The quantum query complexity of elliptic PDE

usual boundary conditions, and constraints u~U(t, z)~Em, where the control space U = U(t, z) is a given subset of Era. Here z =(z 1 . . . . . : ) are state variables and u =(u 1 .. . . , u m) control variables. As stated by P. K. RASHEVSKY [6], very general partial differential equations and systems can be written locally in the form (2). For instance, second order partial differential equations such as z~,+z~s= 1 u) can be immediately written in the form f ( t , s, z 1, zlt, z~,

A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest G(u ⋆) with a linear goal functional G and the solution u ⋆ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

Chapter Three - The BEM for Potential Problems in Two Dimensions

Support-Operator Finite-Difference Algorithms for General Elliptic Problems

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

This paper presents the applicati on of a general purpose partial differential equation solver, called PDEase2D and the associated soil property functions to the analysis of saturated-unsaturated seepage problem. Examples of steady state and transient seepage through an earth dam are used to illustrate the use of PDEase2D in solving saturated-unsaturated seepage problems.

Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?

In the paper, we investigate the basic transmission problems arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body ( Ω + ) is embedded in an unbounded fluid domain ( Ω − ). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω + , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω − , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.

Complexity and tractability for a class of elliptic partial integro-differential equations

Small perturbations in the type of boundary conditions for an elliptic operator

We study the approximation of the integration of multivariate functions classes in the quantum model of computation. We first obtain a lower bound of the n-th minimal query error for integration on anisotropic Sobolev-Slobodezkii classes. Then combining our previous results we determine the optimal bound of n-th minimal query error for anisotropic Hölder-Nikolskii class and Sobolev class. The results show that for these two type of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.