Strong convergence analysis of iterative solvers for random operator equations

Block floating point (BFP) arithmetic is currently seeing a resurgence in interest because it requires less power, less chip area, and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed-and progressive-precision multigrid methods, enabling the solution of linear elliptic partial differential equations (PDEs) in energy-and hardware-efficient integer arithmetic. While most existing applications of BFP arithmetic tend to use small block sizes, the block size here is chosen to be maximal such that matrices and vectors share a single exponent for all entries. This is sometimes also referred to as a scaled fixed-point format. We provide algorithms for BLAS-like routines for BFP arithmetic that ensure exact vector-vector and matrix-vector computations up to a specified precision. Using these algorithms, we study the asymptotic precision requirements to achieve discretization-error-accuracy. We demonstrate that some computations can be performed using as little as 4-bit integers, while the number of bits required to attain a certain target accuracy is similar to that of standard floating point arithmetic. Finally, we present a heuristic for full multigrid in BFP arithmetic based on saturation and truncation that still achieves discretization-error-accuracy without the need for expensive normalization steps of intermediate results.

This is the first in a series of papers about asynchronous parallel algorithms based on domain decomposition method (DDM) for solving partial differential equations on MIMD machines, especially on CM-5. These algorithms are all called Schwarz-Chaotic Over Relaxation (S-COR), the generalization of the Schwarz Alternating Method. In this paper, the linear elliptic partial differential equations is discussed. Numerical results are presented to demonstrate the effectiveness of these decomposition algorithms on the Thinking Machines CM-5 parallel supercomputer.

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.

An irregular grid for the numerical solution of linear elliptic partial differential equations

Solution of high order compact discretized 3D elliptic partial differential equations by an accelerated multigrid method

The Multiscale Spectral Generalized Finite Element Method (MS-GFEM) was developed in recent work by Babuska and Lipton. The method uses optimal local shape functions, optimal in the sense of the Kolmogorov n-width, to approximate solutions to a second order linear elliptic partial differential equation with L-infinity coefficients. In this dissertation an implementation of MS-GFEM over a two subdomain partition of unity is outlined and several numerical experiments are presented. The method is applied to compute local fields inside high contrast particle suspensions. The method's performance is evaluated for various examples with different contrasts between reinforcement particles and matrix material. The numerical experiments are shown to agree with a new theoretical estimate that shows the convergence rate is independent of the elastic properties of particles and matrix materials. A new domain decomposition method based on MS-GFEM is presented. Numerical computations using this iterative method are discussed and the theoretical convergence rate is provided. It is shown that the convergence rate is given by the same near-exponential bound given for MS-GFEM. A systematic method for identifying the worst case load amongst all boundary loads of a fixed energy is introduced. Here the worst case load delivers the largest fraction of input energy into a prescribed subdomain of interest. This leads to an eigenvalue problem, for which the largest eigenvalue is the maximum fraction of energy which concentrates in the subdomain. The associated eigenfunctions are the worst case solutions. These eigenfunctions are related back to the MS-GFEM shape functions and numerical results are presented for several different geometries.

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.

We derive a method of obtaining approximate numerical solution of linear variable-coefficient partial differential equations (PDEs) with two independent variables from Kida's optimum approximation theory. It is shown that a certain generalized filter bank implements linear PDEs. By applying generalized discrete orthogonality of Kida's optimum approximation to this filter bank, we prove that our approximate solution satisfies a given linear PDEs and the corresponding initial or boundary conditions at all given sample points, simultaneously.

This chapter is devoted to the theory of linear elliptic partial differential equations and the related problems of potential theory. The basic concept of the Green’s function and the source solution are introduced and explored. This is followed by a detailed analysis of the Helmholtz equation in one, two, and three dimensions with applications to the diffraction problem for monochromatic waves. The inhomogeneous media case sets the stage for the Helmholtz equation with a variable coefficient and an analysis of waves in waveguides. The latter can be reduced to the celebrated Sturm–Liouville problem, and we study properties of its eigenvalues and eigenfunctions.KeywordsFundamental SolutionPoisson EquationRadiation ConditionHelmholtz EquationComplex AmplitudeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, the solution of classical and generalized linear and nonlinear elliptic partial differential equations of the second order with non-local boundary conditions is studied. The basic concepts and theorems related to the study are included, which includes the main result of this research. Many basic definitions, theorems and observations about Sobolev spaces H^k (Ω) and H_0^k (Ω) are discus-sed. The existence and uniqueness of the solution of the Poisson equation with non-local boundary conditions are considered. The existence and uniqueness of the solution of a second-order quasi-linear elliptic differential equation with non-linear integral boun-dary condition are also discussed. The argument for proving the previous problem is based on Banach's fixed point theorem in the complete metric space, the maxima and minima principle, and the comparison principle. Key word: Sobolev Spaces, Poisson Equation, Quesilinear Ellipitic Differential Equation, Boundary, and Banach's fixed Point Theorem.

Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H2.

Fundamental solutions of degenerate or singular elliptic equations

A fourth-order linear elliptic partial differential equation describing the displacements of a transversely isotropic linear elastic medium is considered. Its symmetries and the symmetries of an inhomogeneous equation with a delta function on the right-hand side are found. The latter symmetries are used to construct an invariant fundamental solution of the original equation in terms of elementary functions.