Partitioned Least Squares

In this paper we propose a variant of the linear least squares model allowing practitioners to partition the input features into groups of variables that they require to contribute similarly to the final result. The output allows practitioners to assess the importance of each group and of each variable in the group. We formally show that the new formulation is not convex and provide two alternative methods to deal with the problem: one non-exact method based on an alternating least squares approach; and one exact method based on a reformulation of the problem using an exponential number of sub-problems whose minimum is guaranteed to be the optimal solution. We formally show the correctness of the exact method and also compare the two solutions showing that the exact solution provides better results in a fraction of the time required by the alternating least squares solution (assuming that the number of partitions is small). For the sake of completeness, we also provide an alternative branch and bound algorithm that can be used in place of the exact method when the number of partitions is too large, and a proof of NP-completeness of the optimization problem introduced in this paper

The efficiency factorization multiplier for the Watson efficiency in partitioned linear models: some examples and a literature review

We consider partitioned linear models where the model matrix X = (X1 : X2) has full column rank, and concentrate on the special case whereX0 1X2 = 0 when we say that the model is orthogonally partitioned. We assume that the underlying covariance matrix is positive definite and introduce the efficiency factorization multiplier which relates the total Watson efficiency of ordinary least squares to the product of the two subset Watson efficiencies. We illustrate our findings with several examples and present a literature review

On a Partitioned Inversion Formula having Useful Applications in Econometrics

In this paper a novel partitioned inversion formula is obtained in terms of the orthogonal complements of off-diagonal blocks, with the emblematic matrix of unit-root econometrics springing up as the leading diagonal block of the inverse. On the one hand, the result paves the way to a stimulating reinterpretation of restricted least-squares estimation and, on the other, to a straightforward derivation of a key-result of time-series econometrics.Partitioned inversion; Restricted least-squares; VAR econometrics

More on partitioned possibly restricted linear regression

This paper deals with the general partitioned linear regression model where the regressor matrix X=\pmatrix{X_1 & X_2\cr} may be deficient in column rank, the dispersion matrix $V$ is possibly singular, \beta^t=\pmatrix{\beta_1^t & \beta_2^t\cr} - being partitioned according to $X$ - is the vector of unknown regression coefficients, and $\beta_2$ is possibly subject to consistent linear equality or inequality restrictions. In particular, we are interested in the set of {\it generalized least squares (GLS) selections} for $\beta_2$. Inspired by Aigner and Balestra [1], as well as by Nurhonen and Puntanen [2], we also consider a specific reduced model and describe a scenario under which the set of GLS selections for $\beta_2$ under the reduced model equals the set of GLS selections for $\beta_2$ under the original full model. The results obtained in [2] and [1] for the unrestricted {\it standard} (full rank) regression model are reobtained as special cases.Gauss-Markov model, singular model, perfect multicollinearity, partitioned linear regression, linear equality constraints, linear inequality constraints, constrained generalized least squares selections, oblique projectors, generalized inverses.

Coupling techniques for partitioned fluid-structure interaction simulations with black-box solvers

In partitioned simulations of fluid‐structure interaction, the flow and the displacement of the structure are calculated separately and coupling iterations between the flow solver and the structural solver are required to calculate the solution of the coupled problem if the interaction is strong. This work is a comparison of three coupling algorithms which use the flow solver and structural solver as a “black box”. Consequently, these algorithms are suitable for implementation in future versions of MpCCI. It is demonstrated that the algorithm of the interface quasi‐Newton technique with an approximation for the inverse of the Jacobian from a least‐squares model is straightforward and that this technique needs a relatively low number of coupling iterations in the simulation of an oscillating flexible beam and the propagation of a pressure wave in a flexible tube

A partitioned quasi-newton solution technique for fluid-structure interaction problems using a coarsened grid to accelerate the convergence of the coupling iterations

Previous stability analyses on Gauss-Seidel coupling iterations in partitioned fluid-structure interaction simulations have demonstrated that Fourier modes with a low wave-number in the difference between the current and correct interface displacement are unstable. To stabilize these modes, the IQN-ILS technique automatically constructs a least-squares model of the flow solver and structural solver. In this work, the multi-level IQN-ILS technique (ML-IQN-ILS) is presented, which uses a coarsened grid of the fluid and structure subdomains to initialize this least-squares model. As the modes that need to be present in this least-squares model have a low wave-number, they can be resolved on a coarsened grid. Therefore, in each time step, a number of cheap coupling iterations is first performed on the coarsened grid to construct the model, followed by a smaller number of coupling iterations on the fine grid. As the iterations on the coarse grid are fast and fewer iterations are performed on the fine grid, the total duration of the simulation decreases compared to a simulation on the fine grid only

Optimization of a piezoelectric fan using fluid-structure interaction simulation

In this paper, the heat transfer from a single heat fin to the air flow in the wake of a piezoelectric fan (piezofan) is optimised. Both the heat fin and the piezofan are positioned in a channel, which has a significant influence on the flow field. The design variable is the frequency of the voltage applied to the piezofan. The heat transfer for different excitation frequencies is calculated using unsteady fluid-structure interaction simulations. To obtain a modular simulation environment, the flow equations and the structural equations are solved separately. However, the equilibrium on the fluid-structure interface is not satisfied automatically in this partitioned approach. Therefore, the interface quasi-Newton technique with an approximation for the inverse of the Jacobian from a least-squares model (IQN-ILS) is used to perform coupling iterations between the flow solver and the structural solver in each time step. With the unsteady fluid-structure interaction model, a surrogate model is constructed. The optimization of the surrogate model yields a frequency close to the first eigenfrequency of the structure