Efficient method solves large quadratic parameter linear system families
A single LU~factorization approach to model reduction for linear system families with a quadratic matrix pencil
Computational Engineering, Finance, and Science
Summary
Solving many related large linear systems where the system changes with a parameter in a quadratic way can be very costly. The authors propose a method that builds a special set of approximations at just one chosen parameter value by using derivatives of solutions. This approach only requires factoring a matrix once, saving a lot of time when solving the whole family of systems. This method, called ODB, can greatly reduce the computing needed compared to traditional techniques.
What this means in practice
- •For computational engineers: Solve parameter-dependent linear systems efficiently in simulations with varying conditions using one matrix factorization step.
- •For control system developers: Quickly analyze families of linear control systems whose parameters vary quadratically, reducing computational overhead in design and testing.
Authors
Mikhail A. Botchev, Leonid Knizhnerman, Nikolai L. Zamarashkin, Dmitry A. Zheltkov
Abstract
To solve a family of large linear systems where the system matrix depends on the family parameter quadratically, we propose a new solution approach. Our method employs the derivatives of solutions with respect to the parameter. It turns out that the accumulated solution derivatives of increasing order, evaluated at a single specially chosen parameter value, provide successively better approximations to solution subspaces for the whole linear system family. We show that the derivatives can be computed by a recurrence, involving, at each step, solving a linear system with the same matrix. Constructing an orthonormal basis for the subspace spanned by the derivatives then leads to an efficient solver where just a single (sparse) LU factorization (or a single preconditioner setup in an iterative scheme) suffices to solve the whole linear system family. We call the new method ODB (orthonormal derivative basis method). As shown in numerical tests, compared to conventional solution methods, the ODB method allows to drastically reduce computational costs.