A residual based error estimate for Leja interpolation of matrix functions

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

Quasi-static and dynamic response of functionally graded viscoelastic plates

By using a non-linear 3-D elastic waves real-time expert system, the new theory of “ Non-linear Real-Time Expert Seismology ” is investigated, for the exploration of the on-shore and off-shore petroleum and gas reserves all over the world. Such a highly innovative and groundbreaking technology is working under Real Time Logic for searching the on-shore and off-shore hydrocarbon reserves developed on the continental crust and in deeper water ranging from 300 to 3000 m, or even deeper. Consequently, this real-time expert system, will be the best device for the exploration of the continental margin areas (shelf, slope and rise) and the very deep waters, as well. The proposed modern technology can be used at any depth of seas and oceans all over the world and for any depth in the subsurface of existing oil reserves. Beyond the above, the various mechanical properties of rock regulating the wave propagation phenomenon appear as spatially varying coefficients in a system of time-dependent hyperbolic partial differential equations. The propagation of the seismic waves through the earth subsurface is described by the wave equation, which is then reduced to a Helmholz differential equation. Furthermore, the Helmholtz differential equation is numerically evaluated by using the Singular Integral Operators Method (S.I.O.M.). Several properties are further analyzed and investigated for the wave equation. Finally, an application is proposed for the determination of the seismic field radiated from a pulsating sphere into an infinite homogeneous medium. Thus, by using the S.I.O.M., then the acoustic pressure radiated from the above pulsating sphere is determined. Key word s : Non-linear real-time expert seismology; Singular integral operators method (S.I.O.M.); Time-dependent hyperbolic partial differential equations; Oil and gas reserves; Petroleum reservoir engineering; Helmholtz differential equation; Real-time expert system; Wave equation

In this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations. With those, well-posedness can be proven for a general initial-boundary value problem. The procedure is exemplified on the linearized Navier--Stokes equations in two and three space dimensions on a general domain.

: This report documents a program for the numerical solution of large sparse systems of algebraic and implicitly defined stiff differential equations. The principal use is intended to be the solution of differential equations arising from time dependent partial differential equations when the finite element method is used to discretize the space domain. The use of compact matrix storage techniques and iteration for the solution of the quasi-Newton iterates in Gear's method makes the program extremely efficient both in terms of storage requirements and execution times. (Author)

The local discontinuous Galerkin method for convection-diffusion-fractional anti-diffusion equations

Ultrathin layer convolution

Dynamic simulation of reversible solid-fluid reactions in nonisothermal porous spheres with Stefan-Maxwell diffusion

In this thesis we analyze and develop methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs). The methods combine a direct solution approach with model reduction via proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). The reduced-order models (ROMs) are used to approximate the high-dimensional dynamic systems originating from a spatial discretization of a PDE. However, when used in an optimization algorithm, conventional POD/DEIM ROMs often lack the ability to give adequate approximations of the gradient. We propose methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces. We distinguish two types of error between quantities evaluated with the high-dimensional model and its ROM approximation in dependency on the optimization variable q: The reconstruction error which is evaluated with the same q0 which is used constructing the ROM and the prediction error which assesses approximations at q with a ROM constructed at q0 different to q. The novel reconstruction results we present include estimates for solutions of the adjoint equation and the sensitivity equations as well as for the gradient of the objective function. Based on the estimates we explain how the POD and DEIM bases should be extended with either adjoint or sensitivity information. The enhanced ROMs allow control of the reconstruction error for the objective and its gradient up to machine precision. Moreover, we propose a POD prediction estimate for the objective of the optimization problem in a neighborhood of q where the ROM is constructed. In case of sensitivity-extended POD and DEIM bases we give an analogous result for solutions of the states. The derivative-extended ROMs are then used to develop adaptive algorithms for the solution of optimal control and parameter estimation problems which results in great runtime improvements for the optimization while ensuring high approximation quality of the solution of the original problem. For the parameter estimation case a novel a posteriori error estimate is proposed which assesses the quality of suboptimal solutions obtained with the ROM. A further fundamental contribution is a discussion of discretize-then-optimize (DTO) vs. optimize-then-discretize (OTD) approaches in the context of MOR for optimization. We analyze advantages and disadvantages of both approaches and discuss to which extent our methods exhibit properties of either strategy. We also give examples of representative optimization problems in which standard POD/DEIM ROMs show an inacceptable behavior and can be successfully solved by derivative-extended ROMs. We have further implemented the developed methods emphasizing an efficient realization which is important for the investigation of the MOR potential. We showcase the practical performance of the proposed algorithms and the superiority of derivative-extended over conventional ROMs on two academic and one industry-relevant application which exhibit a variety of challenges for the model reduction approach in optimization.

Avoiding stability-induced inefficiencies in BDF methods

Non-intrusive surrogate modeling for parametrized time-dependent partial differential equations using convolutional autoencoders

Local derivative post-processing for the discontinuous Galerkin method

In this manuscript, we investigate the approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with fractional Brownian motion and memory in Hilbert space. By using semigroup theory, stochastic analysis techniques and fixed point approach, we derive a new set of sufficient conditions for the approximate controllability of nonlinear stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate our results.

We present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time-dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from \[26], \[27]. It is in particular well suited for problems with time-dependent PDE constraints. Instead of the quasi-normal step in a classical SQP method which results in solving the linearized PDE sufficiently well, in this algorithm a (nonlinear) solver is applied to the current discretization of the PDE. Moreover, different discretizations and solvers for the PDE and the adjoint PDE may be applied. The resulting inexactness of the reduced gradient in the current discretization is controlled within the algorithm. Thus, highly efficient PDE solvers can be coupled with the proposed optimization framework. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process implementable criteria for an adaptive refinement strategy of the current discretization based on error estimators. We prove global convergence to a stationary point of the infinite-dimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using a posteriori error estimators for the state and the adjoint equation. Numerical results for a semilinear parabolic PDE-constrained problem with pointwise control constraints are presented.

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.