Discrete Laplace cycles of period four

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.     

Laplace approximation of transition densities posed as Brownian expectations

We construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron–Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation.     

A Laplace Operator on Semi-Discrete Surfaces

This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace–Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme.     

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.     

Quadrature formulas for the Laplace and Mellin transforms

A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the integral Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfies given conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. The use of asymptotic formulae yields an algorithm to compute the discrete Laplace transform by using only exponentials.     

Trefftz energy method for solving the Cauchy problem of the Laplace equation

The inverse Cauchy problem of Laplace equation is hard to solve numerically, since it is highly ill-posed in the Hadamard sense. With this in mind, we propose a natural regularization technique to overcome the difficulty. In the linear space of the Trefftz bases for solving the Laplace equation, we introduce a novel concept to construct the Trefftz energy bases used in the numerical solution for the inverse Cauchy problem of the Laplace equation in arbitrary star plane domain. The Trefftz energy bases not only satisfy the Laplace equation but also preserve the energy, whose performance is better than the original Trefftz bases. We test the new method by two numerical examples.     

Covariant Laplace Transforms

Among the solutions of the wave equation we make a distinction between the homogeneous and nonhomogeneous waves. For the first ones we use a two-dimensional-Laplace transform and for the second ones a one-dimensional-Laplace transform with respect to the proper time. We discuss the behaviour of these waves and of their Laplace transform under the Lorentz group.     

Laplace invariants for a fourth-order equation with two independent variables

We construct the Laplace invariants for an equation with the leading partial derivative. We write defining equations in terms of the Laplace invariants. We obtain classes of equations admitting four-dimensional Lie algebras.     

Inhomogeneous infinity Laplace equation

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously.     

Preconditioner for a discrete Laplace operator on a condensed grid

In this paper, it is proved that the discrete Laplace operator approximating a Dirichlet boundary value problem for a Poisson equation by a finite element method with piecewise-linear functions on an evenly condensed grid that is topologically equivalent to a rectangular grid (i.e., obtained by shifting the rectangular grid nodes) is equivalent, in the range, to the operator of a 5-point difference scheme on a uniform grid.     

三重介质油藏模型及其Laplace空间解

本文建立了裂缝、洞与井筒连通的三重介质油藏的试井解释模型,利用Laplace变换和构造通解法,研究了这一类线性偏微分方程的边值问题,得到了其Laplace空间解,为裂缝、洞与井筒连通油藏的试井分析提供了坚实的理论基础。     

Boundary Integral Operator for the Fractional Laplace Equation in a Bounded Lipschitz Domain

We study the boundary integral operator induced from fractional Laplace equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace equation.     

A regularization method for approximating the inverse Laplace transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum     

Singular perturbations of Laplace operator and their resolvents

An inner closed (without boundary) smooth manifold of a lower dimension is cut from a multidimensional ball. In this region, invertible restrictions of the Laplace operator are well defined. In particular, the well-posed non-smooth Bitsadze-Samarski? problem for the Laplace equation is defined. Moreover, we obtain M.G. Krein's formula for the trace of the difference of resolvents of the studied operators. We prove assertions on the spectrum of the non-smooth Bitsadze-Samarski? problem.     

On the Laplace invariants of a fourth-order equation

We construct the Laplace invariants of the Bianchi equation with four independent variables and obtain determining equations expressed in terms of these invariants.     

A Laplace transform certified reduced basis method; application to the heat equation and wave equation

We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.     

Solution of boundary value problems for the Laplace equation in a ball bounded by a multilayer film

We derive boundary conditions on multilayer films bounding a ball and consisting of infinitely thin strongly and weakly permeable layers and obtain formulas expressing the solutions of boundary value problems for the Laplace equation in a ball bounded with two-layer films by single quadratures via the solutions of the classical Dirichlet and Neumann problems for the Laplace equation in the ball (without the films).     

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.     

On a generalization of the Neumann problem for the Laplace equation

We investigate solvability of a fractional analogue of the Neumann problem for the Laplace equation. As a boundary operator we consider operators of fractional differentiation in the Hadamard sense. The problem is solved by reduction to an integral Fredholm equation. A theorem on existence and uniqueness of the problem solution is proved.