Directed loop algorithm

The loop, worm and directed loop algorithms can be used for the simulation of a wide class of quantum magnets. They are of interest not only to theoretical physicists, but also to experimentalists who want to fit experimental measurements to theoretical models. The wide applicability of these methods has led to the publication of open-source versions of these algorithms as part of the ALPS project (Algorithms and Libraries for Physics Simulations) [10] on the web page http //alps.comp-phys.org/. 

The worm and directed loop algorithms are applied when magnetic fields are present. Typical examples include the calculation of magnetization curves of quantum magnets [74], the determination of the first order nature of the spin flop transition in two dimensions [75] and the calculation of phase diagrams of dimerized quantum magnets in a magnetic field [76]. 

Inserting complete sets of basis states between the different Hi factors then leads to a similar representation of the partition function and a similar world-line picture as in the world-line Monte Carlo method. Because there is no Trotter decomposition involved, the method is free of time discretization errors from the outset. Early applications of the SSE method employed local updates, but more efficient cluster-type updates have been developed more recently to overcome the critical slowing down. They include the operator-loop update and the previously mentioned directed-loop algorithm. ... 

Worm and Directed Loop Updates The Loop Algorithm in a Magnetic Field... 

Besides size consistency, pair approaches have always exploited special features of two-electron functions as mentioned at the beginning of this review. However, a remarkable convergence of methods has taken place in this respect. Efficient matrix-oriented direct Cl algorithms—which avoid logic in inner loops and are well suited for vector computers —were first... 

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

For the iteration algorithm (5) the optimal estimations (6) are directly used by a second back loop to block B (long dashed line in Fig. 1). 

The procedure Split selects the internal displacement coordinates, q, and momenta, tt, (describing vibrations), the coordinates, r, and velocities, v, of the centers of molecular masses, angular velocities, a>, and directional unit vectors, e, of the molecules from the initial Cartesian coordinates, q, and from momenta, p. Thus, the staring values for algorithm loop are prepared. Step 1 Vibration... 

The performance and large-scale feasibility of a TN algorithm depend on the precise formulation of a truncation criterion and implementation of the inner PCG loop. The PCG process can be terminated when either one of the following conditions is satisfied (1) the residual rk is sufficiently small, (2) the quadratic model qk p ) as defined in Eq. [35] is sufficiently reduced, or (3) a direction of negative curvature d is encountered (i.e., < 0). A negative... 

This paper presents the application of a model based predictive control strategy for the primary stage of the freeze drying process, which has not been tackled until now. A model predictive control framework is provided to minimize the sublimation time. The problem is directly addressed for the non linear distributed parameters system that describes the dynamic of the process. The mathematical model takes in account the main phenomena, including the heat and mass transfer in both the dried and frozen layers, and the moving sublimation front. The obtained results show the efficiency of the control software developed (MPC CB) under Matlab. The MPC( CB based on a modified levenberg-marquardt algorithm allows to control a continuous process in the open or closed loop and to find the optimal constrained control. 

This chapter reports experimental and analytical results for pulsed control of combustion instabilities at both fundamental and subharmonic frequencies. Two suites of control algorithms have been developed one based on least-mean-square (LMS) techniques that is suitable for inner-loop stabilization of combustion instabilities, and one bcised on direct optimization that can be used either for stabilization or outer-loop optimization of combustion process objectives, such as flame compactness or emissions. 

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