MALA updates

The Metropolis-adjusted Langevin algorithm (MALA) updates continuous auxiliary fields using the action derivative to guide a Gaussian proposal, then applies a Metropolis-Hastings accept/reject step. The correction makes the target distribution exact for a finite proposal time step; unlike an unadjusted Langevin update, the time step is a tuning parameter rather than a source of integration bias.

ALF provides three MALA scopes:

Scope	Use it for	Main switch
Sequential	Field-by-field updates during a sequential sweep	`Sequential_MALA = .T.`
Global tau	A selected group of fields on one imaginary-time slice	`Global_tau_MALA_moves = .T.`
Global	A selected group of fields across the full space-time configuration	`Global_MALA_moves = .T.`

All MALA updates are for real, continuous ALF fields (type 3). In particular, the selection routines used by global MALA moves must select only fields that can be updated continuously.

What ALF proposes¶

For a selected field, ALF constructs the drift

D = F_0 + \frac{\operatorname{Re}\!\left(\texttt{Phase}\,F_\mathrm{F}\right)} {\operatorname{Re}\!\left(\texttt{Phase}\right)},

(1)

where $F_0 = \partial S_0/\partial s$ is the bosonic-action derivative and $F_\mathrm{F}$ is the fermionic force. It proposes

s' = s - D\,\delta t_\mathrm{run} + \sqrt{2\delta t_\mathrm{run}}\,\eta, \qquad \eta \sim \mathcal{N}(0,1).

(2)

Because this proposal is asymmetric, ALF includes the forward and reverse Gaussian proposal densities in the Metropolis-Hastings test:

P_\mathrm{acc} = \min\!\left( 1, \frac{T(C' \rightarrow C)\,W(C')} {T(C \rightarrow C')\,W(C)} \right).

(3)

The acceptance step is important: do not use MALA terminology interchangeably with ALF’s unadjusted Langevin = .T. mode.

Adaptive time step and `Max_Force`¶

Each MALA variant has its own nominal time step and force threshold. ALF does not clip the force. Instead, for the largest selected force magnitude $X_\mathrm{max}$ , it uses

\delta t_\mathrm{run} = \delta t\, \min\!\left(1, \frac{\texttt{Max\_Force\_MALA}}{X_\mathrm{max}} \right).

(4)

The running time step is evaluated at both ends of the proposal and included in the proposal-density ratio. Consequently, adaptive stepping remains part of the MALA proposal rather than a force cutoff. A small threshold only makes proposals shorter; it cannot make singular or unbounded forces reliable.

Quick start: sequential MALA¶

Sequential MALA is the most direct option for an existing continuous-field simulation. Put the following in the &VAR_QMC namelist of parameters:

&VAR_QMC
   Sequential                 = .T.
   Sequential_MALA            = .T.
   Delta_t_MALA_sequential    = 0.01d0
   Max_Force_MALA_sequential  = 1.0d0
   NSweep                     = 100
   NBin                       = 100
/

ALF applies the MALA proposal to type-3 fields in the sequential operator range. Nt_sequential_start and Nt_sequential_end can restrict that range when required by the Hamiltonian.

Always explicitly set both MALA parameters. In this checkout, Delta_t_MALA_sequential defaults to zero, and ALF terminates if an enabled MALA mode has a non-positive step size or force threshold.

The Hamiltonian must provide the bosonic force. For sequential MALA, provide the efficient single-field hook Ham_Langevin_HMC_S0_single(Force_0, n, nt). If it is absent, ALF falls back to Ham_Langevin_HMC_S0(Forces_0), extracts one element, and emits a warning because evaluating the full force array for every field is expensive.

Global-tau MALA¶

Global-tau MALA makes a joint proposal for fields on one time slice. It runs within the sequential update path, so Sequential = .T. is required.

&VAR_QMC
   Sequential                      = .T.
   Global_tau_MALA_moves           = .T.
   N_Global_tau_MALA               = 1
   Delta_t_MALA_global_tau         = 0.01d0
   Max_Force_MALA_global_tau       = 1.0d0
/

N_Global_tau_MALA is the number of MALA proposals made on each visited time slice. A worked configuration example is available in A Jupyter notebook for: Metropolis, MALA, HMC and Langevin updates.. Implement the following Hamiltonian hook to choose the fields in a proposal:

subroutine Global_MALA_move_tau(Flip_list, Flip_length, ntau)
   integer, intent(out) :: Flip_list(:)
   integer, intent(out) :: Flip_length
   integer, intent(in) :: ntau

   ! Set Flip_list(1:Flip_length) to type-3 operator indices at ntau.
end subroutine Global_MALA_move_tau

The base implementation selects every operator index and prints a warning. Override it when only a subset is appropriate. The Hamiltonian can also override N_Global_tau_MALA through Overide_global_tau_sampling_parameters.

A worked pyALF example is available in A Jupyter notebook for: Metropolis, MALA, HMC and Langevin updates..

Global MALA¶

Global MALA proposes a selected set of fields across all space and imaginary time, followed by one accept/reject decision for the full proposal. It does not require Sequential = .T..

&VAR_QMC
   Global_MALA_moves          = .T.
   N_Global_MALA_sweeps       = 1
   Delta_t_MALA_global        = 0.01d0
   Max_Force_MALA_global      = 1.0d0
/

N_Global_MALA_sweeps is the number of global MALA proposals performed per simulation sweep. Select the fields in the Hamiltonian with:

subroutine Global_MALA_move(Flip_list)
   integer, intent(out) :: Flip_list(:,:)

   ! Use 1 for selected type-3 fields and 0 for fields left unchanged.
end subroutine Global_MALA_move

The base implementation marks every field. A custom selector is useful when a full space-time proposal is too large or mixes different field types. The global version needs the full bosonic-force routine Ham_Langevin_HMC_S0(Forces_0).