WLS design

Required Functions for WLS

variables:

• nts: width of time stencil, for forward euler nts = 2
• ns: number of time steps in the window
• K: ROM basis size
• N: FOM size

Evauate single residual

This computes the residual at a single step, and is the same as LSPG. Required inputs are:

1. The nts states required for the time discretization, type = rom vectors
2. The time instance (for non-autonomous functions), type=double
3. The time step, type=double

Outputs/required storage is

1. residual, FOM vector of size N

Evauate windowed residual

This computes the residual over the time window. Presumably this will use the "evaluate single residual" function. Required inputs are:

1. The number of time steps, ns, to compute the residual over, type=int
2. The nts + ns - 1 states required for the time discretization, type = rom vectors
3. The time instances (for non-autonomous functions), type=vector of doubles
4. The time steps, type = vector of doubles

Outputs/required storage is

1. windowed residual, extended FOM vector of size N * ns

Evauate single Jacobian

This computes the Jacobian of a single step, and is the same as LSPG Required inputs are:

1. The nts states required for the time discretization, type = rom vectors
2. The time instance (for non-autonomous functions), type=double
3. The time step, type=double
4. The argument of the residual with which to compute the Jacobian of, type=int

Outputs/required storage is

1. Jacobian, FOM matrix of size N * K

Evauate windowed Jacobian

This computes the Jacobian of over the time window. Presumably this will use the "evaluate single Jacobian" function. Required inputs are:

1. The number of time steps, ns, to compute the residual over, type=int
2. The nts + ns - 1 states required for the time discretization, type = rom vectors
3. The time instance (for non-autonomous functions), type=double
4. The time step, type=double
5. The argument of the residual with which to compute the Jacobian of, type=int

Outputs/required storage is

1. Jacobian, extended FOM sparse matrix of size N ns* K ns