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Register a new userInterval Superposition Arithmetic for Guaranteed Parameter Estimation
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The problem of guaranteed parameter estimation (GPE) consists in enclosing the set of all possible parameter values, such that the model predictions match the corresponding measurements within prescribed error bounds. One of the bottlenecks in GPE algorithms is the construction of enclosures for the image-set of factorable functions. In this paper, we introduce a novel set-based computing method called interval superposition arithmetics (ISA) for the construction of enclosures of such image sets and its use in GPE algorithms. The main benefits of using ISA in the context of GPE lie in the improvement of enclosure accuracy and in the implied reduction of number set-membership tests of the set-inversion algorithm.
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State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements, when the parameter vector $y$ is unknown. Fast numerical recovery methods have been proposed based on reduced models which are linear spaces of moderate dimension $n$ which are tailored to approximate the solution manifold $mathcal{M}$ where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches, and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m(mathcal{M})$ of the solution manifold. In this paper we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces $V_k$, each having dimension at most $m$ and leading to different estimators $u_k^*$. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator $u^*$. Our analysis shows that $u^*$ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator $y^*$ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation, that potentially improves the quality of both estimators.
If the non-zero finite floating-point numbers are interpreted as point intervals, then the effect of rounding can be interpreted as computing one of the bounds of the result according to interval arithmetic. We give an interval interpretation for the signed zeros and infinities, so that the undefined operations 0*inf, inf - inf, inf/inf, and 0/0 become defined. In this way no operation remains that gives rise to an error condition. Mathematically questionable features of the floating-point standard become well-defined sets of reals. Interval semantics provides a basis for the verification of numerical algorithms. We derive the results of the newly defined operations and consider the implications for hardware implementation.
Estimating parameters of Partial Differential Equations (PDEs) is of interest in a number of applications such as geophysical and medical imaging. Parameter estimation is commonly phrased as a PDE-constrained optimization problem that can be solved iteratively using gradient-based optimization. A computational bottleneck in such approaches is that the underlying PDEs needs to be solved numerous times before the model is reconstructed with sufficient accuracy. One way to reduce this computational burden is by using Model Order Reduction (MOR) techniques such as the Multiscale Finite Volume Method (MSFV). In this paper, we apply MSFV for solving high-dimensional parameter estimation problems. Given a finite volume discretization of the PDE on a fine mesh, the MSFV method reduces the problem size by computing a parameter-dependent projection onto a nested coarse mesh. A novelty in our work is the integration of MSFV into a PDE-constrained optimization framework, which updates the reduced space in each iteration. We also present a computationally tractable way of differentiating the MOR solution that acknowledges the change of basis. As we demonstrate in our numerical experiments, our method leads to computational savings particularly for large-scale parameter estimation problems and can benefit from parallelization.
We consider a global variable consensus ADMM algorithm for solving large-scale PDE parameter estimation problems asynchronously and in parallel. To this end, we partition the data and distribute the resulting subproblems among the available workers. Since each subproblem can be associated with different forward models and right-hand-sides, this provides ample options for tailoring the method to different applications including multi-source and multi-physics PDE parameter estimation problems. We also consider an asynchronous variant of consensus ADMM to reduce communication and latency. Our key contribution is a novel weighting scheme that empirically increases the progress made in early iterations of the consensus ADMM scheme and is attractive when using a large number of subproblems. This makes consensus ADMM competitive for solving PDE parameter estimation, which incurs immense costs per iteration. The weights in our scheme are related to the uncertainty associated with the solutions of each subproblem. We exemplarily show that the weighting scheme combined with the asynchronous implementation improves the time-to-solution for a 3D single-physics and multiphysics PDE parameter estimation problems.
We present the library Moore, which implements Interval Arithmetic in modern C++. This library is based on a new feature in the C++ language called concepts, which reduces the problems caused by template meta programming, and leads to a new approach for implementing interval arithmetic libraries in C++.
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