This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O((m/N p )3 ) for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and N p is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than N p and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier-Stokes equations and the Boussinesq approximation with Brownian forcing.

This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O((m/N p )3 ) for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and N p is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than N p and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier-Stokes equations and the Boussinesq approximation with Brownian forcing.

We propose a data-driven stochastic method (DSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. An essential ingredient of the proposed method is to construct a data-driven stochastic basis under which the stochastic solutions to the SPDEs enjoy a compact representation for a broad range of forcing functions and/or boundary conditions. Our method consists of offline and online stages. A data-driven stochastic basis is computed in the offline stage using the Karhunen-Loève (KL) expansion. A two-level preconditioning optimization approach and a randomized SVD algorithm are used to reduce the offline computational cost. In the online stage, we solve a relatively small number of coupled deterministic PDEs by projecting the stochastic solution into the data-driven stochastic basis constructed offline. Compared with a generalized polynomial chaos method (gPC), the ratio of the computational complexities between DSM (online stage) and gPC is of order O((m/Np) 2 ). Here m and Np are the numbers of elements in the basis used in DSM and gPC, respectively. Typically we expect m Np when the effective dimension of the stochastic solution is small. A timing model, which takes into account the offline computational cost of DSM, is constructed to demonstrate the efficiency of DSM. Applications of DSM to stochastic elliptic problems show considerable computational savings over traditional methods even with a small number of queries. We also provide a method for an a posteriori error estimate and error correction.

Luby Transform (LT) codes provide efficient erasure protection for information distributed across a multiuser channel with unknown characteristics. For real-time applications, however, the random selection of information symbols within a large message can increase encoding and decoding latency beyond allowable parameters. By using a smaller message size, the encoder can generate a codeword at a faster rate, leading to a higher decoder throughput. We decrease decoding latency by optimizing various parameters of the Robust Soliton distribution. This optimization in terms of delay does not lead to minimizing overhead. However, if a small increase in overhead can be tolerated, particularly for a smaller message size, then the overall encoding and decoding delay can be greatly reduced.

In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electrical circuit, one is faced with three special situations: 1. The system involves external (control) forcing through external (controlled) voltage sources and resistors. 2. The system is constrained via the Kirchhoff current (KCL) and voltage laws (KVL). 3. The Lagrangian is degenerate. Based on a geometric setting, an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. A time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. Dependent on the discretization, the intrinsic degeneracy of the system can be canceled for the discrete variational scheme. In this way, a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits; in particular, a comparison to BDF methods (which are usually the method of choice for the simulation of electric circuits) shows that even for simple LCR circuits, a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator.