A block Krylov subspace time-exact solution method for linear ODE systems

We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$ and $y''=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations

Solution of systems of nonlinear equations Final report

Algorithm and computer program of diagonal discrimination method for computing nonlinear and transcendental function

Computational Methods for Nonlinear Systems Analysis With Applications in Mathematics and Engineering

An investigation into current methods and new approaches for solving systems of nonlinear equations was performed. Nontraditional methods for implementing arc-length type solvers were developed in search of a more robust capability for solving general systems of nonlinear algebraic equations. Processes for construction of parameterized curves representing the many possible solutions to systems of equations versus finding single or point solutions were established. A procedure based on these methods was then developed to identify static equilibrium states for solutions to multi-body-dynamic systems. This methodology provided for a pictorial of the overall solution to a given system, which demonstrated the possibility of multiple candidate equilibrium states for which a procedure for selection of the proper state was proposed. Arc-length solvers were found to identify and more readily trace solution curves as compared to other solvers making such an approach practical. Comparison of proposed methods was made to existing methods found in the literature and commercial software with favorable results. Finally, means for parallel processing of the Jacobian matrix inherent to the arc-length and other nonlinear solvers were investigated, and an efficient approach for implementation was identified. Several case studies were performed to substantiate results. Commercial software was also used in some instances for additional results verification

A preconditioned Krylov subspace approach to a tightly coupled aeromechanical system

A tightly coupled approach is attempted to compute a modest fluid-structure interaction for high subsonic flow through a converging nozzle with deformable walls. A globally convergent Newton statement and a matrix-free GMRES linear equation solver are used to linearize and solve the coupled system of equations without explicitly forming the left hand side jacobian matrix associated with the Newton method. A variable forcing function term is successfully incorporated into the Newton statement to balance inner (linear) and outer (nonlinear) iterations. The fluid-structure system is solved for comparison purposes using a loosely coupled approach. Residual convergence stagnated in the tightly coupled system approach but converged successfully in the loosely coupled approach using the same coding for domain calculations. A novel approach using time derivative preconditioning is incorporated to speed convergence of the GMRES linear equation solver. No algebraic preconditioning is used. The fluid flow equations showed significant improvements using the time derivative preconditioning method but the error term generated in the structural equations overwhelmed the physical solution increment. The Taylor Weak Statement derivation of the finite element form of the fluid flow equations with time derivative preconditioning shows a strong connection to the Streamwise Upwind Petrov Galerkin (SUPG) method. This connection is exploited to develop a theoretical basis for the damping term and the time scale parameter common to the SUPG method

Augmented Block-Arnoldi Recycling CFD Solvers

One of the limitations of recycled GCRO methods is the large amount of computation required to orthogonalize the basis vectors of the newly generated Krylov subspace for the approximate solution when combined with those of the recycle subspace. Recent advancements in low synchronization Gram-Schmidt and generalized minimal residual algorithms, Swirydowicz et al.~\cite{2020-swirydowicz-nlawa}, Carson et al. \cite{Carson2022}, and Lund \cite{Lund2022}, can be incorporated, thereby mitigating the loss of orthogonality of the basis vectors. An augmented Arnoldi formulation of recycling leads to a matrix decomposition and the associated algorithm can also be viewed as a {\it block} Krylov method. Generalizations of both classical and modified block Gram-Schmidt algorithms have been proposed, Carson et al.~\cite{Carson2022}. Here, an inverse compact $WY$ modified Gram-Schmidt algorithm is applied for the inter-block orthogonalization scheme with a block lower triangular correction matrix $T_k$ at iteration $k$. When combined with a weighted (oblique inner product) projection step, the inverse compact $WY$ scheme leads to significant (over 10$\times$ in certain cases) reductions in the number of solver iterations per linear system. The weight is also interpreted in terms of the angle between restart residuals in LGMRES, as defined by Baker et al.\cite{Baker2005}. In many cases, the recycle subspace eigen-spectrum can substitute for a preconditioner

Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Adjoint Systems

Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a high implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver leads to a higher number of iterations to get the same level of convergence as the monolithic one. The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. We also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers for the fluid block equations. We demonstrate the benefit of these techniques by computing the coupled derivatives of an aeroelastic configuration of the ONERA-M6 fixed wing in transonic flow. For this exercise the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. All computations are performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 19% for the nested FGCRO-DR solver.Comment: 42 pages, 21 figure

On Duality and the Bi-Conjugate Gradient Algorithm

It is not uncommon to encounter problems that lead to large, sparse linear systems with coefficient matrices that are invertible and sparse, but have little other structure. In such problems the solution u=A¹ƒ is typically calculated only to acurately compute functionals of the solution, L(u). This paper determines a method that converges rapidly to the functional's value. Specifially, a modified bi-conjugate gradient algorithm is found to generate convergence to the solution of linear functionals, L(u), much more rapidly than convergence to the linear system solution u