Discontinuous Integrands Research Articles

Inversion problem for Radon transforms defined on pseudoconvex sets

This paper is devoted to some questions of inversion for the classical and generalized integral Radon transform. The main question is to determine information about the integrand functions if the values of some integrals are known. A feature of the work of the authors of this message is an analysis of the case when the function is integrated according to hyperplanes in finite-dimensional Euclidean space, and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. At the same time, the number of independent variables describing known integrals are smaller than those of the unknown integrand. We consider discontinuous integrands defined specifically introduced pseudo-convex sets. A Stefan-type problem is posed about finding surfaces discontinuities of the integrand function. The work provides formulas based on the application special integro-differential operators to known data and allowing you to solve the assigned tasks.

On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>\\frac{1}{2}$$\\end{document} with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-2H}$$\\end{document}, which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

Boolean finite cell method for multi-material problems including local enrichment of the Ansatz space

AbstractThe Finite Cell Method (FCM) allows for an efficient and accurate simulation of complex geometries by utilizing an unfitted discretization based on rectangular elements equipped with higher-order shape functions. Since the mesh is not aligned to the geometric features, cut elements arise that are intersected by domain boundaries or internal material interfaces. Hence, for an accurate simulation of multi-material problems, several challenges have to be solved to handle cut elements. On the one hand, special integration schemes have to be used for computing the discontinuous integrands and on the other hand, the weak discontinuity of the displacement field along the material interfaces has to be captured accurately. While for the first issue, a space-tree decomposition is often employed, the latter issue can be solved by utilizing a local enrichment approach, adopted from the extended finite element method. In our contribution, a novel integration scheme for multi-material problems is introduced that, based on the B-FCM formulation for porous media, originally proposed by Abedian and Düster (Comput Mech 59(5):877–886, 2017), extends the standard space-tree decomposition by Boolean operations yielding a significantly reduced computational effort. The proposed multi-material B-FCM approach is combined with the local enrichment technique and tested for several problems involving material interfaces in 2D and 3D. The results show that the number of integration points and the computational time can be reduced by a significant amount, while maintaining the same accuracy as the standard FCM.

On some properties of *-integral

This work continues the author's research on the theory of regulated functions and *-integral. The possibility to express a regulated function as a sum of right-continuous and left-continuous functions (called $rl$-representation) is studied. A limit theorem for the *-integral is proved. It allows approximating discontinuous integrands and integrators by sequences of absolutely continuous functions. A new result on $\delta$-correctness of the solution of an ordinary linear differential equation with generalized functions in coefficients is proved. This solution is defined via a quasi-differential equation. A formula for the total variation of an indefinite *-integral of a $\sigma$-continuous function with respect to a function of bounded variation is given. It generalizes the well-known formula for computing the total variation of an absolutely continuous function. The formula is also interesting in the case of an indefinite $RS$-integral.

Non-negative moment fitting quadrature for cut finite elements and cells undergoing large deformations

Fictitious domain methods, such as the finite cell method, simplify the discretization of a domain significantly. This is because the mesh does not need to conform to the domain of interest. However, because the mesh generation is simplified, broken cells with discontinuous integrands must be integrated using special quadrature schemes. The moment fitting quadrature is a very efficient scheme for integrating broken cells since the number of integration points generated is much lower as compared to the commonly used adaptive octree scheme. However, standard moment fitting rules can lead to integration points with negative weights. Whereas negative weights might not cause any difficulties when solving linear problems, this can change drastically when considering nonlinear problems such as hyperelasticity or elastoplasticity. Then negative weights can lead to a divergence of the Newton-Raphson method applied within the incremental/iterative procedure of the nonlinear computation. In this paper, we extend the moment fitting method with constraints that ensure the generation of positive weights when solving the moment fitting equations. This can be achieved by employing a so-called non-negative least square solver. The performance of the non-negative moment fitting scheme will be illustrated using different numerical examples in hyperelasticity and elastoplasticity.

Systematically differentiating parametric discontinuities

Emerging research in computer graphics, inverse problems, and machine learning requires us to differentiate and optimize parametric discontinuities. These discontinuities appear in object boundaries, occlusion, contact, and sudden change over time. In many domains, such as rendering and physics simulation, we differentiate the parameters of models that are expressed as integrals over discontinuous functions. Ignoring the discontinuities during differentiation often has a significant impact on the optimization process. Previous approaches either apply specialized hand-derived solutions, smooth out the discontinuities, or rely on incorrect automatic differentiation. We propose a systematic approach to differentiating integrals with discontinuous integrands, by developing a new differentiable programming language. We introduce integration as a language primitive and account for the Dirac delta contribution from differentiating parametric discontinuities in the integrand. We formally define the language semantics and prove the correctness and closure under the differentiation, allowing the generation of gradients and higher-order derivatives. We also build a system, Teg, implementing these semantics. Our approach is widely applicable to a variety of tasks, including image stylization, fitting shader parameters, trajectory optimization, and optimizing physical designs.

Reparameterizing discontinuous integrands for differentiable rendering

Differentiable rendering has recently opened the door to a number of challenging inverse problems involving photorealistic images, such as computational material design and scattering-aware reconstruction of geometry and materials from photographs. Differentiable rendering algorithms strive to estimate partial derivatives of pixels in a rendered image with respect to scene parameters, which is difficult because visibility changes are inherently non-differentiable. We propose a new technique for differentiating path-traced images with respect to scene parameters that affect visibility, including the position of cameras, light sources, and vertices in triangle meshes. Our algorithm computes the gradients of illumination integrals by applying changes of variables that remove or strongly reduce the dependence of the position of discontinuities on differentiable scene parameters. The underlying parameterization is created on the fly for each integral and enables accurate gradient estimates using standard Monte Carlo sampling in conjunction with automatic differentiation. Importantly, our approach does not rely on sampling silhouette edges, which has been a bottleneck in previous work and tends to produce high-variance gradients when important edges are found with insufficient probability in scenes with complex visibility and high-resolution geometry. We show that our method only requires a few samples to produce gradients with low bias and variance for challenging cases such as glossy reflections and shadows. Finally, we use our differentiable path tracer to reconstruct the 3D geometry and materials of several real-world objects from a set of reference photographs.

On the Error Rate of Conditional Quasi--Monte Carlo for Discontinuous Functions

This paper studies the rate of convergence for conditional quasi--Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of ...

Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube

This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $[0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only $o(n^{-1/2})$ for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of $O(n^{-1/2-1/(4d-2)+\epsilon})$ for arbitrarily small $\epsilon>0$. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains $O(n^{-1+\epsilon})$ if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.

Improved stratification for Metropolis light transport

The approximation error of Monte Carlo methods depends on two factors, how accurately the target density of the samples mimics the integrand and how well the finite number of samples are distributed with the target density. The first factor can be reduced by importance sampling, the second by stratification. The Metropolis algorithm is particularly effective in importance sampling, but is poor concerning stratification if mutation strategies are not carefully designed. This means that although the asymptotic distribution of the samples well mimics most of the factors of the integrand, the empirical distribution of the finite number of samples can be far from the desired distribution. This paper examines this issue and proposes a simple mutation scheme for Primary Sample Space Metropolis Light Transport (PSSMLT), which improves the stratum of samples and thus reduces the integration error. Unlike other approaches, the proposed method does not sacrifice importance sampling for better stratification and uses only the local importance value without requiring derivatives that would be problematic for discontinuous integrands caused by occlusions and texture patterns. The method is simple to implement and to integrate into photo-realistic renderers.

Numerical integration of discontinuous functions: moment fitting and smart octree

A fast and simple grid generation can be achieved by non-standard discretization methods where the mesh does not conform to the boundary or the internal interfaces of the problem. However, this simplification leads to discontinuous integrands for intersected elements and, therefore, standard quadrature rules do not perform well anymore. Consequently, special methods are required for the numerical integration. To this end, we present two approaches to obtain quadrature rules for arbitrary domains. The first approach is based on an extension of the moment fitting method combined with an optimization strategy for the position and weights of the quadrature points. In the second approach, we apply the smart octree, which generates curved sub-cells for the integration mesh. To demonstrate the performance of the proposed methods, we consider several numerical examples, showing that the methods lead to efficient quadrature rules, resulting in less integration points and in high accuracy.

Conditional Monte Carlo: A Change-of-Variables Approach

Conditional Monte Carlo (CMC) has been widely used for sensitivity estimation with discontinuous integrands as a standard simulation technique. A major limitation of using CMC in this context is that finding conditioning variables to ensure continuity and tractability of the resulting conditional expectation is often problem dependent and may be difficult. In this paper, we attempt to circumvent this difficulty by proposing a change-of-variables approach to CMC, leading to efficient sensitivity estimators under mild conditions that are satisfied by a wide class of discontinuous integrands. These estimators do not rely on the structure of the simulation models and are less problem dependent. The value of the proposed approach is exemplified through applications in sensitivity estimation for financial options and gradient estimation of chance-constrained optimization problems.

Efficient and accurate numerical quadrature for immersed boundary methods

One question in the context of immersed boundary or fictitious domain methods is how to compute discontinuous integrands in cut elements accurately. A frequently used method is to apply a composed Gaussian quadrature based on a spacetree subdivision. Although this approach works robustly on any geometry, the resulting integration mesh yields a low order representation of the boundary. If high order shape functions are employed to approximate the solution, this lack of geometric approximation power prevents exponential convergence in the asymptotic range. In this paper we present an algorithmic subdivision approach that aims to be as robust as the spacetree decomposition even for close-to-degenerate cases—but remains geometrically accurate at the same time. Based on 2D numerical examples, we will show that optimal convergence rates can be obtained with a nearly optimal number of integration points.

Calculation of self-field coefficients for 2D magnetostatic systems

Physics and engineering students are introduced to the notion of a demagnetizing field in classical electromagnetism courses. This concept involves a formalism based on an integral formulation for calculating the coefficients of the demagnetizing tensor, i.e., a pure geometric quantity. For self-fields, the observation point is located inside the integration region which in turn leads to discontinuous integrands. Therefore, in order to avoid mathematical inconsistencies, special care must be taken when evaluating self-field coefficients, referred to here as self-terms. Given the complexity of this approach, in particular in 3D, it is certainly interesting from a pedagogical stand point to employ 2D systems as a first step for describing these kinds of coefficients. Thus, in this paper, the generalization of self-terms of the demagnetizing tensor is proven for 2D magnetostatic systems. Nonetheless, the structure of this proof pertains to many other situations given the fact that discontinuous integrands commonly arise in physics (e.g. integral solutions of PDEs which use a Green’s function).

Numerical Integration for Interactive Terms in Mesh Superposition Method by Auto-Mesh Generation

The interactive terms in the mesh superposition method are calculated approximately because the integrand has a discontinuity along the mesh lines in a superimposed global mesh. In this paper, the accurate numerical integration technique for a discontinuous integrand by the Delaunay triangulation is proposed. Integral ranges including discontinuous integrands are divided into triangles only with a continuum integrand. The exact solution is sum of the integral values calculated in all triangles. Appropriate triangle edges can be regenerated by the swapping algorithm even though generated triangles contain the inappropriate boundary lines which bring about a discontinuity. A comparison of the accuracy of numerical integration between the proposed technique and conventional approximate means and an application to a sensitivity calculation are investigated in the first example. The second example illustrates the swapping algorithm for the adjustment of generated triangles.

A relaxation result in BV for integral functionals with discontinuous integrands

We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

On some sharp conditions for lower semicontinuity in $L^1$

Let $\Omega $ be an open set of $\mathbb{R}^{n}$ and let $f:\Omega \times \mathbb{R}\times \mathbb{R}^{n}$ be a nonnegative continuous function, convex with respect to $\xi \in \mathbb{R}^{n}$. Following the well known theory originated by Serrin Serrin [14] in 1961, we deal with the lower semicontinuity of the integral $F\left( u,\Omega \right) =\int_{\Omega }f\left( x,u(x),Du(x)\right) \,dx$ with respect to the $L_{\text{loc} }^{1}\left( \Omega \right) $ strong convergence. Only recently it has been discovered that dependence of $f\left( x,s,\xi \right) $ on the $x$ variable plays a crucial role in the lower semicontinuity. In this paper we propose a mild assumption on $x$ that allows us to consider discontinuous integrands too. More precisely, we assume that $f\left( x,s,\xi \right) $ is a nonnegative Carathéodory function, convex with respect to $\xi $ , continuous in $\left( s,\xi \right) $ and such that $f(\cdot ,s,\xi )\in W_{\text{loc}}^{1,1}\left( \Omega \right) $ for every $s\in \mathbb{R}$ and $\xi \in \mathbb{R}^{n}$, with the $L^{1}$ norm of $f_{x}(\cdot ,s,\xi )$ locally bounded. We also discuss some other conditions on $x$; in particular we prove that Hölder continuity of $f$ with respect to $x$ is not sufficient for lower semicontinuity, even in the one dimensional case, thus giving an answer to a problem posed by the authors in [12]. Finally, we investigate the lower semicontinuity of the integral $F\left( u,\Omega \right) $, with respect to the strong norm topology of $L_{\text{ loc}}^{1}\left( \Omega \right) $, in the vector-valued case, i.e., when $ f:\Omega \times \mathbb{R}^{m}\times \mathbb{R}^{m\times n}\rightarrow \mathbb{R}$ for some $n\geq 1$ and $m>1$.

Multi-Dimensional Quadrature of Singular and Discontinuous Functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.

Variational Problems with Nonconvex, Noncoercive, Highly Discontinuous Integrands: Characterization and Existence of Minimizers

We consider the functional $F(v)=\int_a^b f(t,v^\prime(t)) dt $ in ${\cal H}_p=\{ v\in W^{1,p}: v(a)=0, v(b)=d\}$, $p\in [1,+\infty]$. Under only the assumption that the integrand is ${\cal L}\otimes {\cal B}_n$-measurable, we prove characterizations of strong and weak minimizers both in terms of the minimizers of the relaxed functional and by means of the Euler--Lagrange inclusion. As an application, we provide necessary and sufficient conditions for the existence of the minimum, expressed in terms of a limitation on the width of the slope d.

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects.