293,321 research outputs found

    On optimal solution error covariances in variational data assimilation problems

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    The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection-diffusion model

    Optimal solution error covariance in highly nonlinear problems of variational data assimilation

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    The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem (see, e.g.[1]) to find the initial condition, boundary conditions or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can be approximated by the inverse Hessian of the cost functional of an auxiliary data assimilation problem ([2], [3]). The relationship between the optimal solution error covariance matrix and the Hessian of the auxiliary control problem is discussed for different degrees of validity of the tangent linear hypothesis. For problems with strongly nonlinear dynamics a new statistical method based on computation of a sample of inverse Hessians is suggested. This method relies on the efficient computation of the inverse Hessian by means of iterative methods (Lanczos and quasi-Newton BFGS) with preconditioning. The method allows us to get a sensible approximation of the posterior covariance matrix with a small sample size. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term. The first author acknowledges the funding through the project 09-01-00284 of the Russian Foundation for Basic Research, and the FCP program "Kadry"

    Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs

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    Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Gal\'{a}n recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion channel models, such as the Hodgkin-Huxley or other conductance based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. We consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Gal\'{a}n's approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.Comment: Added one reference, typos corrected in Equation 6 and Appendix C, added the assumption that the graph is irreducible to the main theorem (results unchanged

    An analysis of the practical DPG method

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    In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree pp on each mesh element. Earlier works showed that there is a "trial-to-test" operator TT, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator TT is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply TT. In practical computations, TT is approximated using polynomials of some degree r>pr > p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that rp+Nr\ge p+N, where NN is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.Comment: Mathematics of Computation, 201

    Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

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    Petrov–Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. A concise proposal on how to overcome these shortcomings has been raised during the last decade by the Discontinuous Petrov–Galerkin (DPG) methodology. However, DPG has also some limitations and difficulties: the method requires ultraweak variational formulations, obtained through a hybridization process, which is not trivial to implement at the discrete level. Our motivation is to offer a simpler alternative for the case of parametric PDEs, which can be used with any variational formulation. Indeed, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently in an online stage, for a given range of parameters. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE parameter, to the matrix of coefficients of optimal test functions (in some basis expansion) associated with that PDE parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE parameters). When solving online the resulting (compressed) Petrov–Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix–vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure is as fast as an (unstable) Galerkin approach. We illustrate our findings by means of 2D–3D Eriksson–Johnson problems, together with 2D Helmholtz equation

    θ-D Approximation Technique for Nonlinear Optimal Speed Control Design of Surface-Mounted PMSM Drives

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    This paper proposes nonlinear optimal controller and observer schemes based on a θ-D approximation approach for surface-mounted permanent magnet synchronous motors (PMSMs). By applying the θ-D method in both the controller and observer designs, the unsolvable Hamilton–Jacobi–Bellman equations are switched to an algebraic Riccati equation and statedependent Lyapunov equations (SDLEs). Then, through selecting the suitable coefficient matrices, the SDLEs become algebraic, so the complex matrix operation technique, i.e., the Kronecker product applied in the previous papers to solve the SDLEs is eliminated. Moreover, the proposed technique not only solves the problem of controlling the large initial states, but also avoids the excessive online computations. By utilizing a more accurate approximation method, the proposed control system achieves superior control performance (e.g., faster transient response, more robustness under the parameter uncertainties and load torque variations) compared to the state-dependent Riccati equation-based control method and conventional PI controlmethod. The proposed observer-based control methodology is tested with an experimental setup of a PMSM servo drive using a Texas Instruments TMS320F28335 DSP. Finally, the experimental results are shown for proving the effectiveness of the proposed control approac

    θ-D Approximation Technique for Nonlinear Optimal Speed Control Design of Surface-Mounted PMSM Drives

    Get PDF
    This paper proposes nonlinear optimal controller and observer schemes based on a θ-D approximation approach for surface-mounted permanent magnet synchronous motors (PMSMs). By applying the θ-D method in both the controller and observer designs, the unsolvable Hamilton–Jacobi–Bellman equations are switched to an algebraic Riccati equation and statedependent Lyapunov equations (SDLEs). Then, through selecting the suitable coefficient matrices, the SDLEs become algebraic, so the complex matrix operation technique, i.e., the Kronecker product applied in the previous papers to solve the SDLEs is eliminated. Moreover, the proposed technique not only solves the problem of controlling the large initial states, but also avoids the excessive online computations. By utilizing a more accurate approximation method, the proposed control system achieves superior control performance (e.g., faster transient response, more robustness under the parameter uncertainties and load torque variations) compared to the state-dependent Riccati equation-based control method and conventional PI controlmethod. The proposed observer-based control methodology is tested with an experimental setup of a PMSM servo drive using a Texas Instruments TMS320F28335 DSP. Finally, the experimental results are shown for proving the effectiveness of the proposed control approac
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