Second Derivative Estimates Research Articles

Simultaneous estimation of multiple order phase derivatives using deep learning method in digital holographic interferometry

For non-contact deformation testing, digital holographic interferometry is a prominent optical technique where the first and second order interference phase derivatives directly embed information about the strain and curvature distributions of a deformed object. Hence, reliable extraction of multiple order phase derivatives is of great practical significance; however, this problem is marred by several challenges such as the need of multiple differentiation operations, complex shearing operations and performance degradation due to noise. In this paper, we introduce a deep learning approach for the direct and simultaneous estimation of first and second order phase derivatives in digital holographic interferometry. Our method's performance is demonstrated via rigorous numerical simulations exhibiting wide range of additive white Gaussian noise and speckle noise. Moreover, we substantiate the practical efficacy of our proposed method for processing deformation fringes acquired via digital holographic interferometry.

Multiquadric based RBF-HFD approximation formulas and convergence properties

RBF-FD formulas have received special attention due to their superior accuracy, well-conditioning, and ease of implementation in solving partial differential equations. To further improve the accuracy of RBF-FD formulas, without increasing the stencil size RBF-HFD formulas are proposed in the literature. In this article, we obtain leading/higher order analytical expressions for weights and local truncation errors associated with various such formulas using the multiquadric radial function. We also study the convergence properties of these formulas. Symbolic computation in Mathematica is employed for analytically solving linear systems for the unknown weights in the RBF-HFD formulas. Symmetry properties of central difference approximations help reduce the number of unknown weights and thereby the size of the linear system. We compute the analytical expressions of weights and local truncation errors for first and second-derivative approximation formulas till tenth order; and for two-dimensional Laplacian operator approximation formulas till sixth order. The obtained formulas are free from ill-conditioning. It is observed that the RBF-HFD formula weights converge to corresponding order classical compact FD scheme weights when the shape parameter (ϵ) tends to zero. The local truncation error has global minimum for an optimal value of the shape parameter. Further, the optimal shape parameter is independent of nodal distance but depends on choice of test function and its derivative values at a reference node.

Boundary-optimized summation-by-parts operators for finite difference approximations of second derivatives with variable coefficients

Boundary-optimized summation-by-parts (SBP) finite difference operators for second derivatives with variable coefficients are presented. The operators achieve increased accuracy by utilizing non-equispaced grid points close to the boundaries of the grid. Using the optimized operators we formulate SBP schemes for the acoustic and elastic operators defined directly on curvilinear multiblock domains. Numerical studies of the acoustic and elastic wave equations demonstrate that, compared to traditional SBP difference operators, the new operators provide increased accuracy for surface waves as well as block interfaces in multiblock grids. For instance, simulations of Rayleigh waves demonstrate that the boundary-optimized operators more than halve the runtime required for a given error tolerance.

Implicit summation by parts operators for finite difference approximations of first and second derivatives

Implicit finite difference approximations are derived for both the first and second derivatives. The boundary closures are based on the banded-norm summation-by-parts framework and the boundary conditions are imposed using a weak (penalty) enforcement. Up to 8th order global convergence is achieved. The finite difference approximations lead to implicit ODE systems. Spectral resolution characteristics are achieved by proper tuning of the internal difference stencils. The accuracy and stability properties are demonstrated for linear hyperbolic problems in 1D and the 2D compressible Euler equations.

On the σ2-Nirenberg problem on [formula omitted

We establish theorems on the existence and compactness of solutions to the σ2-Nirenberg problem on the standard sphere S2. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ2-Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ2-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L∞ a priori estimates for solutions of the σ2-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ2-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.

Implicit Summation by Parts Operators for Finite Difference Approximations of First and Second Derivatives

Implicit Summation by Parts Operators for Finite Difference Approximations of First and Second Derivatives

Convergent Finite Difference Methods for Fully Nonlinear Elliptic Equations in Three Dimensions

We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully placed along the boundary at high resolution. We introduce and analyze a least-squares approach to building consistent, monotone approximations of second directional derivatives on these grids. We then show how to efficiently approximate functions of the eigenvalues of the Hessian through a multi-level discretization of orthogonal coordinate frames in \(\mathbb {R}^3\). The resulting schemes are monotone and fit within many recently developed convergence frameworks for fully nonlinear elliptic equations including non-classical Dirichlet problems that admit discontinuous solutions, Monge–Ampère type equations in optimal transport, and eigenvalue problems involving nonlinear elliptic operators. Computational examples demonstrate the success of this method on a wide range of challenging examples.

A general SPH framework for transient seepage flows through unsaturated porous media considering anisotropic diffusion

The Smoothed Particle Hydrodynamics (SPH) method has become a popular numerical tool to solve many challenging engineering applications. However, its application to solving anisotropic transient unsaturated–saturated seepage flow problems is still absent in the literature. Among several challenges involved in solving this important problem, the lack of a robust and general SPH formulation to obtain accurate approximations of second derivatives on disordered particle system is the key obstacle preventing this to happen. In this paper, we first propose a general SPH formulation for the second derivatives, capable of considering anisotropic diffusions, and demonstrate that the new formulation outperforms existing SPH formulations for the second derivatives and achieves high accuracy on the highly disordered particle systems. Subsequently, we utilise this newly developed SPH formulation to tackle the transient seepage flows through porous media for the first time, considering anisotropic flows and complete time-dependent transition from unsaturated to saturated states and vice-versa. This naturally leads to a general SPH framework for solving transient seepage problems in unsaturated/saturated porous media using one set of Lagrangian particles (or Lagrangian discretisation). Robust boundary conditions and their implementations for solving general seepage problems in SPH are also proposed, enabling the proposed SPH model to automatically capture seepage surface (including the free surface of unconfined seepage) without any difficulty or ad hoc treatment. Moreover, a simple method to generate a random particle system suitable for SPH simulations on large scale problems using the Voronoi tessellation technique is proposed to study transient seepage problems for the first time. Several verification examples are presented to demonstrate the capabilities of the proposed SPH framework in solving the complex seepage flow problems, including those on multi-resolution particle configurations. Excellent agreements with a wide range of numerical and theoretical solutions are achieved, suggesting that the newly proposed SPH model can be applied to solve complex transient seepage flow problems. This marks an important milestone for the future development of SPH to solve fully coupled flow deformation problems involved large deformation and failure of porous media, which are challenging for the most state-of-the-art mesh-based numerical methods.

Second order estimates for fully nonlinear elliptic equations with gradient terms on closed Riemannian manifolds

Establishing a priori estimates for second derivatives is a key ingredient in the study of fully nonlinear elliptic equations,which is the focus of this paper. We consider a class of second order fully nonlinear equations on closed Riemannian manifolds. In terms of the tangent cone at infinity to the level sets of an associated function which may depend on the solution and its gradient, we introduce a condition to derive second order estimates for solutions of the equation.

Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

In the Lagrangian meshless (particle) methods, such as the smoothed particle hydrodynamics (SPH), moving particle semi-implicit (MPS) method and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R), the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities (such as the viscous stresses). In some meshless applications, the Laplacians are also needed as stabilisation operators to enhance the pressure calculation. The particles in the Lagrangian methods move following the material velocity, yielding a disordered (random) particle distribution even though they may be distributed uniformly in the initial state. Different schemes have been developed for a direct estimation of second derivatives using finite difference, kernel integrations and weighted/moving least square method. Some of the schemes suffer from a poor convergent rate. Some have a better convergent rate but require inversions of high order matrices, yielding high computational costs. This paper presents a quadric semi-analytical finite-difference interpolation (QSFDI) scheme, which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations. The convergence, accuracy and robustness of the present schemes are compared with the existing schemes. It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures, particularly for estimating the Laplacian of given functions.

On the difference method for approximation of second order derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

We present and justify finite difference schemes with the 14-point averaging operator for the second derivatives of the solution of the Dirichlet problem for Laplace?s equations on a rectangular parallelepiped. The boundary functions ?j on the faces ?j,j = 1,2,..., 6 of the parallelepiped are supposed to have fifth derivatives belonging to the H?lder classes C5?, 0 < ? < 1. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. It is proved that the proposed difference schemes for the approximation of the pure and mixed second derivatives converge uniformly with order O(h3+?), 0 < ? < 1 and O(h3), respectively. Numerical experiments are illustrated to support the theoretical results.

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

An Improved CSPM Approach for Accurate Second-Derivative Approximations with SPH

We compare several approximations for second derivatives with Smoothed Particle Hydrodynamics (SPH). A first-order consistent approximation, derived from the zeroth-order consistent Corrective Smoothed Particle Method (CSPM), is proposed. The accuracy of the new method (ICSPM) is similar to that of the Finite Particle Method (FPM) and Modified Smoothed Particle Hydrodynamics (MSPH), but it is computationally less expensive. We demonstrate the accuracy of our method by studying heat conduction in a slab with discontinuous conductivity coefficients. We use both uniformly and pseudo-randomly distributed particles.

On the Neumann Problem for Monge-Ampére Type Equations

AbstractIn this paper, we study the global regularity for regular Monge-Ampère type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. The techniques build upon the delicate and intricate treatment of the standard Monge-Ampère case by Lions, Trudinger, and Urbas in 1986 and the recent barrier constructions and second derivative bounds by Jiang, Trudinger, and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

Estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications

In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDEs around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform exterior ball condition. We prove rather sharp continuity results for these shape derivatives with respect to Sobolev norms of the boundary-traces of the displacements. With respect to previous results of this kind, the approach is quite different and is valid in any dimension N ≥ 2 . It is based on sharp regularity results for Poisson-type equations in such nonsmooth domains. We also enlarge the class of functionals and PDEs for which these estimates apply. Applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement.

Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds

We derive a priori estimates for second order derivatives of solutions to a wide class of fully nonlinear elliptic equations on Riemannian manifolds. There had been significant work in this direction, especially in connection with important geometric problems and other applications, but one had to make use of the special structures or needed extra assumptions which are more technical in nature to overcome various difficulties. In this paper we are able to remove most of the technical assumptions and derive the estimates under conditions which are close to optimal. These estimates enable one to prove existence results which are new even for bounded domains in Euclidean space.

Generalized Summation-by-Parts Operators for the Second Derivative

The generalization of summation-by-parts operators for the first derivative of Del Rey Fernández, Boom, and Zingg [J. Comput. Phys., 266 (2014), pp. 214--239] is extended to approximations of second derivatives with a constant or variable coefficient. This enables the construction of second-derivative operators with one or more of the following characteristics: (i) nonrepeating interior point operators, (ii) nonuniform nodal distributions, and (iii) exclusion of one or both boundary nodes. Definitions are proposed that give rise to generalized summation-by-parts operators that result in consistent, conservative, and stable discretizations of partial differential equations with or without mixed derivatives. It is proven that approximations to the second derivative with a variable coefficient can be constructed using the constituent matrices of the constant-coefficient operator. Moreover, for operators with a repeating interior point operator, a decomposition is proposed that makes the application of such operators particularly straightforward. A number of novel operators are constructed, including operators on the Chebyshev--Gauss quadrature nodes and operators that have a repeating interior point operator but nonuniform nodal spacing near boundaries. The various operators are compared to the application of the first-derivative operator twice in the context of the linear convection-diffusion equation with a variable coefficient.

Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations I: Bellman Equations with Constant Coefficients

We consider a stochastic optimal control problem in a domain, in which the diffusion coefficients, drift coefficients, and discount factor are independent of the spatial variables. Under appropriate assumptions, for k=0,1, when the terminal and running payoffs are globally $C^{k,1}$, we establish the interior $C^{k,1}$-smoothness of the value function, which yields the existence and uniqueness of the locally $C^{k,1}$-solution to the associated Dirichlet problem for the possibly degenerate elliptic Bellman equation with constant coefficients. Estimates for first and second derivatives of the solution are also obtained.

Numerical solution of a diffusion problem by exponentially fitted finite difference methods.

This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.

NDL-v2.0: A new version of the numerical differentiation library for parallel architectures

NDL-v2.0: A new version of the numerical differentiation library for parallel architectures