Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

Deep mixed residual method for solving PDE-constrained optimization problems

As modern field effect transistors (FETs) scale down into the sub-10-nm regime, the approaching end of CMOS scaling becomes evident. Recently, we have performed a series of three-dimensional, fully quantum-mechanical, charge-self-consistent transport simulations and optimizations to show [1] that all FETs, irrespective of their channel material, will reach a fundamental downscaling limit around 4-5 nm technology node, due to unacceptably high levels of thermally induced errors. To investigate alternatives to FET/CMOS scaling, many emerging devices and technologies have been under active research in recent years. Among them, oxide-based memristors have become one of the strongest candidates to replace flash, and possibly DRAM and SRAM in the near future [2]. Moreover, memristors have a high potential to enable beyond-CMOS technology advances in novel architectures for high performance computing [3]. However, despite the spectacular progress in experimental demonstration of oxide memristors, their microscopic transport theory and even some key operational principles, essential to advance experimental progress, remain uncertain [4]. To facilitate the understanding of physical switching mechanisms and accelerate experimental development in oxide memristors, we have developed a 3D fully-coupled electrical and thermal transport model [5] that solves simultaneously the time-dependent continuity equations for electrons, holes, and oxygen vacancies, the time-dependent heat equation including Joule heating sources, and the Poisson equation for all the charged species. The model captures all the most important processes that drive memristive switching, including field drift due to the electrical potential gradient, Fick diffusion due to the concentration gradient, and the Soret effect due to the temperature gradient. To properly simulate the movement of oxygen vacancies, the rigid point ion model [5] is used, in which the vacancy velocity exponentially depends on the temperature and the electric field when the field exceeds a critical value. The fully-coupled equations and relevant physical models are implemented in CHARON, a multi-dimensional parallel TCAD code developed at Sandia National Labs. CHARONis built upon the open-source Trilinos packages [6], which support finite element and finite volume discretization methods, state-of-the-art nonlinear and linear solvers, MPI parallel capability, and many more. It can simulate arbitrary device geometry and supports multi-physics capability (i.e., allowing for solving different equations in different regions). The coupled model has been applied to simulate the switching process in a 3D filamentary tantalum oxide memristor shown in Fig. 1(a). The layer material and thickness come from an experimental device published in Ref. [7]. Making use of the multi-physics capability in CHARON, we solve the coupled Poisson, electron, oxygen vacancy, and heat equations in the active Ta and conduction filament (CF) regions, with vacancies confined in Ta and CF regions, while in the Pt and Ta2O5 regions, due to absence of mobile vacancies, the coupled Poisson, electron, and heat equations are solved, with Pt treated as a highly doped n-type semiconductor using immobile dopants, and Ta2O5 as an undoped semiconductor. Extensive simulations show that both field drift and thermal processes play crucial roles in determining the switching dynamics of the bipolar TaOx device. Specifically, (i) during OFF switching, when a sufficient negative voltage is applied to the top Pt electrode, Joule heating causes the increase of temperature with the hottest spot located in the CF, and consequently, oxygen vacancies move away from CF into Ta under the influence of both heating and field, which leads to a vacancy density gap and possible depletion of vacancies in the entire CF at high negative voltage, resulting in the OFF state; (ii) during ON switching, a sufficient positive voltage causes Joule heating and the increase of temperature, and increased temperature causes vacancies moving from Ta into CF to refill the vacancy gap, resulting in the ON state. Interestingly, we show that the thermally activated field-dominated switching in the considered devices can lead to more than three orders of magnitude speed-up in switching time, with only 50% voltage/current increase. Simulated resistances during OFF and ON switching as a function of time and voltage show very good qualitative agreement with experimental measurement in Ref. [7]. Furthermore, under a triangular voltage sweep, the simulated current-voltage hysteresis curve is shown in Fig. 1(b), which also shows good qualitative agreement with the measured data in Ref. [8]. The generality of the proposed model allows to simulate and investigate the underlying physical switching mechanisms in a wide spectrum of oxide memristors ranging from field-dominant, field-thermal-driven to thermal-dominant memristive devices. Figure 1

Distributed control and constraint preconditioners

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations.

In this paper we study the regularity theory in Orlicz spaces for the Poisson and heat equations.

Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions

In this paper, a class of additive block triangular preconditioners are constructed for solving block two-by-two linear systems with symmetric positive (semi-)definite sub-matrices. Convergence analysis of the related splitting iteration method shows that it is almost unconditionally convergent and behaves problem independent with a convergence rate less than 0.5 under a practical parameter choice. Optimization of the preconditioned matrices, which have real and tight eigenvalue distributions, shows that it can result in an upper bound less than 2 for the condition number of the preconditioned matrices. Moreover, we also give a special consideration about the feasibility of the proposed preconditioner for solving more general problems with indefinite sub-matrices. Numerical experiments based on examples arising from complex symmetric linear systems and PDE-constrained optimization problems are presented to show the robustness and effectiveness of the proposed preconditioners compared with some other existing preconditioners.

PurposeThe aim of the paper is to achieve textbook multigrid efficiency for some flow problems.Design/methodology/approachThe steady incompressible Euler equations are decoupled into elliptic and hyperbolic subsystems. Numerous classical FAS‐MG algorithms are implemented and tested for convergence. A full multigrid algorithm that costs less than 10 work units (WUs) is sufficient to reduce the algebraic error below the discretization error. A new algorithm “NUVMGP” is introduced. A two‐step iterative procedure is adopted. First, given the pressure gradient, the convection equations are solved on the computational grid for the velocity components by performing one Gauss‐Seidel iteration ordered in the flow direction. second, a linear multigrid (MG) cycle for Poisson's equation is performed to update pressure values.FindingsIt is found that algorithm “NUVMGP‐FMG” requires less than 6 WU to attain the target solution. The convergence rates are independent on both the mesh size and the approximation order.Research limitations/implicationsLexicographic Gauss‐Seidel using downstream ordering is a good solver for the advection terms and provides excellent smoothing rates for relaxation. But it is complicated to maintain downstream ordering in case the flow directions change with location.Originality/valueAlthough the scope of this work is limited to rectangular domains, finite difference schemes, and incompressible Euler equation, the same approaches can be extended for other flow problems. However, such relatively simple problems may provide deep understanding of the ideal convergence behavior of MG and accumulate experience to detect unacceptable performance and regain the optimal one.

The purpose of this work is to develop a numerical code for modeling the performance of thin film plasmonic solar cells performing a detailed analysis of both the light interaction with the solar cell and the resulting charging transport due to the photovoltaic effect. The continuity and Poisson equations were modeled in each layer of thin film silicon solar cells using finite difference method. In each layer the system of equations were solved numerically. The mesh size was chosen to be smaller than the Debye length. Because of the large bandgap of amorphous silicon the intrinsic carrier concentration is lower in thin film silicon solar cells. Different parameters like energy bands, electron and hole concentrations, electric field, voltage, current densities etc. were calculated at each mesh point. The generation rate for carrier transport modeling was calculated from full wave electromagnetic simulations.

In [1], the solution of the heat equation on a polygonal domain is decomposed into regular and singular parts. Here, it is shown that the solution is globally regular by using a suitable weighted Sobolev space. This result is used to design an optimally convergent semi‐discrete finite element method (FEM) where the mesh size is suitably refined. As full discrete schemes, the Fourier series and the non‐standard finite difference methods coupled with the FEM are presented.

In this work we present a programming philosophy and a template code for achieving computational scalability when using multiple graphics processing units (GPUs) in the numerical solution of any mathematical system of equations found in continuum dynamic simulations. The programming philosophy exploits the principal characteristics of the GPU hardware, with emphasis in the delivering of threads with massive memory fetches, intense calculations using local registers and limited writes to global memory. The philosophy requires explicit formulas for calculations for which domain decomposition is trivial. The domains are decomposed in regions that use the local central processing unit (CPU) to communicate common interfaces using the message passing interface (MPI). A template code for the heat equation is established and tested for scalability. The novelty is that we show a series of codes, constructed from the basic template, that solve all the basic model equations found in continuum dynamics, and present illustrative results. The model equations are the heat equation, the Poisson equation, the shallow-water equations, the flow in porous media equations and the vorticity equations.

In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form sum_{t=1}^{p}A_{t}XB_{t}=C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

This book is designed for senior undergraduate and graduate students pursuing courses in mathematics, physics, engineering and biology. The text begins with a study of ordinary differential equations. The concepts of first- and second-order equations are covered initially. It moves further to linear systems, series solutions, regular Sturm–Liouville theory, boundary value problems and qualitative theory. Thereafter, partial differential equations are explored. Topics such as first-order partial differential equations, classification of partial differential equations and Laplace and Poisson equations are also discussed in detail. The book concludes with heat equation, one-dimensional wave equation and wave equation in higher dimensions. It highlights the importance of analysis, linear algebra and geometry in the study of differential equations. It provides sufficient theoretical material at the beginning of each chapter, which will enable students to better understand the concepts and begin solving problems straightaway.

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE-constrained shape optimization problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so-called smoothing parameter that allows a smoothness adjustment of the shape during the optimization process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.