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We propose a computationally efficient method to construct nonparametric, heteroscedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. Our approach provides an alternative to the now-standard conformal prediction for uncertainty quantification, with novel theoretical insights and computational advantages. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, has strong non-asymptotic coverage properties, and is easy to implement using standard convex programs. Our approach can be viewed as a novel variance interpolation with confidence and further leverages techniques from semi-definite programming and sum-of-squares optimization. Theoretical and numerical performances for the proposed approach for uncertainty quantification are analyzed., 21 pages, 4 figures

Investigation of spectral (collocation or Galerkin) methods for the solution approximation of different classes of optimal control problems have had been increased in recent years. In this research work, we proposed and analyzed an applicable Legendre spectral collocation approach, based on an efficient space of fractional basis functions, to compute numerically the solution of fractional optimal control problems. We first introduce a fractional Lagrange interpolation and then approximate the unknown control and state by this new set of fractional interpolation functions. The problem is discretized in terms of shifted Legendre–Gauss collocation points and two approaches are suggested to calculate the exact fractional differentiation matrix. Also, we prove that the approximate solutions of the obtained discretized problem converge to the optimal solution of the main problem by assuming some mild conditions. The method is implemented for three numerical test problems to show the claimed efficiency and capability.

Numerical approach of two-derivative Runge-Kutta type method with three-stage fifth-order (TDRKT3(5)) is developed and proposed for solving a special type of third-order delay differential equations (DDEs) with constant delay. An algorithm based on Newton interpolation and hybrid with the TDRKT method is built to approximate the solution of third-order DDEs. In this paper, three-stage fifth-order called TDRKT3(5) method with single third derivative and multiple evaluations of the fourth derivative is highlighted to solve third-order pantograph type delay differential equations directly with the aid of the Newton interpolation method. Stability analysis of TDRKT3(5) method is investigated. The numerical experiments illustrate high efficiency and validity of the new method for solving a special class of third-order DDEs and some future works are recommended by extending proposed method to solve fractional and singularly perturbed delay differential equations.

In the process industry, it is essential to establish a data-driven soft sensor to predict the key variable that is difficult to online measure directly. The accuracy performance of data-driven soft sensors relies heavily on data. Unfortunately, it is hard to acquire sufficient and informative data from the samples with limited number, which is called as the small sample problem. For handling the small sample problem, it is a good solution to generating virtual samples according to the distribution of original data. This paper proposes an enhanced method of virtual sample generation utilizing manifold features to develop soft sensors using small data. First, T-Distribution Stochastic Neighbor Embedding (t-SNE) is utilized to extract the features of input data. The main idea of generating virtual samples is to use the interpolation algorithm to obtain virtual t-SNE input features and then the random forest algorithm is utilized to get the virtual outputs using virtual t-SNE input features. Finally, virtual samples using the proposed t-SNE based virtual sample generation (t-SNE-VSG) can be achieved. For the sake of confirming the effectiveness and feasibility of the presented t-SNE-VSG, a standard data set is first used. What is more, a small data set from an actual industrial process of Purified Terephthalic Acid is used to establish a soft sensor model. The results from simulations show that the accuracy performance of the soft sensor established with small data can be effectively improved by adding the virtual samples generated by t-SNE-VSG. In addition, t-SNE-VSG achieves superior accuracy to state-of-the-art virtual sample generation methods.

Establishing correct correspondences between two images should consider both local and global spatial context. Given putative correspondences of feature points in two views, in this paper, we propose Order-Aware Network, which infers the probabilities of correspondences being inliers and regresses the relative pose encoded by the essential or fundamental matrix. Specifically, this proposed network is built hierarchically and comprises three operations. First, to capture the local context of sparse correspondences, the network clusters unordered input correspondences by learning a soft assignment matrix. These clusters are in canonical order and invariant to input permutations. Next, the clusters are spatially correlated to encode the global context of correspondences. After that, the context-encoded clusters are interpolated back to the original size and position to build a hierarchical architecture. We intensively experiment on both outdoor and indoor datasets. The accuracy of the two-view geometry and correspondences are significantly improved over the state-of-the-arts. Besides, based on the proposed method and advanced local feature, we won the first place in CVPR 2019 image matching workshop challenge and also achieve state-of-the-art results in the Visual Localization benchmark. Code is available at https://github.com/zjhthu/OANet.

We propose a generic and scalable method for computing global solutions of nonlinear, high-dimensional dynamic stochastic economic models. First, within an MPI–TBB parallel time-iteration framework, we approximate economic policy functions using an adaptive, high-dimensional model representation scheme, combined with adaptive sparse grids. With increasing dimensions, the number of points in this efficiently-chosen combination of low-dimensional grids grows much more slowly than standard tensor product grids, sparse grids, or even adaptive sparse grids. Moreover, the adaptivity within the individual component functions adds an additional layer of sparsity, since grid points are added only where they are most needed — that is to say, in regions of the computational domain with steep gradients or at non-differentiabilities. Second, we introduce a performant vectorization scheme of the interpolation compute kernel. Third, we validate our claims with numerical experiments conducted on “Piz Daint" (Cray XC50) at the Swiss National Super-computing Center. We observe significant speedups over the state-of-the-art techniques, and almost ideal strong scaling up to at least 1, 000 compute nodes. Fourth, to demonstrate the broad applicability of our method, we compute global solutions to two different versions of a dynamic stochastic economic model: a high-dimensional international real business cycle model with capital adjustment costs, and with or without irreversible investment. We solve these models up to 300 continuous state variables globally.

In X-ray computed tomography (CT) an important objective is to reduce the radiation dose without significantly degrading the image quality. Compressed sensing (CS) enables the radiation dose to be reduced by producing diagnostic images from a limited number of projections. However, conventional CS-based algorithms are computationally intensive and time-consuming. We propose a new algorithm that accelerates the CS-based reconstruction by using a fast pseudopolar Fourier based Radon transform and rebinning the diverging fan beams to parallel beams. The reconstruction process is analyzed using a maximum-a-posterior approach, which is transformed into a weighted CS problem. The weights involved in the proposed model are calculated based on the statistical characteristics of the reconstruction process, which is formulated in terms of the measurement noise and rebinning interpolation error. Therefore, the proposed method not only accelerates the reconstruction, but also removes the rebinning and interpolation errors. Simulation results are shown for phantoms and a patient. For example, a 512 × 512 Shepp-Logan phantom when reconstructed from 128 rebinned projections using a conventional CS method had 10% error, whereas with the proposed method the reconstruction error was less than 1%. Moreover, computation times of less than 30 sec were obtained using a standard desktop computer without numerical optimization.

In this paper, we propose two Maple procedures and some related utilities to determine the maximum curvature of a cubic Bézier-spline curve that interpolates an ordered set of points in R2 or R3. The procedures are designed from closed-form formulas for such open and closed curves.

One of the major challenges of coupled problems is to manage nonconforming meshes at the interface between two models and/or domains, due to different numerical schemes or domains discretizations employed. Moreover, very often complex submodels depend on (e.g., physical or geometrical) parameters. Understanding how outputs of interest are affected by parameter variations thus plays a key role to gain useful insights on the problem's physics; however, expensive repeated solutions of the problem using high-fidelity, full-order models are often unaffordable. In this paper, we propose a parametric reduced order modeling (ROM) technique for parametrized one-way coupled problems made by a first independent model, the master model, and a second model, the slave model, that depends on the master model through Dirichlet interface conditions. We combine a reduced basis (RB) method, applied to each subproblems, with the discretized empirical interpolation method (DEIM) to efficiently interpolate or project Dirichlet data across conforming and non-conforming meshes at the domains interface, building a low-dimensional representation of the overall coupled problem. The proposed technique is then numerically verified by considering a series of test cases involving both steady and unsteady problems, and deriving a-posteriori error estimates on the solution of the coupled problem in both cases. This work arises from the need to solve staggered cardiac electrophysiological models and represents the first step towards the setting of ROM techniques for the more general two-way Dirichlet-Neumann coupled problems solved with domain decomposition sub-structuring methods, when interface non-conformity is involved., Comment: 36 pages, 23 figures, 4 tables

We present a systematic way to implement higher order Whitney forms in numerical methods based on discrete exterior calculus. Given a simplicial mesh, we first refine the mesh into smaller simplices which can be used to define higher order Whitney forms. Cochains on this refined mesh can then be interpolated using higher order Whitney forms. Hence, when the refined mesh is used with methods based on discrete exterior calculus, the solution can be expressed as a higher order Whitney form. We present algorithms for the three required steps: refining the mesh, solving the coefficients of the interpolant, and evaluating the interpolant at a given point. With our algorithms, the order of the Whitney forms one wishes to use can be given as a parameter so that the same code covers all orders, which is a significant improvement on previous implementations. Our algorithms are applicable with all methods in which the degrees of freedom are integrals over mesh simplices — that is, when the solution is a cochain on a simplicial mesh. They can also be used when one simply wishes to approximate differential forms in finite-dimensional spaces. Numerical examples validate the generality of our algorithms.

The description of a physical problem through a model necessarily involves the introduction of parameters. Hence, one wishes to have a solution of the problem that is a function of all these parameters: a parametric solution. However, the construction of such parametric solutions exhibiting localization in space is only ensured by costly and time-consuming tests, which can be both numerical or experimental. Numerical methodologies used classically imply enormous computational efforts for exploring the design space. Therefore, parametric solutions obtained using advanced nonlinear regressions are an essential tool to address this challenge. However, classical regression techniques, even the most advanced ones, can lead to non physical interpolation in some fields such as fluid dynamics, where the solution localizes in different regions depending on the problem parameters choice. In this context, Optimal Transport (OT) offers a mathematical approach to measure distances and interpolate between general objects in a, sometimes, more physical way than the classical interpolation approach. Thus, OT has become fundamental in some fields such as statistics or computer vision, and it is being increasingly used in fields such as computational mechanics. However, the OT problem is usually computationally costly to solve and not adapted to be accessed in an online manner. Therefore, the aim of this paper is combining advanced nonlinear regressions with Optimal Transport in order to implement a parametric real-time model based on OT. To this purpose, a parametric model is built offline relying on Model Order Reduction and OT, leading to a real-time interpolation tool following Optimal Transport theory. Such a tool is of major interest in design processes, but also within the digital twin rationale.

Due to recent developments in the additive manufacturing industry, topology optimization is nowadays a powerful computational tool that allows to design feasible lightweight components. Although the Finite Element Method (FEM) is the most applied discretization technique, meshless methods are currently established as accurate numerical methods with relevant advantages in several engineering fields. Nonetheless, the state-of-the-art of meshless methods in topology optimization is still scarce. This work develops the combination of a bi-direction structural optimization (BESO) algorithm with the Natural Neighbour Radial Point Interpolation Method (NNRPIM), a meshless method combining the natural neighbours geometric concept with the RPI shape functions. First, several benchmark examples are solved to evaluate the algorithm capability under several algorithm parameters. The proposed methodology is then implemented to design new automotive lightweight components. The results from the numerical applications demonstrate that the NNRPIM is a solid technique to be incorporated in optimization algorithms. Additionally, innovative automotive industry designs for additive manufacturing can be obtained using the presented approach.

Adaptive-Network-based Fuzzy Inference System, ANFIS, offers a popular and powerful fuzzy inference mechanism. As with many other advanced data driven techniques, developing an effective ANFIS typically requires sufficient training data. However, in many real-world applications it is not always straightforward to obtain a large amount of representative data that covers the entire problem space to accomplish the required training, seriously restricting the performance of a learned ANFIS. This paper introduces a new ANFIS learning approach through an evolutionary process, which is able to generate an ANFIS with only a small number of training data in a certain problem region, by interpolating well trained ANFISs in the neighbouring regions. Such a process works by firstly producing an initial population of candidate fuzzy rules in the region of data shortage, through interpolating a rule dictionary constructed from trained ANFISs in the neighbourhood regions. The crossover and mutation operations over these candidate rules are then executed in an effort to attain candidates of improved performance. When this genetic learning process terminates the chromosomes in the final population either collectively form or each individually represent a learned ANFIS, depending on whether a single fuzzy rule or a set of fuzzy rules representing an entire ANFIS is implemented with a chromosome within the evolving population. Comparative experimental evaluations on both synthetic and real world datasets are carried out, demonstrating that in spite of data shortage, the proposed interpolation approach is able to produce ANFIS models that significantly outperform those trained using existing learning mechanisms.

This paper applies a unified process to calculate ”exact” (consistent) finite-element (FE) matrices for framed structures having nonprismatic elements and including shear-deformation and rotational-inertia effects. In this process, the exact expressions for the element stiffness and nodal-load coefficients result from applying the principle of virtual forces (PVF) at the element level. Rigidity values, determined at a certain number of cross sections in the frame element, are employed to describe how the corresponding rigidities vary along its length. For that, interpolation polynomials of different orders are considered. Exact Timoshenko’s shape functions, built under the most general cases of rigidity variation, are used for evaluating the mass matrix coefficients. In the applications, complex 2D frames with nonprismatic elements are considered to simulate bridge structures under seismic excitation and a generic harmonic load. Comparisons with highly accurate response time-histories obtained by employing ANSYS (3D) solid-FE models are effected to verify the robustness of the proposed formulation.

Interpolation is defined as an estimate of a known value. Extensive interpolation is an attempt to determine the approximate value of an analytic function that is unknown or alternatively a complex function whose analytic equation cannot be obtained. You can combine the use of math and math to analyze the price of an item. This study describes Newton's method and the polynomial method. Therefore, interpolation with Newton's method has an error value that is smaller than the error value of the Lagrange interpolation. The results in this research case study when x = 2.5 using 10 data performed using Newton's polynomial interpolation method have a result of 1.70956 where this value is lower than the value of the analysis using the Lagrange method which is 3.2163.

By adopting the scaling functions of the B-spline wavelet on the interval (BSWI) as interpolation functions, a wavelet-based boundary element method (BEM) model is constructed to compute the band structures of two-dimensional photonic crystals (2D PCs), which are composed of square or triangular lattices with arbitrarily shaped inclusions. The boundary integral equations of both the matrix and inclusion are developed in a unit cell because of the structural periodicity. In order to make the curve boundary be compatible well, geometric boundaries are interpolated by employing second order BSWI scaling functions while arbitrary order scaling functions are used to approximate boundary variables. For any given angular frequency, an effective technique is provided to generate matrix values related to the boundary shape. Moreover, singular integral problem involved in the presented wavelet-based BEM is considered. Then, combining the Bloch theorem and the interface conditions, a linear eigenvalue equation related to the Bloch wave vector is obtained. Numerical examples are given to verify the performance of the wavelet-based BEM developed herein compared with the conventional BEM.

In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems., Comment: 55 pages

The radial basis function interpolation is an effective method for approximation on scattered data. However, this interpolation method suffers from the contradiction between the accuracy and the numerical stability. With considering the distribution of the interpolation points, a bounded random shape variation scheme for the radial basis function is developed to circumvent the problem of numerical stability. In this scheme, the shape variation is bounded by the value that is determined by the maximum distance and the minimal distance which are applied to describe the average density of the interpolation centers. Within this bound, the shape of the MQ is modified through a random scheme. With applying this bounded randomly variable shape scheme, the accuracy and the stability of the MQ interpolation are balanced. Comparisons on the accuracy and the condition number of the interpolation matrix between the constant shaped MQ interpolation and this bounded randomly variable shaped MQ interpolation have been made to verify the conclusion. Furthermore, this scheme is integrated in the dual reciprocity boundary element method in the analysis of three dimensional elastic problems. Results of the numerical examples demonstrated that the developed scheme improved the accuracy of the dual reciprocity boundary element method stably.

In the current work, the stable node-based smoothed radial point interpolation method (SNS-RPIM) was used to investigate the dynamic responses of magneto-electro-elastic (MEE) beams under hygrothermal environment with the generalized Newmark method. Based on the generalized smoothed Galerkin weak form and the gradient smoothing technique, the basic equations for hygro-thermo-magneto-electro-elastic (HTMEE) coupling problems were derived. The stable item of SNS-RPIM made the stiffness ‘close-to-exact’ and cured the ‘over-soft’ of the model. Through the numerical examples, the effect of empirical constants, temperature variation and moisture concentration on the elastic stiffness coefficients and generalized displacement was investigated. Besides, numerical results verified the accuracy, convergence and correctness of the present method compared with the traditional finite element method when analyzing the HTMEE coupling problems. Moreover, SNS-RPIM was very stable and accurate for extremely distributed meshes, which can have an excellent application for future studies of the coupled multi-physical problems.

The meshless numerical manifold method (MNMM) inherits two covers of the numerical manifold method. A mathematical cover is composed of nodes' influence domains and a physical cover consists of physical patches, which are produced through cutting mathematical cover by physical boundaries. Because two covers are adopted, MNMM can naturally and uniquely solve both the continuous and discontinuous problems under Galerkin's variational framework. However, Galerkin's meshless method needs background integration grids to realize solving, which often does not match the nodes' influence domains, so the accuracy of numerical integration is reduced. Consider that MNMM allows even distribution of nodes and the physical cover contains the characteristics of the boundary and nodes' influence domains, the study presents a new numerical integration strategy to ensure that the background integration grids match the nodes' influence domains. The method can be applied to continuous and discontinuous problems, and is proved to be equivalent to the influence domain integration. At the same time, a reasonable arrangement of mathematical nodes is made to assure the background integration grid that it is accordant and straightforward. In this way, the number of physical patches of each integral point is the same, which improves the accuracy of interpolation calculation. The effectiveness of the proposed method is verified by numerical examples of both continuous and discontinuous problems.

The third-order and fourth-order semi-discretization methods (3rd SDM and 4th SDM) are presented to predict milling stability based on precise integration. Firstly, the milling system is mathematically represented by a delay differential equation (DDE) and re-expressed in the state space. After dividing the tooth passing period into free and forced vibration time intervals, the forced vibration time interval is equally discretized into many small-time intervals. Then, the DDE is integrated on any small-time interval and solved in the framework of the semi-discretization algorithm. Subsequently, the 3rd SDM and 4th SDM are derived for predicting the stability limits in milling. In the 3rd SDM and 4th SDM, the third-order and fourth-order interpolation polynomials are used to approximate the delayed term, respectively. The periodic-coefficient matrix is approximated by linear interpolation polynomial. The convergence rates of the one degree of freedom (one-DOF) milling system and the milling system with multiple modes are conducted. The results indicate the convergence rates of the 3rd SDM and 4th SDM are higher than those of the benchmark methods. Besides, no matter for large or low radial immersion conditions, the stability lobe diagrams (SLDs) obtained by the proposed 3rd SDM and 4thSDM are closer to the reference than those obtained by the benchmark methods. The computational efficiency of the 3rd SDM and 4th SDM is also evaluated. Compared with the benchmark methods, the presented 3rd SDM and 4th SDM are superior in computational efficiency. Generally, the main advantage of the proposed methods is the improvement of computational efficiency. The proposed 3rd SDM and 4th SDM can use a smaller discrete number for achieving the desired accuracy to reduce the size of the state transition matrix and improve the computational efficiency.

This paper designs a fifth-order weighted compact nonlinear scheme (WCNS) on a five-point stencil to solve one- and two-dimensional Hamilton-Jacobi equations. The five-point WCNS is used to compute the left and right limits of first-order spatial derivatives of the HJ equations in the Lax-Friedrichs monotone numerical Hamiltonian. The WENO-Z type interpolation for cell-edge values of the solutions is used to suppress numerical oscillations which may appear near discontinuities. Five- and seven-point WENO-Z schemes for Hamilton-Jacobi equations are also designed for comparisons. The performance of the WCNS and the two WENO-Z schemes is demonstrated by several numerical examples in one-dimensional and two-dimensional cases.

Groundwater flow problems are generally solved using analytical or numerical methods. Though analytical solutions are exact and preferable, they are not available for complex field problems. Hence numerical methods such as Finite Element and Finite Difference methods are used to solve complex groundwater problems. These conventional mesh/ grid-based numerical methods need construction of a detailed mesh/ grid. On the other hand, the meshless approach creates a system of algebraic equations on a collection of distributed nodes in the problem area and the boundary. As a result, it is easy to incorporate any modifications to the model at a later time by simply adding nodes to the domain. In this study a weak form meshless method known as local radial point interpolation method (LRPIM) which uses radial basis functions for approximation or interpolation is developed to solve the groundwater flow problems in a confined aquifer. The results obtained from the LRPIM model has been compared with other numerical methods for benchmark and real field problems, and are found to be satisfactory. Implementation of the essential boundary conditions was relatively easier in LRPIM and gave good accuracy for the problems considered. LRPIM can potentially be used as an alternative to the other conventional methods, especially where the domain boundary is irregular or varying with time.

Multidimensional interpolation is commonly encountered in numerical methods such as the Finite Element Method (FEM) the Finite Volume Method (FVM) used for solving partial differential equations. It is a general practice in numerical methods to discretize a two (three) dimensional domain into large number of small areas (volumes) known as elements in FEM volumes in FVM. These methods assume a functional form for variable within each sub domain based on the nodal values solve for the variables at the nodes. Development of multidimensional interpolation has also applications in computer graphics where surfaces are represented by Bezier curves NURBS (Non-uniform rational B-splines). The present chapter intends to introduce the readers to multi-dimensional interpolation, with simple examples, to be made use of in later chapters.

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In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

An inverse meshfree approach is presented for thermo-mechanical load identification of a fractured body. To this end, the crack propagation path is used as the input of the inverse analysis. This study can be used for finding the thermo-mechanical loads that have led to the fracture of a body, and is applicable to situations where the direct measurement of the boundary loadings on some part of a body is not possible. This highly ill-posed problem is solved using the meshfree radial point interpolation method (RPIM) with a global optimization approach. The inspected crack path is compared to the one obtained numerically and their difference is set as a cost function. By finding the minimum of this cost function, the actual loading conditions that have led to the fracture of the body is obtained. The density clustering method is used as the global optimization method in which the damped Gauss-Newton technique is used as the local search algorithm. To demonstrate the performance of the presented meshfree inverse approach, two examples are provided.

Regardless of the bone remodeling type (i.e. targeted or non targeted), disturbances in this process can have serious repercussions in bone diseases’ progression. Therefore, research in this field has been continuous in recent decades, producing clinically relevant results. Regarding non targeted remodeling, distinct computational models were developed to describe the process. However, they only offer a time analysis, lacking mathematical models able to simulate the process in space. Thus, this work combines, for the first time, advanced numerical discretization techniques with non targeted remodeling models. The mathematical formulation used is based on Komarova’s model, correlating the remodeling activity of osteoclasts and osteoblasts with their autocrine and paracrine signaling pathways. For the numerical example, a two-dimensional bone patch was studied, allowing a spatio-temporal analysis of bone mass, cell density of osteoclasts and osteoblasts during an event of bone remodeling. A biological stimulus was applied to trigger remodeling and the obtained bone mass variation is consistent with the applied stimulus. Also, results are supported by the histomorphometric data available in the literature. The numerical methods used in this work are the Finite Element Method (FEM) and the Radial Point Interpolation Method (RPIM). The inclusion of the meshless method RPIM conveys a great advantage for future work since the model will be combined with medical images obtained with CT scans and MRI.

We propose an alternative generator architecture for generative adversarial networks, borrowing from style transfer literature. The new architecture leads to an automatically learned, unsupervised separation of high-level attributes (e.g., pose and identity when trained on human faces) and stochastic variation in the generated images (e.g., freckles, hair), and it enables intuitive, scale-specific control of the synthesis. The new generator improves the state-of-the-art in terms of traditional distribution quality metrics, leads to demonstrably better interpolation properties, and also better disentangles the latent factors of variation. To quantify interpolation quality and disentanglement, we propose two new, automated methods that are applicable to any generator architecture. Finally, we introduce a new, highly varied and high-quality dataset of human faces., CVPR 2019 final version

This paper describes a new efficient approach based on the concept of reduced basis for large deformation analysis. The domain problem is discretized using the meshfree particle radial point interpolation method (RPIM), which inherently possesses the Kronecker’s delta property. Meshless numerical integration is evaluated by the Cartesian transformation method (CTM), which enhances the performance of the RPIM. In addition, we also introduce a new approach to further improve the capability of the current CTM in evaluation of numerical integration for problems with complex geometries, i.e., by incorporation of the non-uniform rational B-splines function (NURBS) into the CTM. The emphasis of the paper is on the nonlinear nature of the large deformation problems, which are often solved by an iterative scheme. Conventional Newton–Raphson technique usually requires high cost due to the fact that several load steps are usually performed, and multiple iterations are needed in each load step. This low computational efficiency can be overcome, as proposed in this work, by using the so-called combined approximation, which approximates the full-size solution by a set of reduced bases. In other words, reduction of the problem size can be obtained, leading to reduction of the computational time, while accuracy is almost preserved.

An algorithm for the fast computation of multi-static scattering patterns, produced by low-contrast inhomogeneous volumetric objects, which satisfy the Born approximation assumption, is presented. The algorithm relies on hierarchical interpolation and aggregation of optimally sampled phase-compensated partial contributions to the scattered field integral. The Born approximation enables the efficient simultaneous computation for ranges of incident and observation angles, as well as excitation frequencies. This leads to a reduction in the asymptotic computational complexity by several polynomial orders of the problem’s acoustical size. The algorithm’s error controllability and favorable computation time are demonstrated via representative examples.

Model order reduction of electronic devices and on-chip interconnects plays an important role in determining the performance of very large scale IC (Integrated Circuit) designs. The ever shrinking process technology continues to increase complexity of these systems with the current Intel device technology node standing at 10 nm. In the analysis of high-speed IC interconnects, the reduced model is often required to approximate the original system in the desired frequency range. In this paper, an efficient Statistically Inspired Multi-shift Arnoldi (SIMA) Method is proposed, which dynamically selects normally distributed shifts to determine the projection matrix. The SIMA is implemented by developing an iterative algorithm for approximation of large-scale systems as an eigenvalue problem using statistically generated interpolation points. The motivation of selecting these points weighted with the Gaussian kernel is derived from the fact that there exist many natural phenomenon, which follow normal distribution. Simulation results have shown better accuracy for the proposed method as compared to other existing model order reduction techniques.

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method.

This paper is concerned with conforming finite element discretizations for quasilinear elliptic problems in divergence form, of a class that generalizes the p-Laplace equation and allows to show existence and uniqueness of the continuous and discrete problems. We derive discretization error estimates under general regularity assumptions for the solution and using high order polynomial spaces, resulting in convergence rates that are then verified numerically. A key idea of this error analysis is to consider carefully the relation between the natural W 1 , p -seminorm and a specific quasinorm introduced in the literature. In particular, we are able to derive interpolation estimates in this quasinorm from known interpolation estimates in the W 1 , p -seminorm. We also give a simplified proof of known near-best approximation results in W 1 , p -seminorm starting from the corresponding result in the mentioned quasinorm.

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C 1 -rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form p i ( θ ) / q i ( θ ) , where p i ( θ ) and q i ( θ ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C 3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C 1 -RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.