 2016 (article)

am ics

PDF DOI [BibTex]

am ics Laidig, D., Trimpe, S., Seel, T. Event-based Sampling for Reducing Communication Load in Realtime Human Motion Analysis by Wireless Inertial Sensor Networks Current Directions in Biomedical Engineering, 2(1):711-714, De Gruyter, 2016 (article)

PDF DOI [BibTex]

Thumb xl screen shot 2016 02 22 at 11.46.41
The GRASP Taxonomy of Human Grasp Types

Feix, T., Romero, J., Schmiedmayer, H., Dollar, A., Kragic, D.

Human-Machine Systems, IEEE Transactions on, 46(1):66-77, 2016 (article)

ps

publisher website pdf DOI Project Page [BibTex]

ps Feix, T., Romero, J., Schmiedmayer, H., Dollar, A., Kragic, D. The GRASP Taxonomy of Human Grasp Types Human-Machine Systems, IEEE Transactions on, 46(1):66-77, 2016 (article)

publisher website pdf DOI Project Page [BibTex]

Thumb xl pami
Map-Based Probabilistic Visual Self-Localization

Brubaker, M. A., Geiger, A., Urtasun, R.

IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI), 2016 (article)

Abstract
Accurate and efficient self-localization is a critical problem for autonomous systems. This paper describes an affordable solution to vehicle self-localization which uses odometry computed from two video cameras and road maps as the sole inputs. The core of the method is a probabilistic model for which an efficient approximate inference algorithm is derived. The inference algorithm is able to utilize distributed computation in order to meet the real-time requirements of autonomous systems in some instances. Because of the probabilistic nature of the model the method is capable of coping with various sources of uncertainty including noise in the visual odometry and inherent ambiguities in the map (e.g., in a Manhattan world). By exploiting freely available, community developed maps and visual odometry measurements, the proposed method is able to localize a vehicle to 4m on average after 52 seconds of driving on maps which contain more than 2,150km of drivable roads.

avg ps

pdf Project Page [BibTex]

avg ps Brubaker, M. A., Geiger, A., Urtasun, R. Map-Based Probabilistic Visual Self-Localization IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI), 2016 (article)

pdf Project Page [BibTex]

2015

Thumb xl grassmanteaser
Scalable Robust Principal Component Analysis using Grassmann Averages

Hauberg, S., Feragen, A., Enficiaud, R., Black, M.

IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), December 2015 (article)

Abstract
In large datasets, manual data verification is impossible, and we must expect the number of outliers to increase with data size. While principal component analysis (PCA) can reduce data size, and scalable solutions exist, it is well-known that outliers can arbitrarily corrupt the results. Unfortunately, state-of-the-art approaches for robust PCA are not scalable. We note that in a zero-mean dataset, each observation spans a one-dimensional subspace, giving a point on the Grassmann manifold. We show that the average subspace corresponds to the leading principal component for Gaussian data. We provide a simple algorithm for computing this Grassmann Average (GA), and show that the subspace estimate is less sensitive to outliers than PCA for general distributions. Because averages can be efficiently computed, we immediately gain scalability. We exploit robust averaging to formulate the Robust Grassmann Average (RGA) as a form of robust PCA. The resulting Trimmed Grassmann Average (TGA) is appropriate for computer vision because it is robust to pixel outliers. The algorithm has linear computational complexity and minimal memory requirements. We demonstrate TGA for background modeling, video restoration, and shadow removal. We show scalability by performing robust PCA on the entire Star Wars IV movie; a task beyond any current method. Source code is available online.

ps sf

preprint pdf from publisher supplemental Project Page [BibTex]

2015

ps sf Hauberg, S., Feragen, A., Enficiaud, R., Black, M. Scalable Robust Principal Component Analysis using Grassmann Averages IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), December 2015 (article)

preprint pdf from publisher supplemental Project Page [BibTex]

no image
Distributed Event-based State Estimation

Trimpe, S.

Max Planck Institute for Intelligent Systems, November 2015 (techreport)

Abstract
An event-based state estimation approach for reducing communication in a networked control system is proposed. Multiple distributed sensor-actuator-agents observe a dynamic process and sporadically exchange their measurements and inputs over a bus network. Based on these data, each agent estimates the full state of the dynamic system, which may exhibit arbitrary inter-agent couplings. Local event-based protocols ensure that data is transmitted only when necessary to meet a desired estimation accuracy. This event-based scheme is shown to mimic a centralized Luenberger observer design up to guaranteed bounds, and stability is proven in the sense of bounded estimation errors for bounded disturbances. The stability result extends to the distributed control system that results when the local state estimates are used for distributed feedback control. Simulation results highlight the benefit of the event-based approach over classical periodic ones in reducing communication requirements.

am ics

arXiv [BibTex]

am ics Trimpe, S. Distributed Event-based State Estimation Max Planck Institute for Intelligent Systems, November 2015 (techreport)

arXiv [BibTex]

Thumb xl splitbodieswebteaser2
SMPL: A Skinned Multi-Person Linear Model

Loper, M., Mahmood, N., Romero, J., Pons-Moll, G., Black, M. J.

ACM Trans. Graphics (Proc. SIGGRAPH Asia), 34(6):248:1-248:16, ACM, New York, NY, October 2015 (article)

Abstract
We present a learned model of human body shape and pose-dependent shape variation that is more accurate than previous models and is compatible with existing graphics pipelines. Our Skinned Multi-Person Linear model (SMPL) is a skinned vertex-based model that accurately represents a wide variety of body shapes in natural human poses. The parameters of the model are learned from data including the rest pose template, blend weights, pose-dependent blend shapes, identity-dependent blend shapes, and a regressor from vertices to joint locations. Unlike previous models, the pose-dependent blend shapes are a linear function of the elements of the pose rotation matrices. This simple formulation enables training the entire model from a relatively large number of aligned 3D meshes of different people in different poses. We quantitatively evaluate variants of SMPL using linear or dual-quaternion blend skinning and show that both are more accurate than a Blend-SCAPE model trained on the same data. We also extend SMPL to realistically model dynamic soft-tissue deformations. Because it is based on blend skinning, SMPL is compatible with existing rendering engines and we make it available for research purposes.

ps

pdf video code/model errata DOI Project Page Project Page [BibTex]

ps Loper, M., Mahmood, N., Romero, J., Pons-Moll, G., Black, M. J. SMPL: A Skinned Multi-Person Linear Model ACM Trans. Graphics (Proc. SIGGRAPH Asia), 34(6):248:1-248:16, ACM, New York, NY, October 2015 (article)

pdf video code/model errata DOI Project Page Project Page [BibTex]

Thumb xl dynateaser
Dyna: A Model of Dynamic Human Shape in Motion

Pons-Moll, G., Romero, J., Mahmood, N., Black, M. J.

ACM Transactions on Graphics, (Proc. SIGGRAPH), 34(4):120:1-120:14, ACM, August 2015 (article)

Abstract
To look human, digital full-body avatars need to have soft tissue deformations like those of real people. We learn a model of soft-tissue deformations from examples using a high-resolution 4D capture system and a method that accurately registers a template mesh to sequences of 3D scans. Using over 40,000 scans of ten subjects, we learn how soft tissue motion causes mesh triangles to deform relative to a base 3D body model. Our Dyna model uses a low-dimensional linear subspace to approximate soft-tissue deformation and relates the subspace coefficients to the changing pose of the body. Dyna uses a second-order auto-regressive model that predicts soft-tissue deformations based on previous deformations, the velocity and acceleration of the body, and the angular velocities and accelerations of the limbs. Dyna also models how deformations vary with a person’s body mass index (BMI), producing different deformations for people with different shapes. Dyna realistically represents the dynamics of soft tissue for previously unseen subjects and motions. We provide tools for animators to modify the deformations and apply them to new stylized characters.

ps

pdf preprint video data DOI Project Page Project Page [BibTex]

ps Pons-Moll, G., Romero, J., Mahmood, N., Black, M. J. Dyna: A Model of Dynamic Human Shape in Motion ACM Transactions on Graphics, (Proc. SIGGRAPH), 34(4):120:1-120:14, ACM, August 2015 (article)

pdf preprint video data DOI Project Page Project Page [BibTex]

Thumb xl objs2acts
Linking Objects to Actions: Encoding of Target Object and Grasping Strategy in Primate Ventral Premotor Cortex

Vargas-Irwin, C. E., Franquemont, L., Black, M. J., Donoghue, J. P.

Journal of Neuroscience, 35(30):10888-10897, July 2015 (article)

Abstract
Neural activity in ventral premotor cortex (PMv) has been associated with the process of matching perceived objects with the motor commands needed to grasp them. It remains unclear how PMv networks can flexibly link percepts of objects affording multiple grasp options into a final desired hand action. Here, we use a relational encoding approach to track the functional state of PMv neuronal ensembles in macaque monkeys through the process of passive viewing, grip planning, and grasping movement execution. We used objects affording multiple possible grip strategies. The task included separate instructed delay periods for object presentation and grip instruction. This approach allowed us to distinguish responses elicited by the visual presentation of the objects from those associated with selecting a given motor plan for grasping. We show that PMv continuously incorporates information related to object shape and grip strategy as it becomes available, revealing a transition from a set of ensemble states initially most closely related to objects, to a new set of ensemble patterns reflecting unique object-grip combinations. These results suggest that PMv dynamically combines percepts, gradually navigating toward activity patterns associated with specific volitional actions, rather than directly mapping perceptual object properties onto categorical grip representations. Our results support the idea that PMv is part of a network that dynamically computes motor plans from perceptual information. Significance Statement: The present work demonstrates that the activity of groups of neurons in primate ventral premotor cortex reflects information related to visually presented objects, as well as the motor strategy used to grasp them, linking individual objects to multiple possible grips. PMv could provide useful control signals for neuroprosthetic assistive devices designed to interact with objects in a flexible way.

ps

publisher link DOI Project Page [BibTex]

ps Vargas-Irwin, C. E., Franquemont, L., Black, M. J., Donoghue, J. P. Linking Objects to Actions: Encoding of Target Object and Grasping Strategy in Primate Ventral Premotor Cortex Journal of Neuroscience, 35(30):10888-10897, July 2015 (article)

publisher link DOI Project Page [BibTex]

Thumb xl screen shot 2015 10 14 at 08.57.57
Multi-view and 3D Deformable Part Models

Pepik, B., Stark, M., Gehler, P., Schiele, B.

Pattern Analysis and Machine Intelligence, 37(11):14, IEEE, March 2015 (article)

Abstract
As objects are inherently 3-dimensional, they have been modeled in 3D in the early days of computer vision. Due to the ambiguities arising from mapping 2D features to 3D models, 3D object representations have been neglected and 2D feature-based models are the predominant paradigm in object detection nowadays. While such models have achieved outstanding bounding box detection performance, they come with limited expressiveness, as they are clearly limited in their capability of reasoning about 3D shape or viewpoints. In this work, we bring the worlds of 3D and 2D object representations closer, by building an object detector which leverages the expressive power of 3D object representations while at the same time can be robustly matched to image evidence. To that end, we gradually extend the successful deformable part model [1] to include viewpoint information and part-level 3D geometry information, resulting in several different models with different level of expressiveness. We end up with a 3D object model, consisting of multiple object parts represented in 3D and a continuous appearance model. We experimentally verify that our models, while providing richer object hypotheses than the 2D object models, provide consistently better joint object localization and viewpoint estimation than the state-of-the-art multi-view and 3D object detectors on various benchmarks (KITTI [2], 3D object classes [3], Pascal3D+ [4], Pascal VOC 2007 [5], EPFL multi-view cars [6]).

ps

DOI Project Page [BibTex]

ps Pepik, B., Stark, M., Gehler, P., Schiele, B. Multi-view and 3D Deformable Part Models Pattern Analysis and Machine Intelligence, 37(11):14, IEEE, March 2015 (article)

DOI Project Page [BibTex]

Thumb xl ssimssmall
Spike train SIMilarity Space (SSIMS): A framework for single neuron and ensemble data analysis

Vargas-Irwin, C. E., Brandman, D. M., Zimmermann, J. B., Donoghue, J. P., Black, M. J.

Neural Computation, 27(1):1-31, MIT Press, January 2015 (article)

Abstract
We present a method to evaluate the relative similarity of neural spiking patterns by combining spike train distance metrics with dimensionality reduction. Spike train distance metrics provide an estimate of similarity between activity patterns at multiple temporal resolutions. Vectors of pair-wise distances are used to represent the intrinsic relationships between multiple activity patterns at the level of single units or neuronal ensembles. Dimensionality reduction is then used to project the data into concise representations suitable for clustering analysis as well as exploratory visualization. Algorithm performance and robustness are evaluated using multielectrode ensemble activity data recorded in behaving primates. We demonstrate how Spike train SIMilarity Space (SSIMS) analysis captures the relationship between goal directions for an 8-directional reaching task and successfully segregates grasp types in a 3D grasping task in the absence of kinematic information. The algorithm enables exploration of virtually any type of neural spiking (time series) data, providing similarity-based clustering of neural activity states with minimal assumptions about potential information encoding models.

ps

pdf: publisher site pdf: author's proof DOI Project Page [BibTex]

ps Vargas-Irwin, C. E., Brandman, D. M., Zimmermann, J. B., Donoghue, J. P., Black, M. J. Spike train SIMilarity Space (SSIMS): A framework for single neuron and ensemble data analysis Neural Computation, 27(1):1-31, MIT Press, January 2015 (article)

pdf: publisher site pdf: author's proof DOI Project Page [BibTex]

no image
Probabilistic Interpretation of Linear Solvers

Hennig, P.

SIAM Journal on Optimization, 25(1):234-260, 2015 (article)

ei pn

Web PDF link (url) DOI [BibTex]

ei pn Hennig, P. Probabilistic Interpretation of Linear Solvers SIAM Journal on Optimization, 25(1):234-260, 2015 (article)

Web PDF link (url) DOI [BibTex]

Thumb xl thumb teaser mrg
Metric Regression Forests for Correspondence Estimation

Pons-Moll, G., Taylor, J., Shotton, J., Hertzmann, A., Fitzgibbon, A.

International Journal of Computer Vision, pages: 1-13, 2015 (article)

ps

springer PDF Project Page [BibTex]

ps Pons-Moll, G., Taylor, J., Shotton, J., Hertzmann, A., Fitzgibbon, A. Metric Regression Forests for Correspondence Estimation International Journal of Computer Vision, pages: 1-13, 2015 (article)

springer PDF Project Page [BibTex]

no image
Probabilistic numerics and uncertainty in computations

Hennig, P., Osborne, M. A., Girolami, M.

Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2179), 2015 (article)

Abstract
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data have led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimizers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

ei pn

PDF DOI [BibTex]

ei pn Hennig, P., Osborne, M. A., Girolami, M. Probabilistic numerics and uncertainty in computations Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2179), 2015 (article)

PDF DOI [BibTex]