Analysis of target data-dependent greedy kernel algorithms: Convergence rates for ff-, f⋅Pf \cdot P- and f/Pf/P-greedy

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory especially when compared to the case of the data-independent $P$-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. The rates are confirmed by a number of examples. These results are made possible by a new analysis of greedy algorithms in general Hilbert spaces

Biomechanical surrogate modelling using stabilized vectorial greedy kernel methods

Greedy kernel approximation algorithms are successful techniques for sparse and accurate data-based modelling and function approximation. Based on a recent idea of stabilization of such algorithms in the scalar output case, we here consider the vectorial extension built on VKOGA. We introduce the so called $\gamma$-restricted VKOGA, comment on analytical properties and present numerical evaluation on data from a clinically relevant application, the modelling of the human spine. The experiments show that the new stabilized algorithms result in improved accuracy and stability over the non-stabilized algorithms

Structured Deep Kernel Networks for Data-Driven Closure Terms of Turbulent Flows

Standard kernel methods for machine learning usually struggle when dealing with large datasets. We review a recently introduced Structured Deep Kernel Network (SDKN) approach that is capable of dealing with high-dimensional and huge datasets - and enjoys typical standard machine learning approximation properties. We extend the SDKN to combine it with standard machine learning modules and compare it with Neural Networks on the scientific challenge of data-driven prediction of closure terms of turbulent flows. We show experimentally that the SDKNs are capable of dealing with large datasets and achieve near-perfect accuracy on the given application

Slope-space integrals for specular next event estimation

International audienceMonte Carlo light transport simulations often lack robustness in scenes containing specular or near-specular materials. Widely used uni- and bidirectional sampling strategies tend to find light paths involving such materials with insufficient probability, producing unusable images that are contaminated by significant variance.This article addresses the problem of sampling a light path connecting two given scene points via a single specular reflection or refraction, extending the range of scenes that can be robustly handled by unbiased path sampling techniques. Our technique enables efficient rendering of challenging transport phenomena caused by such paths, such as underwater caustics or caustics involving glossy metallic objects.We derive analytic expressions that predict the total radiance due to a single reflective or refractive triangle with a microfacet BSDF and we show that this reduces to the well known Lambert boundary integral for irradiance. We subsequently show how this can be leveraged to efficiently sample connections on meshes comprised of vast numbers of triangles.Our derivation builds on the theory of off-center microfacets and involves integrals in the space of surface slopes.Our approach straightforwardly applies to the related problem of rendering glints with high-resolution normal maps describing specular microstructure. Our formulation alleviates problems raised by singularities in filtering integrals and enables a generalization of previous work to perfectly specular materials. We also extend previous work to the case of GGX distributions and introduce new techniques to improve accuracy and performance

A new Certified Hierarchical and Adaptive RB-ML-ROM Surrogate Model for Parametrized PDEs

We present a new surrogate modeling technique for efficient approximation of input-output maps governed by parametrized PDEs. The model is hierarchical as it is built on a full order model (FOM), reduced order model (ROM) and machine-learning (ML) model chain. The model is adaptive in the sense that the ROM and ML model are adapted on-the-fly during a sequence of parametric requests to the model. To allow for a certification of the model hierarchy, as well as to control the adaptation process, we employ rigorous a posteriori error estimates for the ROM and ML models. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. We demonstrate the efficiency of the modeling chain on a Monte Carlo and a parameter-optimization example. Here, the ROM is instantiated by Reduced Basis Methods and the ML model is given by a neural network or a VKOGA kernel model.Comment: 27 pages, 5 figure

The Layer Laboratory: A Calculus for Additive and Subtractive Composition of Anisotropic Surface Reflectance

We present a versatile computational framework for modeling the reflective and transmissive properties of arbitrarily layered anisotropic material structures. Given a set of input layers, our model synthesizes an effective BSDF of the entire structure, which accounts for all orders of internal scattering and is efficient to sample and evaluate in modern rendering systems.Our technique builds on the insight that reflectance data is sparse when expanded into a suitable frequency-space representation, and that this property extends to the class of anisotropic materials. This sparsity enables an efficient matrix calculus that admits the entire space of BSDFs and considerably expands the scope of prior work on layered material modeling. We show how both measured data and the popular class of microfacet models can be expressed in our representation, and how the presence of anisotropy leads to a weak coupling between Fourier orders in frequency space.In addition to additive composition, our models supports subtractive composition, a fascinating new operation that reconstructs the BSDF of a material that can only be observed indirectly through another layer with known reflectance properties. The operation produces a new BSDF of the desired layer as if measured in isolation. Subtractive composition can be interpreted as a type of deconvolution that removes both internal scattering and blurring due to transmission through the known layer.We experimentally demonstrate the accuracy and scope of our model and validate both additive and subtractive composition using measurements of real-world layered materials. Both implementation and data will be released to ensure full reproducibility of all of our results.(1

Specular Manifold Sampling for Rendering High-Frequency Caustics and Glints

Scattering from specular surfaces produces complex optical effects that are frequently encountered in realistic scenes: intricate caustics due to focused reflection, multiple refraction, and high-frequency glints from specular microstructure. Yet, despite their importance and considerable research to this end, sampling of light paths that cause these effects remains a formidable challenge.In this article, we propose a surprisingly simple and general sampling strategy for specular light paths including the above examples, unifying the previously disjoint areas of caustic and glint rendering into a single framework. Given two path vertices, our algorithm stochastically finds a specular subpath connecting the endpoints. In contrast to prior work, our method supports high-frequency normal- or displacement-mapped geometry, samples specular-diffuse-specular ("SDS") paths, and is compatible with standard Monte Carlo methods including unidirectional path tracing. Both unbiased and biased variants of our approach can be constructed, the latter often significantly reducing variance, which may be appealing in applied settings (e.g. visual effects). We demonstrate our method on a range of challenging scenes and evaluate it against state-of-the-art methods for rendering caustics and glints

Analysis of Target Data-Dependent Greedy Kernel Algorithms: Convergence Rates for f-, f⋅P- and f/P-Greedy

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples