A massively parallel exponential integrator for advection-diffusion models

This work considers the Real Leja Points Method (ReLPM) for the exponential integration of large-scale sparse systems of ODEs, generated by Finite Element or Finite Difference discretizations of 3-D advection-diffusion models. We present an efficient parallel implementation of ReLPM for polynomial interpolation of the matrix exponential propagators. A scalability analysis of the most important computational kernel inside the code, the parallel sparse matrix\u2013vector product, has been performed, as well as an experimental study of the communication overhead. As a result of this study an optimized parallel sparse matrix\u2013vector product routine has been implemented. The resulting code shows good scaling behavior even when using more than one thousand processors. The numerical results presented on a number of very large test cases gives experimental evidence that ReLPM is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

Efficient numerical schemes for porous media flow

Partial di erential equations (PDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Due to the uncertainty in the input data, stochastic partial di erential equations (SPDEs) have become popular as a modelling tool in the last century. As the exact solutions are unknown, developing e cient numerical methods for simulating PDEs and SPDEs is a very important while challenging research topic. In this thesis we develop e cient numerical schemes for deterministic and stochastic porous media ows. More schemes are based on the computing of the matrix exponential functions of the non diagonal matrices, we use new e cient techniques: the real fast L eja points and the Krylov subspace techniques. For the deterministic ow and transport problem, we consider two deterministic exponential integrator schemes: the exponential time di erential stepping of order one (ETD1) and the exponential Euler midpoint (EEM) with nite volume method for discretization in space. We give the time and space convergence proof for the ETD1 scheme and illustrate with simulations in two and three dimensions that the exponential integrators are e - cient and accurate for advection dominated deterministic transport ow in heterogeneous anisotropic porous media compared to standard semi implicit and implicit schemes. For the stochastic ow and transport problem, we consider the general parabolic SPDEs in a Hilbert space, using the nite element method for discretization in space (although nite di erence or nite volume can be used as well). We use a linear functional of the noise and the standard Brownian increments to develop and give convergence proofs of three new e cient and accurate schemes for additive noise, one called the modi ed semi{ implicit Euler-Maruyama scheme and two stochastic exponential integrator schemes, and two stochastic exponential integrator schemes for multiplicative and additive noise. The schemes are applied to two dimensional ow and transport

Oblique projection for scalable rank-adaptive reduced-order modeling of nonlinear stochastic PDEs with time-dependent bases

Time-dependent basis reduced order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values, and (iv) error accumulation due to fixed rank. To this end, we present a scalable method based on oblique projections for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive, and highly parallelizable. These favorable properties are achieved via low-rank approximation of the time discrete MDE. Using the discrete empirical interpolation method (DEIM), a low-rank decomposition is computed at each iteration of the time stepping scheme, enabling a near-optimal approximation at a fraction of the cost. We coin the new approach TDB-CUR since it is equivalent to a CUR decomposition based on sparse row and column samples of the MDE. We also propose a rank-adaptive procedure to control the error on-the-fly. Numerical results demonstrate the accuracy, efficiency, and robustness of the new method for a diverse set of problems

Implementation of exponential Rosenbrock-type methods

In this paper, we present a variable step size implementation of exponential Rosenbrock-type methods of orders 2, 3 and 4. These integrators require the evaluation of exponential and related functions of the Jacobian matrix. To this aim, the Real Leja Points Method is used. It is shown that the properties of this method combine well with the particular requirements of Rosenbrock-type integrators. We verify our implementation with some numerical experiments in MATLAB, where we solve semilinear parabolic PDEs in one and two space dimensions. We further present some numerical experiments in FORTRAN. where we compare out-method with other methods from literature. We find a great potential Of Our method for non-normal matrices. Such matrices typically arise in parabolic problems with large advection in combination with moderate diffusion and mildly stiff reactions