CVODES: The Sensitivity-Enabled ODE Solver in SUNDIALS

2005 ◽  
Author(s):  
Radu Serban ◽  
Alan C. Hindmarsh
Keyword(s):  
System Integration ◽  
Adjoint System ◽  
Problem Solver ◽  
Variable Order ◽  
Initial Value ◽  
Adjoint Sensitivity ◽  
Forward Solution ◽  
Independent Manner ◽  
Differentiation Formula ◽  
Variable Step

CVODES, which is part of the SUNDIALS software suite, is a stiff and nonstiff ordinary differential equation initial value problem solver with sensitivity analysis capabilities. CVODES is written in a data-independent manner, with a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods. It shares with the other SUNDIALS solvers several common modules, most notably the generic kernel of vector operations and a set of generic linear solvers and preconditioners. CVODES solves the IVP by one of two methods — backward differentiation formula or Adams-Moulton — both implemented in a variable-step, variable-order form. The forward sensitivity module in CVODES implements the simultaneous corrector method, as well as two flavors of staggered corrector methods. Its adjoint sensitivity module provides a combination of checkpointing and cubic Hermite interpolation for the efficient generation of the forward solution during the adjoint system integration. We describe the current capabilities of CVODES, its design principles, and its user interface, and provide an example problem to illustrate the performance of CVODES.

scholarly journals An Accurate Block Solver for Stiff Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
H. Musa ◽  
M. B. Suleiman ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Variable Step ◽  
Compute Solution ◽  
The Stability ◽  
Size Mode

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

scholarly journals Adjoint-based exact Hessian computation

2021 ◽  
Author(s):  
Shin-ichi Ito ◽  
Takeru Matsuda ◽  
Yuto Miyatake
Keyword(s):  
Krylov Subspace ◽  
Hessian Matrix ◽  
Adjoint System ◽  
Coefficient Matrix ◽  
Scalar Function ◽  
Second Order ◽  
Subspace Method ◽  
Initial Value ◽  
Research Fields ◽  
Memory Efficiency

AbstractWe consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge–Kutta methods play an essential role.

A variable-step variable-order algorithm for systems of stiff odes

1997 ◽  
Vol 63 (1-2) ◽  
pp. 149-157 ◽  
Author(s):  
T. Van Hecke ◽  
G. Vanden G Berghe ◽  
M. Van Daele ◽  
H De Meyer
Keyword(s):  
Variable Order ◽  
Stiff Odes ◽  
Variable Step ◽  
Order Algorithm

Nonlinear Schrodinger Equation-Based Adjoint Sensitivity Analysis

2021 ◽  
Vol 35 (11) ◽  
pp. 1342-1343
Author(s):  
Mahmoud Maghrabi ◽  
Mohamed Bakr ◽  
Shiva Kumar
Keyword(s):  
Sensitivity Analysis ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Adjoint System ◽  
Nonlinear Schrodinger Equation ◽  
Design Parameters ◽  
Adjoint Sensitivity Analysis ◽  
Adjoint Sensitivity ◽  
Nonlinear Schrödinger

A general nonlinear adjoint sensitivity analysis (ASA) approach for the time-dependent nonlinear Schrodinger equation (NLSE) is presented. The proposed algorithm estimates the sensitivities of a desired objective function with respect to all design parameters using only one extra adjoint system simulation. The approach efficiency is shown here through a numerical example.

ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

2021 ◽  
Vol 5 (2) ◽  
pp. 442-446
Author(s):  
Muhammad Abdullahi ◽  
Hamisu Musa
Keyword(s):  
Initial Value Problem ◽  
Initial Value Problems ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
First Order ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Necessary And Sufficient

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

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