Structure-preserving space–time POD reduction method for solving variable-coefficient parabolic equations

Space–time localized radial basis function collocation method for solving parabolic and hyperbolic equations

A new space–time finite element method for the solution of parabolic partial differential equations is introduced. In a mesh and degree-dependent norm, it is first shown that the discrete bilinear form for the space–time problem is both coercive and continuous, yielding existence and uniqueness of the associated discrete solution. In a second step, error estimates in this mesh-dependent norm are derived. In particular, we show that combining low-order elements for the space variable together with an hp-approximation of the problem with respect to the temporal variable allows us to decrease the optimal convergence rates for the approximation of elliptic problems only by a logarithmic factor. For simultaneous space–time hp-discretization in both, the spatial as well as the temporal variable, overall exponential convergence in mesh-degree dependent norms on the space–time cylinder is proved, under analytic regularity assumptions on the solution with respect to the spatial variable. Numerical results for linear model problems confirming exponential convergence are presented.

Various boundary value conditions have been endowed for the fourth‐order differential equations. In the current work, a class of the fourth‐order parabolic equations with the third Neumann boundary conditions is concerned, where the values of the second and third spatial derivatives of the unknown function are given at the boundary. A novel average is defined to get the compact approximation near the boundary. Then a compact difference scheme is derived by using the weighted average and the method of order reduction. Due to the special and highly accurate discretization at the boundary, the related terms have to be handled skillfully during the analysis. By the energy method, the unique solvability, unconditional stability, and convergence of the derived compact difference scheme are strictly proved. The analytical difficulties caused by the boundary approximation are successfully overcome. As far as we know, this is the first time that the global pointwise fourth‐order convergence in space of the difference approach for this problem is achieved. In addition, the generalization to solve the case with a space‐dependent reaction coefficient is discussed. Finally, two numerical examples are computed to verify the accuracy of proposed numerical schemes.

Windowed space–time least-squares Petrov–Galerkin model order reduction for nonlinear dynamical systems

SummaryThis paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.

Kansa method for the solution of a parabolic equation with an unknown spacewise-dependent coefficient subject to an extra measurement

Space–time fractional stochastic partial differential equations

We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

A fast attribute reduction method for large formal decision contexts

Simultaneous space–time adaptive wavelet solution of nonlinear parabolic differential equations

We review how Geroch’s reduction method is extended from Ricci-flat to Einstein spacetimes. The Ehlers–Geroch SL(2,R) group is still present in the three-dimensional sigma-model that captures the dynamics, but only a subgroup of it is solution-generating. Holography provides an alternative three-dimensional perspective to integrability properties of Einstein’s equations in asymptotically anti-de Sitter spacetimes. These properties emerge as conditions on the boundary data (metric and energy–momentum tensor) ensuring that the hydrodynamic derivative expansion be resummed into an exact four-dimensional Einstein geometry. The conditions at hand are in- variant under a set of transformations dubbed holographic U-duality group. The latter fills the gap left by the Ehlers–Geroch group in Einstein spaces, and allows for solution-generating maps mixing e.g. the mass and the nut charge.

We propose a scheme for the solution of a mixed problem for a parabolic differential equation with coefficients that are generalized derivatives of functions of bounded variation. We seek the solution of this problem by the method of reduction. According to this method, the solution of the proposed problem is reduced to the solution of two problems: (i) a quasistationary boundary-value problem with input boundary conditions and (ii) a mixed problem with trivial boundary conditions. The first of these problems is solved by introducing the quasiderivative. For the solution of the second problem, we use the Fourier method and the expansion in eigenfunctions of a certain boundary-value problem for a quasidifferential equation of the second-order. The obtained results can be used, in particular, for the investigation of the processes of heat transfer in multilayer plates, hollow cylinders, and spheres.

Subject of this work is the development of concepts for the efficient numerical solution of optimization problems governed by parabolic partial differential equations. Optimization problems of this type arise for instance from the optimal control of physical processes and from the identification of unknown parameters in mathematical models describing such processes. For their numerical treatment, these generically infinite-dimensional optimal control and parameter estimation problems have to be discretized by finite-dimensional approximations. This discretization process causes errors which have to be taken into account to obtain reliable numerical results. Focal point of the thesis at hand is the assessment of these discretization errors by a priori and especially a posteriori error analyses. Thereby, we consider Galerkin finite element discretizations of the state and the control variable in space and time. For the a priori analysis, we concentrate on the case of linear-quadratic optimal control problems. In this configuration, we prove error estimates of optimal order with respect to all involved discretization parameters. The a posteriori error estimation techniques are developed for a general class of nonlinear optimization problems. They provide separated and evaluable estimates for the errors caused by the different parts of the discretization and yield refinement indicators, which can be used for the automatic choice of suitable discrete spaces. The usage of adaptive refinement techniques within a strategy for balancing the several error contributions leads to efficient discretizations for the continuous problems. The presented results and developed concepts are substantiated by various numerical examples including large scale optimization problems motivated by concrete applications from engineering and chemistry.

We are interested in the optimal control problem of a parabolic equation with no state constraints, where the quadratic cost functional involves a final observation and the control variable is a Dirichlet boundary condition. Practical considerations lead us to use Dirichlet controls with no more regularity than square integrability, which arise some technical difficulties in the mathematical analysis. After setting the state equation and the related adjoint equation we fit the two coupled equations into a mixed space–time variational framework. The resulting saddle-point problem turns out to be well posed. We then use a Robin penalization on the Dirichlet control which enables us to re-transcript the mixed problem in a form better suited to numerical computations. We analyze and establish the convergence when the penalty parameter tends to zero, first without additional smoothness assumptions on the optimal control and then for smooth controls.

Parallel two-level space–time hybrid Schwarz method for solving linear parabolic equations