Showing papers on "Discretization published in 2016"
TL;DR: A new computing paradigm is developed, which is referred to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, thus bypassing the empirical material modeling step of conventional computing altogether.
458 citations
01 Aug 2016
TL;DR: A new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability is presented, showing potentiality, generality, and superiority of the method as compared with other well-known methods.
Abstract: Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical systems is to use fuzzy differential equations. In this paper, we present a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability. The analytic and approximate solutions are given with series form in terms of their parametric form in the space $$W_2^2 [a,b]\oplus W_2^2 [a,b].$$W22[a,b]źW22[a,b]. The method used in this paper has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this paper show potentiality, generality, and superiority of our method as compared with other well-known methods.
330 citations
Proceedings Article•
05 Dec 2016TL;DR: In this article, the dual optimal transport problem can be re-cast as the maximization of an expectation, which results in a smooth dual optimization problem which can be addressed with algorithms that have a provably faster convergence.
Abstract: Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of stochastic optimization algorithms to cope with large-scale problems routinely encountered in machine learning applications. These methods are able to manipulate arbitrary distributions (either discrete or continuous) by simply requiring to be able to draw samples from them, which is the typical setup in high-dimensional learning problems. This alleviates the need to discretize these densities, while giving access to provably convergent methods that output the correct distance without discretization error. These algorithms rely on two main ideas: (a) the dual OT problem can be re-cast as the maximization of an expectation; (b) entropic regularization of the primal OT problem results in a smooth dual optimization optimization which can be addressed with algorithms that have a provably faster convergence. We instantiate these ideas in three different computational setups: (i) when comparing a discrete distribution to another, we show that incremental stochastic optimization schemes can beat the current state of the art finite dimensional OT solver (Sinkhorn's algorithm) ; (ii) when comparing a discrete distribution to a continuous density, a re-formulation (semi-discrete) of the dual program is amenable to averaged stochastic gradient descent, leading to better performance than approximately solving the problem by discretization ; (iii) when dealing with two continuous densities, we propose a stochastic gradient descent over a reproducing kernel Hilbert space (RKHS). This is currently the only known method to solve this problem, and is more efficient than discretizing beforehand the two densities. We backup these claims on a set of discrete, semi-discrete and continuous benchmark problems.
276 citations
TL;DR: Robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications are developed.
235 citations
11 Apr 2016
TL;DR: SCOTS a software tool for the automatic controller synthesis for nonlinear control systems based on symbolic models, also known as discrete abstractions, which natively provides algorithms to synthesize controllers with respect to invariance and reachability specifications.
Abstract: We introduce SCOTS a software tool for the automatic controller synthesis for nonlinear control systems based on symbolic models, also known as discrete abstractions. The tool accepts a differential equation as the description of a nonlinear control system. It uses a Lipschitz type estimate on the right-hand-side of the differential equation together with a number of discretization parameters to compute a symbolic model that is related with the original control system via a feedback refinement relation. The tool supports the computation of minimal and maximal fixed points and thus natively provides algorithms to synthesize controllers with respect to invariance and reachability specifications. The atomic propositions, which are used to formulate the specifications, are allowed to be defined in terms of finite unions and intersections of polytopes as well as ellipsoids. While the main computations are done in C++, the tool contains a Matlab interface to simulate the closed loop system and to visualize the abstract state space together with the atomic propositions. We illustrate the performance of the tool with two examples from the literature. The tool and all conducted experiments are available at www.hcs.ei.tum.de.
200 citations
TL;DR: An efficient discretization method for the solution of the unsteady incompressible Navier–Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation is presented and the performance on two and three dimensional benchmark problems is demonstrated.
196 citations
TL;DR: This work introduces a family of Markov chain Monte Carlo samplers that can adapt to the particular structure of a posterior distribution over functions that may be useful for a large class of high-dimensional problems where the target probability measure has a density with respect to a Gaussian reference measure.
189 citations
TL;DR: In this article, a fully implicit formulation of coupled flow and geomechanics for fractured three-dimensional subsurface formations is presented, where flow equations are discretized using a finite-volume method, and the poroelasticity equations are Discrete Fracture Model (DFM) is used to model deformation.
Abstract: We present a fully implicit formulation of coupled flow and geomechanics for fractured three-dimensional subsurface formations. The Reservoir Characterization Model (RCM) consists of a computational grid, in which the fractures are represented explicitly. The Discrete Fracture Model (DFM) has been widely used to model the flow and transport in natural geological porous formations. Here, we extend the DFM approach to model deformation. The flow equations are discretized using a finite-volume method, and the poroelasticity equations are discretized using a Galerkin finite-element approximation. The two discretizations—flow and mechanics—share the same three-dimensional unstructured grid. The mechanical behavior of the fractures is modeled as a contact problem between two computational planes. The set of fully coupled nonlinear equations is solved implicitly. The implementation is validated for two problems with analytical solutions. The methodology is then applied to a shale-gas production scenario where a synthetic reservoir with 100 natural fractures is produced using a hydraulically fractured horizontal well.
167 citations
TL;DR: It is shown that both proposed continuous and discrete controllers can ensure that the system output tracks the target signal with a small bounded error and the other closed-loop signals remain bounded.
Abstract: In this paper, tracking control problems are investigated for a class of uncertain nonlinear systems in lower triangular form First, a state-feedback controller is designed by using adaptive backstepping technique and the universal approximation ability of fuzzy logic systems During the design procedure, a developed method with less computation is proposed by constructing one maximum adaptive parameter Furthermore, adaptive controllers with nonsymmetric dead-zone are also designed for the systems Then, a sampled-data control scheme is presented to discretize the obtained continuous-time controller by using the forward Euler method It is shown that both proposed continuous and discrete controllers can ensure that the system output tracks the target signal with a small bounded error and the other closed-loop signals remain bounded Two simulation examples are presented to verify the effectiveness and applicability of the proposed new design techniques
164 citations
TL;DR: In this paper, a new three-dimensional numerical wave tank is developed for the calculation of wave propagation and wave hydrodynamics by solving the incompressible Navier-Stokes equations.
163 citations
01 Jan 2016
TL;DR: The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method.
Abstract: Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.
TL;DR: This article proposes to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduces fast and robust solvers and reduces the memory consumption of the Moulinec–Suquet algorithms by 50%.
Abstract: Summary
In this article, we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers. Our method shares some properties with the FFT-based homogenization technique of Moulinec and Suquet, which has received widespread attention recently because of its robustness and computational speed. These similarities include the use of FFT and the resulting performing solvers. The staggered grid discretization, however, offers three crucial improvements. Firstly, solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by Moulinec–Suquet's discretization. Secondly, the iteration numbers of our solvers are bounded independently of the grid size and the contrast. In particular, our solvers converge for three-dimensional porous structures, which cannot be handled by Moulinec–Suquet's method. Thirdly, the finite difference discretization allows for algorithmic variants with lower memory consumption. More precisely, it is possible to reduce the memory consumption of the Moulinec–Suquet algorithms by 50%. We underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale. Copyright © 2015 John Wiley & Sons, Ltd.
TL;DR: This paper considers the line spectral estimation problem and proposes an iterative reweighted method which jointly estimates the sparse signals and the unknown parameters associated with the true dictionary, and achieves super resolution and outperforms other state-of-the-art methods in many cases of practical interest.
Abstract: Conventional compressed sensing theory assumes signals have sparse representations in a known dictionary. Nevertheless, in many practical applications such as line spectral estimation, the sparsifying dictionary is usually characterized by a set of unknown parameters in a continuous domain. To apply the conventional compressed sensing technique to such applications, the continuous parameter space has to be discretized to a finite set of grid points, based on which a “nominal dictionary” is constructed for sparse signal recovery. Discretization, however, inevitably incurs errors since the true parameters do not necessarily lie on the discretized grid. This error, also referred to as grid mismatch, leads to deteriorated recovery performance. In this paper, we consider the line spectral estimation problem and propose an iterative reweighted method which jointly estimates the sparse signals and the unknown parameters associated with the true dictionary. The proposed algorithm is developed by iteratively decreasing a surrogate function majorizing a given log-sum objective function, leading to a gradual and interweaved iterative process to refine the unknown parameters and the sparse signal. A simple yet effective scheme is developed for adaptively updating the regularization parameter that controls the tradeoff between the sparsity of the solution and the data fitting error. Theoretical analysis is conducted to justify the proposed method. Simulation results show that the proposed algorithm achieves super resolution and outperforms other state-of-the-art methods in many cases of practical interest.
TL;DR: Tesseroids as discussed by the authors is a set of command-line programs to perform forward modeling of gravitational fields in spherical coordinates. But it does not support the integration of tesseroid prisms.
Abstract: We have developed the open-source software Tesseroids, a set of command-line programs to perform forward modeling of gravitational fields in spherical coordinates. The software is implemented in the C programming language and uses tesseroids (spherical prisms) for the discretization of the subsurface mass distribution. The gravitational fields of tesseroids are calculated numerically using the Gauss-Legendre quadrature (GLQ). We have improved upon an adaptive discretization algorithm to guarantee the accuracy of the GLQ integration. Our implementation of adaptive discretization uses a “stack-based” algorithm instead of recursion to achieve more control over execution errors and corner cases. The algorithm is controlled by a scalar value called the distance-size ratio (D) that determines the accuracy of the integration as well as the computation time. We have determined optimal values of D for the gravitational potential, gravitational acceleration, and gravity gradient tensor by comparing the comp...
TL;DR: In this article, the convergence of the L-scheme/Newton method was theoretically proved and the convergence performance of the other linearization methods was discussed. But the convergence was not analyzed.
Abstract: This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards’ equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards’ equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L-scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L-scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L-scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.
TL;DR: An efficient numerical method within the framework of variational formulation is employed to study the buckling and vibration of axially-compressed functionally graded carbon nanotube-reinforced composite (FG-CNTRC) conical shells.
Abstract: An efficient numerical method within the framework of variational formulation is employed to study the buckling and vibration of axially-compressed functionally graded carbon nanotube-reinforced composite (FG-CNTRC) conical shells. The effective material properties of functionally graded composite conical shell are estimated based on the extended rule of mixture. To derive the governing equations, the matrix form of Hamilton's principle is first presented on the basis of the first order shear deformation theory. Then, employing the generalized differential quadrature (GDQ) method in axial direction and periodic differential operators in circumferential direction, the numerical differential and integral operators are introduced to perform the discretization process. The comparison study is carried out to verify the accuracy and efficiency of the proposed method. Numerical results indicate that the volume fraction and types of distribution of CNTs have considerable effects on the buckling and vibration characteristics of FG-CNTRC conical shells subjected to axial loadings.
TL;DR: This paper analyzes several iterative solution schemes for solving matrix systems that result from discretization and linearization of the governing equations and highlights the fundamental connections that underlie their effectiveness.
TL;DR: This is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation and the theoretical result is validated by a number of numerical tests.
Abstract: In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the ($3-\alpha$)-order convergence of the time scheme, where $\alpha$ is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation.
TL;DR: This work designs an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations and proves that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced.
TL;DR: This article proposes a distributed implementation of one of the most well‐known discretizers based on Information Theory, obtaining better results than the one produced by the entropy minimization discretizer proposed by Fayyad and Irani.
Abstract: Discretization of numerical data is one of the most influential data preprocessing tasks in knowledge discovery and data mining. The purpose of attribute discretization is to find concise data representations as categories which are adequate for the learning task retaining as much information in the original continuous attribute as possible. In this article, we present an updated overview of discretization techniques in conjunction with a complete taxonomy of the leading discretizers. Despite the great impact of discretization as data preprocessing technique, few elementary approaches have been developed in the literature for Big Data. The purpose of this article is twofold: a comprehensive taxonomy of discretization techniques to help the practitioners in the use of the algorithms is presented; the article aims is to demonstrate that standard discretization methods can be parallelized in Big Data platforms such as Apache Spark, boosting both performance and accuracy. We thus propose a distributed implementation of one of the most well-known discretizers based on Information Theory, obtaining better results than the one produced by: the entropy minimization discretizer proposed by Fayyad and Irani. Our scheme goes beyond a simple parallelization and it is intended to be the first to face the Big Data challenge. WIREs Data Mining Knowl Discov 2016, 6:5-21. doi: 10.1002/widm.1173
Posted Content•
TL;DR: A new class of stochastic optimization algorithms to cope with large-scale problems routinely encountered in machine learning applications, based on entropic regularization of the primal OT problem, which results in a smooth dual optimization optimization which can be addressed with algorithms that have a provably faster convergence.
Abstract: Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of stochastic optimization algorithms to cope with large-scale problems routinely encountered in machine learning applications. These methods are able to manipulate arbitrary distributions (either discrete or continuous) by simply requiring to be able to draw samples from them, which is the typical setup in high-dimensional learning problems. This alleviates the need to discretize these densities, while giving access to provably convergent methods that output the correct distance without discretization error. These algorithms rely on two main ideas: (a) the dual OT problem can be re-cast as the maximization of an expectation ; (b) entropic regularization of the primal OT problem results in a smooth dual optimization optimization which can be addressed with algorithms that have a provably faster convergence. We instantiate these ideas in three different setups: (i) when comparing a discrete distribution to another, we show that incremental stochastic optimization schemes can beat Sinkhorn's algorithm, the current state-of-the-art finite dimensional OT solver; (ii) when comparing a discrete distribution to a continuous density, a semi-discrete reformulation of the dual program is amenable to averaged stochastic gradient descent, leading to better performance than approximately solving the problem by discretization ; (iii) when dealing with two continuous densities, we propose a stochastic gradient descent over a reproducing kernel Hilbert space (RKHS). This is currently the only known method to solve this problem, apart from computing OT on finite samples. We backup these claims on a set of discrete, semi-discrete and continuous benchmark problems.
TL;DR: Several safe-guarding techniques are incorporated into the algorithm, namely virtual control and trust regions, which add another layer of algorithmic robustness and convergence results will be independent from any numerical schemes used for discretization.
Abstract: This paper presents an algorithm to solve non-convex optimal control problems, where non-convexity can arise from nonlinear dynamics, and non-convex state and control constraints. This paper assumes that the state and control constraints are already convex or convexified, the proposed algorithm convexifies the nonlinear dynamics, via a linearization, in a successive manner. Thus at each succession, a convex optimal control subproblem is solved. Since the dynamics are linearized and other constraints are convex, after a discretization, the subproblem can be expressed as a finite dimensional convex programming subproblem. Since convex optimization problems can be solved very efficiently, especially with custom solvers, this subproblem can be solved in time-critical applications, such as real-time path planning for autonomous vehicles. Several safe-guarding techniques are incorporated into the algorithm, namely virtual control and trust regions, which add another layer of algorithmic robustness. A convergence analysis is presented in continuous- time setting. By doing so, our convergence results will be independent from any numerical schemes used for discretization. Numerical simulations are performed for an illustrative trajectory optimization example.
TL;DR: A comprehensive framework for modeling flow in porous media containing thin, discrete features, which could be high-permeability fractures or low-Permeability deformation bands, is presented and is compatible with any connectivity-based simulator.
TL;DR: In this paper, the authors present a parallel multigrid method for parabolic equations based on arbitrarily high order discontinuous Galerkin discretizations in time and a finite element discretization in space.
Abstract: We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters, a precise criterion for semi-coarsening in time or full coarsening, and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties.
TL;DR: Based on the space-time spectral method, a high-order scheme is proposed in this paper, where the Legendre polynomials are adopted in temporal discretization and the Fourier-like basis functions are constructed for spatial discretisation.
TL;DR: An implicit fully-discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time is proposed and proven: stability and error estimates are proved for this scheme.
Abstract: We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition. We propose and analyze an implicit fully discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time. We prove stability and error estimates for this scheme.
TL;DR: An original algorithm for state-of-charge estimation using the pseudo two-dimensional model that circumvents the degeneracy problems associated with high-dimensional state estimation and avoids the repetitive solution of implicit equations by defining a ‘tether’ particle.
TL;DR: In this paper, a stronger stability under the infinity norm for the implicit-explicit discretization in time and central finite difference in space was established for the Allen-Chan equations.
Abstract: It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.
TL;DR: In this paper, a class of maximum principle preserving schemes was studied for the generalized Allen-Cahn equation with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear degenerate mobility.
Abstract: This paper is concerned with the generalized Allen-Cahn equation with a nonlinear mobility that may be degenerate, which also includes an advection term appearing in many phase- field models for multi-component fluid flows. A class of maximum principle preserving schemes will be studied for the generalized Allen-Cahn equation, with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear degenerate mobility. For time discretization, the standard semi-implicit scheme as well as the stabilized semi-implicit scheme will be adopted, while for space discretization, the central finite difference is used for approximating the diffusion term and the upwind scheme is employed for the advection term. We establish the maximum principle for both semi-discrete (in time) and fully discretized schemes. We also provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are carried out to verify our theoretical results.
TL;DR: The numerical efficiency can be greatly improved, since the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be implemented with the help of the FFT in a pseudo-spectral setting.
Abstract: We present a second order energy stable numerical scheme for the two and three dimensional Cahn---Hilliard equation, with Fourier pseudo-spectral approximation in space. A convex splitting treatment assures the unique solvability and unconditional energy stability of the scheme. Meanwhile, the implicit treatment of the nonlinear term makes a direct nonlinear solver impractical, due to the global nature of the pseudo-spectral spatial discretization. We propose a homogeneous linear iteration algorithm to overcome this difficulty, in which an $$O(s^2)$$O(s2) (where s the time step size) artificial diffusion term, a Douglas---Dupont-type regularization, is introduced. As a consequence, the numerical efficiency can be greatly improved, since the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be implemented with the help of the FFT in a pseudo-spectral setting. Moreover, a careful nonlinear analysis shows a contraction mapping property of this linear iteration, in the discrete $$\ell ^4$$l4 norm, with discrete Sobolev inequalities applied. Moreover, a bound of numerical solution in $$\ell ^\infty $$lź norm is also provided at a theoretical level. The efficiency of the linear iteration solver is demonstrated in our numerical experiments. Some numerical simulation results are presented, showing the energy decay rate for the Cahn---Hilliard flow with different values of $$\varepsilon $$ź.