odeSensitivity
Description
Use odeSensitivity
objects to perform sensitivity analysis on a
system of ordinary differential equations (ODEs) or differential algebraic equations (DAEs).
Sensitivity analysis examines how changes in the values of parameters in the differential
equations affect the calculated solutions. If an equation is sensitive to the value of a
particular parameter, then small changes in the parameter value can produce large changes in
the solution.
Create an ode
object to
represent the ODE or DAE problem, and specify an odeSensitivity
object as the
value of the Sensitivity
property to perform sensitivity analysis.
Creation
Description
creates an
S
= odeSensitivityodeSensitivity
object with empty properties.
specifies one or more property values using name-value arguments. For example, you can
specify the equation parameters to include in the sensitivity analysis by using the
S
= odeSensitivity(Name=Value
)ParameterIndices
property.
Properties
ParameterIndices
— Parameter indices for sensitivity analysis
scalar index | vector of indices
Parameter indices for sensitivity analysis, specified as a scalar index or vector of
indices. ParameterIndices
contains indices into the numeric
Parameters
property of the ode
object to specify
which parameters to include in the sensitivity analysis.
If you do not specify a value for ParameterIndices
, then the
ode
object performs sensitivity analysis on all parameters.
Example:
S = odeSensitivity(ParameterIndices=1:2)
specifies that the first and
second parameters are used for sensitivity analysis.
InitialValue
— Initial sensitivity of solution
matrix
Initial sensitivity of solution, specified as a matrix. The matrix has one row for
each equation in the system, and one column for each sensitivity parameter as defined by
the ParameterIndices
property. Each (i,j)
value
in the matrix specifies the initial sensitivity of equation i
with
respect to sensitivity parameter j
.
If you do not specify a value for InitialValue
, then the
ode
object uses a matrix of zeros.
Example: S = odeSensitivity(InitialValue=M)
specifies a matrix
M
of initial sensitivities.
InitialSlope
— Initial slope for DAE problems
vector
Initial slope for DAE problems, specified as a vector. Use this property with the
idas
solver when solving DAEs. The vector specifies the initial
slope ds/dt
for each sensitivity parameter as defined by the
ParameterIndices
property.
If the specified initial slopes are not consistent with the
InitialTime
, InitialValue
, and
MassMatrix
properties of the ode
object, then
the solver treats the slopes as guesses and attempts to compute consistent values for
the initial slopes that are close to the guesses before continuing to solve the
problem.
If you do not specify a value for InitialSlope
, then the
ode
object uses a vector of zeros.
Example:
S = odeSensitivity(InitialSlope=[0.1 0.2])
specifies initial slopes
for two sensitivity parameters.
Jacobian
— Jacobian matrix with respect to parameters
odeJacobian
object | matrix | function handle
Jacobian matrix with respect to parameters, specified as an odeJacobian
object, matrix, or handle to a function that evaluates the Jacobian. The Jacobian is a
matrix of partial derivatives of the functions that define the system of differential
equations with respect to one or more parameters defined in the
Parameters
property of the ode
object. The
Jacobian matrix has one row for each equation in the system and one column for each
sensitivity parameter as defined by the ParameterIndices
property.
The Jacobian is used for all sensitivity analyses, regardless of solver choice. If you do not provide the Jacobian, then the ODE solver approximates it numerically using finite differences.
For large systems of equations where it is not feasible to provide the entire analytic Jacobian, you can specify the sparsity pattern of the Jacobian matrix instead. The solver uses the sparsity pattern to calculate a sparse Jacobian.
You can specify the value of the Jacobian
property as:
An
odeJacobian
object, which can represent either the Jacobian matrix or its sparsity pattern.A constant matrix with calculated values for the Jacobian elements.
A handle to a function that computes the matrix elements and that accepts two input arguments,
dfdp = Fjac(t,y)
. To give the function access to parameter values in theParameters
property, specify a third input argument in the function definition,dfdp = Fjac(t,y,p)
.
If you specify a matrix or function handle, then MATLAB® converts it to an odeJacobian
object.
Example: S = odeSensitivity(Jacobian=@Fjac)
specifies the function
Fjac
that evaluates the Jacobian matrix.
Example: S = odeSensitivity(Jacobian=[0 1; -2 1])
specifies a
constant Jacobian matrix.
Example: S =
odeSensitivity(Jacobian=odeJacobian(SparsityPattern=S))
specifies the
Jacobian sparsity pattern using sparse matrix S
.
Examples
Examine Parameter Sensitivity
Solve an ODE system with two equations and two parameters, and perform sensitivity analysis on the parameters.
Create an ode
object to represent this system of equations.
Specify the initial conditions as and , and parameter values of and . To enable sensitivity analysis of the parameters, set the Sensitivity
property of the ode
object to an odeSensitivity
object.
p = [0.05 1.5]; F = ode(ODEFcn=@(t,y,p) [p(1)*y(1)-y(2); -p(2)*y(2)], ... InitialValue=[2 3], ... Parameters=p, ... Sensitivity=odeSensitivity)
F = ode with properties: Problem definition ODEFcn: @(t,y,p)[p(1)*y(1)-y(2);-p(2)*y(2)] Parameters: [0.0500 1.5000] InitialTime: 0 InitialValue: [2 3] Sensitivity: [1x1 odeSensitivity] EquationType: standard Solver properties AbsoluteTolerance: 1.0000e-06 RelativeTolerance: 1.0000e-03 Solver: auto SelectedSolver: cvodesnonstiff Show all properties
Because the equations are nonstiff and sensitivity analysis is enabled, the ode
object automatically chooses the cvodesnonstiff
solver for this problem.
Solve the ODE over the time interval [0 5]
, and plot the solution for each component.
S = solve(F,0,5)
S = ODEResults with properties: Time: [0 2.9540e-09 2.9543e-05 2.2465e-04 4.1976e-04 0.0024 0.0080 0.0137 0.0245 0.0353 0.0611 0.0869 0.1499 0.2129 0.3169 0.4208 0.5248 0.7204 0.9161 1.1118 1.3075 1.5031 1.6988 1.8945 2.0901 2.2858 2.4815 2.6772 2.8728 ... ] (1x40 double) Solution: [2x40 double] Sensitivity: [2x2x40 double]
plot(S.Time,S.Solution(1,:),"-o",S.Time,S.Solution(2,:),"-o") legend("y1","y2")
The values in S.Sensitivity
are partial derivatives of the equations with respect to the parameters. To examine the effects of the parameter values during the integration, plot the sensitivity values.
figure hold on plot(S.Time,squeeze(S.Sensitivity(1,1,:)),"-o") plot(S.Time,squeeze(S.Sensitivity(1,2,:)),"-o") plot(S.Time,squeeze(S.Sensitivity(2,1,:)),"-o") plot(S.Time,squeeze(S.Sensitivity(2,2,:)),"-o") legend("p1,eq1","p2,eq1","p1,eq2","p2,eq2") hold off
Version History
Introduced in R2024a
See Also
ode
| odeJacobian
| odeMassMatrix
| odeEvent
| ODEResults
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