It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to the boundary steps. For $s$-stage methods, we prove convergence of order $s$ for the state variables if the adjoint method satisfies the conditions for order $s\!-\!1$, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of the first author et al. [J. Comput. Appl. Math., 262:73-86, 2014] and discuss the construction of 3-stage methods for the order pair (3,2) in detail including some matrix background for the combined forward and adjoint order conditions. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to BDF. Three 3-stage methods are constructed which show the expected orders in numerical tests.
We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally numerically stable. Tangents of systems of linear equations can become instable as well as both tangents and adjoints of systems of nonlinear equations, which extends to optima of convex unconstrained objectives. Sufficient conditions for numerical stability are derived.
Differentiate the Solver, Not the Equation: Reverse-Sweep Adjoints for Block Implicit Simulation
Stanford University（斯坦福大学）；University of California, Los Angeles（加州大学洛杉矶分校）；LightSpeed Studios, Tencent America（腾讯美国光速工作室）；The University of Utah（犹他大学）
In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac 12$ and Poisson random measure, more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time, more precisely linear implicit method, exponential integrator and …
We develop a numerical solver for three-dimensional wave propagation in coupled poroelastic-elastic media, based on a high-order discontinuous Galerkin (DG) method, with the Biot poroelastic wave equation formulated as a first order conservative velocity/strain hyperbolic system. To derive an upwind numerical flux, we find an exact solution to the Riemann problem, including the poroelastic-elastic interface; we also consider attenuation mechanisms both in Biot's low- and high-frequency regimes. Using either a low-storage explicit or implicit-explicit (IMEX) Runge-Kutta scheme, according to the…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise under more relaxed conditions. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. This extends the current results in the literature to not necessary self-adjoint operator with more general boundary conditions. As a consequence key part of the proof does not rely on the spectral decomposition of the linear operator. We achieve optimal convergence order…
This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs while modeling real world phenomena. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet well understood. The non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the t…
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
A high-order partitioned solver for general multiphysics problems and its applications in optimization
Stanford University（斯坦福大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Berkeley（加州大学伯克利分校）；University of Notre Dame（圣母大学）
Analysis of Robin-boundary control for the Boussinesq equations
SKLMS & NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；School of Mathematics, Xi’an University of Technology（西安理工大学数学学院）；School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）；School of Mathematical Sciences, Guizhou Normal University（贵州师范大学数学科学学院）
We consider the Cauchy problem for a second-order evolution equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of operator A by its conjugate A*. Time approximations are carried out so that the transition to a new level in time was associated with a separate solution of problems for operators A and A*, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (correctness) of operator-difference…
