Schur Complement Decomposition Research Articles

The electromagnetic scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media

We study the electromagnetic scattering problem from multiple arbitrarily shaped cavities filled with inhomogeneous anisotropic media. A local-refined Petrov-Galerkin finite element interface (PGFEI) method is developed for solving the scattering problem. By introducing the transparent boundary condition, the unbounded scattering problem can be reduced into bounded domain problem with coupled boundary conditions on the apertures of multiple cavities. The multiple cavity scattering in transverse magnetic (TM) and transverse electric (TE) polarizations are both investigated based on uniform non-body-fitted meshes. We adopt the level-set functions to capture the irregular cavity wall and the interfaces of inhomogeneous media. The special basis is constructed to satisfy the jump conditions across the interfaces of different materials further. The local-refined algorithm is proposed to enhance the accuracy of the solutions around the discontinuous interfaces. The generation of the grid points facilitates a special numbering scheme which enables a stepwise Schur complement decomposition of the discrete linear system derived by the PGFEI method. It could be an alternative method for solving the scattering problems which have similar structures. The proposed method can handle matrix coefficients caused by the permittivity and permeability tensor of anisotropic media. Numerical experiments illustrate the validity and efficiency of the method for solving the scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media in TM and TE polarization.

Scalable Parallel Nonlinear Optimization with PyNumero and Parapint

We describe PyNumero, an open-source, object-oriented programming framework in Python that supports rapid development of performant parallel algorithms for structured nonlinear programming problems (NLP’s) using the Message Passing Interface (MPI). PyNumero provides three fundamental building blocks for developing NLP algorithms: a fast interface for calculating first and second derivatives with the AMPL Solver Library (ASL), a number of interfaces to efficient linear solvers, and block-structured vectors and matrices based on NumPy, SciPy, and MPI that support distributed parallel storage and computation. PyNumero’s design enables efficient, parallel algorithm development using high-level Python syntax while keeping expensive numerical calculations in fast, compiled implementations based on languages like C and Fortran. To demonstrate the utility of PyNumero, we also present Parapint, a Python package built on PyNumero for parallel solution of dynamic optimization problems. Parapint includes a parallel interior-point solver based on Schur-Complement decomposition. We illustrate the effectiveness of PyNumero for developing parallel algorithms with both code examples and scalability analyses for parallel matrix-vector dot products, parallel solution of structured systems of linear equations using Schur-Complement decomposition, and the parallel solution of a two-dimensional PDE optimal control problem. Our numerical results show nearly perfect scaling to more than 1,000 cores for large matrix-vector dot products and structured linear systems. Moreover, we obtain more than 354 times speedup for the optimal control example. History: Accepted by Alice Smith, EIC/Ted Ralphs, Area Editor/Software Tools. Funding: This work was funded in part by the Institute for the Design of Advanced Energy Systems (IDAES) with funding from the Office of Fossil Energy, Cross-Cutting Research, U.S. Department of Energy. This work was also funded by Sandia National Laboratories Laboratory Directed Research and Development (LDRD) program. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1272 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0285 ) at ( http://dx.doi.org/10.5281/zenodo.7192328 ).

Stochastic Optimal Device Sizing Model for Zero Energy Buildings: A Parallel Computing Solution

Byconsuming35% of the final global energy, buildings are among major contributors to greenhouse gas emissions and global warming. If the economic justification challenge is addressed, zero energy buildings (ZEB), which are defined as buildings that generate as much renewable-based energy as they consume annually, can be a promising solution for energy efficiency improvement in the building sector. In this article, we propose a stochastic ZEB device sizing model considering uncertain parameters (i.e., building's electrical and thermal demands, solar irradiation, and outdoor temperature) and their correlation. The proposed model finds the optimal size of the thermal and electrical devices considering their nonlinear behaviors and temperature-dependent dynamics. Furthermore, two parallel computing-based solution algorithms (i.e., parallelism in the algebraic level using Schur complement decomposition and parallelism in the problem (scenario) level by progressive hedging) are proposed to solve the stochastic model. Using the real historical data, numerical studies on the Woodward library building, located on the University of British Columbia campus, illustrate the efficacy of the solution algorithms to handle large-scale nonlinear programming models with more than 32.8 million variables, which is the largest one reported in the literature so far.

Serial advanced-multi-step nonlinear model predictive control using an extended sensitivity method

Nonlinear model predictive control (NMPC) can directly handle multi-input multi-output nonlinear systems and explicitly consider input and state constraints. However, the computational load for nonlinear programming (NLP) of large-scale systems limits the range of possible applications and degrades NMPC performance. An NLP sensitivity based approach, advanced-step NMPC, has been developed to address the online computational load. In addition, for cases where the NLP solving time exceeds one sampling time, two types of advanced-multi-step NMPC (amsNMPC), parallel and serial, have been proposed. However, in previous studies, a serial amsNMPC could not be applied to large-scale problems because of the size of extended Karush–Kuhn–Tucker matrix and its Schur complement decomposition, and the robustness was analyzed under a conservative assumption for memory effects. In this paper, we propose a serial amsNMPC using an extended sensitivity method to increase the online computation speed further. We successfully apply it to a large-scale air separation unit using the sparse matrix handling packages of Python, Pyomo, and k_aug tools. Furthermore, an auxiliary NLP formulation is defined to analyze the robustness. Using this with the key properties of an extended sensitivity matrix, we can prove robustness while avoiding the memory effects term.

An energetically balanced, quasi-Newton integrator for non-hydrostatic vertical atmospheric dynamics

An energetically balanced, implicit integrator for non-hydrostatic vertical atmospheric dynamics on the sphere is presented. The integrator allows for the exact balance of energy exchanges in space and time for vertical atmospheric motions by preserving the skew-symmetry of the non-canonical Hamiltonian formulation of the compressible Euler equations. The performance of the integrator is accelerated by a preconditioning strategy that reduces the dimensionality of the inner linear system. Here we reduce the four component velocity, density, density weighted potential temperature and Exner pressure system into a single equation for the density weighted potential temperature via repeated Schur complement decomposition and the careful selection of coupling terms. As currently formulated, the integrator is based on a horizontal-vertical spatial splitting that does not permit bottom topography. The integrator is validated for standard test cases for baroclinic instability and a non-hydrostatic gravity wave on the sphere and a rising bubble in a high-resolution planar geometry, and shows robust convergence across all of these regimes.

Scalable preconditioning of block-structured linear algebra systems using ADMM

We study the solution of block-structured linear algebra systems arising in optimization by using iterative solution techniques. These systems are the core computational bottleneck of many problems of interest such as parameter estimation, optimal control, network optimization, and stochastic programming. Our approach uses a Krylov solver (GMRES) that is preconditioned with an alternating method of multipliers (ADMM). We show that this ADMM-GMRES approach overcomes well-known scalability issues of Schur complement decomposition in problems that exhibit a high degree of coupling. The effectiveness of the approach is demonstrated using linear systems that arise in stochastic optimal power flow problems and that contain up to 2 million total variables and 4000 coupling variables. We find that ADMM-GMRES is nearly an order of magnitude faster than Schur complement decomposition. Moreover, we demonstrate that the approach is robust to the selection of the augmented Lagrangian penalty parameter, which is a key advantage over the direct use of ADMM.

Parallel cyclic reduction strategies for linear systems that arise in dynamic optimization problems

Dynamic optimization problems are constrained by differential and algebraic equations and are found everywhere in science and engineering. A well-established method to solve these types of problems is direct transcription, where the differential equations are replaced with discretized approximations based on finite-difference or Runge–Kutta schemes. However, for problems with thousands of state variables and discretization points, direct transcription may result in nonlinear optimization problems which are too large for general-purpose optimization solvers to handle. Also, when an interior-point solver is applied, the dominant computational cost is solving the linear systems resulting from the Newton step. For large-scale nonlinear programming problems, these linear systems may become prohibitively expensive to solve. Furthermore, the systems also become too large to formulate and store in memory of a standard computer. On the other hand, direct transcription can exploit sparsity and structure of the linear systems in order to overcome these challenges. In this paper we investigate and compare two parallel linear decomposition algorithms, Cyclic Reduction (CR) and Schur complement decomposition, which take advantage of structure and sparsity. We describe the numerical conditioning of the CR algorithm when applied to the linear systems arising from dynamic optimization problems, and then compare CR with Schur complement decomposition on a number of test problems. Finally, we propose conditions under which each should be used, and describe future research directions.

A parallel structure exploiting nonlinear programming algorithm for multiperiod dynamic optimization

This article develops a sequential quadratic programming (SQP) algorithm that utilizes a parallel interior-point method (IPM) for the QP subproblems. Our approach is able to efficiently decompose and solve large-scale multiperiod nonlinear programming (NLP) formulations with embedded dynamic model representations, through the use of an explicit Schur-complement decomposition within the IPM. The algorithm implementation makes use of a computing environment that uses the parallel distributed computing message passing interface (MPI) and specialized vector-matrix class representations, as implemented in the third-party software package, OOPS. The proposed approach is assessed, with a focus on computational speedup, using several benchmark examples involving applications of parameter estimation and design under uncertainty which utilize static and dynamic models. Results indicate significant improvements in the NLP solution speedup when moving from a serial full-space direct factorization approach, to a serial Schur-complement decomposition, to a parallelized Schur-complement decomposition for the primal-dual linear system solution within the IPM.

Fluorescence molecular tomographic image reconstruction based on reduced measurement data

The analysis of fluorescence molecular tomography is important for medical diagnosis and treatment. Although the quality of reconstructed results can be improved with the increasing number of measurement data, the scale of the matrices involved in the reconstruction of fluorescence molecular tomography will also become larger, which may slow down the reconstruction process. A new method is proposed where measurement data are reduced according to the rows of the Jacobian matrix and the projection residual error. To further accelerate the reconstruction process, the global inverse problem is solved with level-by-level Schur complement decomposition. Simulation results demonstrate that the speed of the reconstruction process can be improved with the proposed algorithm.

An interior-point method for efficient solution of block-structured NLP problems using an implicit Schur-complement decomposition

In this work, we address optimization of large-scale, nonlinear, block-structured problems with a significant number of coupling variables. Solving these problems using interior-point methods requires the solution of a linear system that has a block-angular structure at each iteration. Parallel solution is possible using a Schur-complement decomposition. In an explicit Schur-complement decomposition, the computational cost of forming and factorizing the Schur-complement is prohibitive for problems with many coupling variables. In this paper, we show that this bottleneck can be overcome by solving the Schur-complement equations implicitly, using a quasi-Newton preconditioned conjugate gradient method. This new algorithm avoids explicit formation and factorization of the Schur-complement. The computational efficiency of this algorithm is compared with the serial full-space approach, and the serial and parallel explicit Schur-complement approach. These results show that the PCG Schur-complement approach dramatically reduces the computational cost for problems with many coupling variables.

Application of the iterative solution method with Schur complement reduction to mixed finite elements based in a tetrahedral discretization

The iterative solution method for mixed finite element methods is applied to a 3-D domain partitioned with tetrahedral elements. For the particular discretization technique of first partitioning the domain with hexahedral cells, and then subsequently partitioning cells with five tetrahedral elements, a Schur complement decomposition is devised wherein the actual number of equations solved is reduced by 80%. Although this Schur complement reduction requires a fair amount of computational overhead, its application within the iterative solution method can reduce overall solution time by about 44%, depending on closure criterion and other factors.

An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization

We present a scalable approach and implementation for solving stochastic optimization problems on high-performance computers. In this work we revisit the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution. These computations consist of solving sparse linear systems with multiple sparse right-hand sides and are needed in our Schur-complement decomposition approach to compute the contribution of each scenario to the Schur matrix. Our novel approach uses an incomplete augmented factorization implemented within the PARDISO linear solver and an outer BiCGStab iteration to efficiently absorb pivot perturbations occurring during factorization. This approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right-hand sides. We report on the performance of the approach on high-performance computers when solving stochastic unit commitment problems of unprecedented size (billions of variables and constraints) that arise in the optimization and control of electrical power grids. Our numerical experiments suggest that supercomputers can be efficiently used to solve power grid stochastic optimization problems with thousands of scenarios under the strict “real-time” requirements of power grid operators. To our knowledge, this has not been possible prior to the present work.

Estimation error for blind Gaussian time series prediction

We tackle the issue of the blind prediction of a Gaussian time series. For this, we construct a projection operator built by plugging an empirical covariance estimator into a Schur complement decomposition of the projector. This operator is then used to compute the predictor. Rates of convergence of the estimates are given.

Optimal operation of cryogenic air separation systems with demand uncertainty and contractual obligations

Cryogenic air separation is an efficient technology for supplying large quantities of nitrogen, argon, and oxygen to chemical, petroleum and manufacturing customers. However, numerous uncertainties make effective operation of these complex processes difficult. This work addresses the problem of determining an optimal operating strategy to maximize the total profit of a cryogenic air separation process while considering demand uncertainty and contractual obligations. A rigorous process model is included as constraints in a nonlinear programming formulation. Uncertain demands are assumed to be normally distributed with known mean and standard deviation, and expected profit in the objective function is evaluated using the standard loss function. A probabilistic fill-rate expression, also based on the loss function, is used to model the contractual obligations by providing a lower bound on the expected product sales. In the single period case with one customer satisfaction constraint, the nonlinear programming formulation can be solved efficiently using the general purpose nonlinear optimization package, Ipopt. This formulation is then extended to include multiple time periods, the potential for product storage, and customer satisfaction constraints on multiple products. To solve the large-scale nonlinear programming formulation that considers a seven-day operating horizon, a tailored parallel nonlinear programming algorithm is used. This approach makes use of a Schur complement decomposition strategy to exploit the block structure of the problem and allow efficient solution in parallel. Using these tools, we solve for a set of optimal operating strategies over the complete space of different fill rates. This produces planning figures that identify key trade-offs between profitability and contractual obligations.

Image reconstruction of fluorescent molecular tomography based on the tree structured Schur complement decomposition

BackgroundThe inverse problem of fluorescent molecular tomography (FMT) often involves complex large-scale matrix operations, which may lead to unacceptable computational errors and complexity. In this research, a tree structured Schur complement decomposition strategy is proposed to accelerate the reconstruction process and reduce the computational complexity. Additionally, an adaptive regularization scheme is developed to improve the ill-posedness of the inverse problem.MethodsThe global system is decomposed level by level with the Schur complement system along two paths in the tree structure. The resultant subsystems are solved in combination with the biconjugate gradient method. The mesh for the inverse problem is generated incorporating the prior information. During the reconstruction, the regularization parameters are adaptive not only to the spatial variations but also to the variations of the objective function to tackle the ill-posed nature of the inverse problem.ResultsSimulation results demonstrate that the strategy of the tree structured Schur complement decomposition obviously outperforms the previous methods, such as the conventional Conjugate-Gradient (CG) and the Schur CG methods, in both reconstruction accuracy and speed. As compared with the Tikhonov regularization method, the adaptive regularization scheme can significantly improve ill-posedness of the inverse problem.ConclusionsThe methods proposed in this paper can significantly improve the reconstructed image quality of FMT and accelerate the reconstruction process.

A scalable parallel Poisson solver for three-dimensional problems with one periodic direction

A code for the direct numerical simulation (DNS) of incompressible flows with one periodic direction has been developed. It provides a fairly good performance on both Beowulf clusters and supercomputers. Since the code is fully explicit, from a parallel point-of-view, the main bottleneck is the Poisson equation. To solve it, a Fourier diagonalization is applied in the periodic direction to decompose the original 3D system into a set of mutually independent 2D systems. Then, different strategies can be used to solved them. In the previous version of the code, that was conceived for low-cost PC clusters with poor network performance, a Direct Schur-complement Decomposition (DSD) algorithm was used to solve them. Such a method, that is very efficient for PC clusters, cannot be used with an arbitrarily large number of processors and mesh sizes, mainly due to the RAM memory requirements. To do so, a new version of the solver is presented in this paper. It is based on the DSD algorithm that is used as a preconditioner for a Conjugate Gradient method. Numerical experiments showing the scalability and the flexibility of the method on both the MareNostrum supercomputer and a PC cluster with a conventional 100 Mbits/s network are presented and discussed. Finally, illustrative DNS results of an air-filled differentially heated cavity at Ra = 10 11 are also presented.

Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems

Multi-scenario optimization is a convenient way to formulate and solve multi-set parameter estimation problems that arise from errors-in-variables-measured (EVM) formulations. These large-scale problems lead to nonlinear programs (NLPs) with specialized structure that can be exploited by the NLP solver in order to obtained more efficient solutions. Here we adapt the IPOPT barrier nonlinear programming algorithm to provide efficient parallel solution of multi-scenario problems. The recently developed object oriented framework, IPOPT 3.2, has been specifically designed to allow specialized linear algebra in order to exploit problem specific structure. This study discusses high-level design principles of IPOPT 3.2 and develops a parallel Schur complement decomposition approach for large-scale multi-scenario optimization problems. A large-scale case study example for the identification of an industrial low-density polyethylene (LDPE) reactor model is presented. The effectiveness of the approach is demonstrated through the solution of parameter estimation problems with over 4100 ordinary differential equations, 16,000 algebraic equations and 2100 degrees of freedom in a distributed cluster.

Full-zone hole dispersion relations in Si using Schur-complement decomposition.

An efficient method for computing semiconductor valence-band energy-wave-vector dispersion relations throughout the Brillouin zone has been developed and applied to Si. The method uses a sequence of Schur-complement matrix determinant decompositions to restrict a large underlying k\ensuremath{\cdot}p secular determinant to a lower-dimensional subspace, thereby avoiding the usual perturbative reduction. The effective secular equation thus obtained generalizes the Kane cubic to apply throughout the Brillouin zone in bulk diamond-type semiconductors. The presence of spin-orbit coupling among the valence bands is efficiently modeled throughout the zone. The method has also led to the determination of a new set of values for k\ensuremath{\cdot}p momentum matrix elements in Si. The computed dispersion relations for Si reproduce the results of k\ensuremath{\cdot}p perturbation theory for small k, and also show very good global agreement with empirical pseudopotential calculations.