In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for generic regular flow maps if and only if the observation matrix H satisfies detectability conditions: the rank of H must be at least as great as the number of nonnegative Lyapunov exponents of the underlying attractor. Numerical experiments illustrate the exponential convergence of the method and the sharpness of the theory for the case of Lorenz '96 and Burgers equations with incomplete and noisy observations.
A new concept in intraoperative decision support for tumour delineation is proposed showing that artificial intelligence provides categorising information and interpretation from the images captured during fluorescence-guided colorectal cancer operations. This is potentially applicable to all cancer subtypes and is pertinent to new fluorophore development.A new concept in intraoperative decision support for tumour delineation is proposed showing that artificial intelligence provides categorising information and interpretation from the images captured during fluorescence-guided colorectal cancer operations. This is potentially applicable to all cancer subtypes and is pertinent to new fluorophore development. A new concept in intraoperative decision support for tumour delineation is proposed showing that artificial intelligence provides categorising information and interpretation from the images captured during fluorescence-guided colorectal cancer operations. This is potentially applicable to all cancer subtypes and is pertinent to new fluorophore development.
Shows promise
Abstract. In this paper we propose a state estimation method for linear parabolic partial differential equations (PDE) that accounts for errors in the model, truncation, and observations. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a differential-algebraic equation (DAE). The original estimation problem for the PDE is then recast as a state estimation problem for the constructed DAE using a linear continuous minimax filter. We construct a numerical time integrator that preserves the monotonic decay of a nonstationary Lyapunov function along the solution. To conclude we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a 2D fluid.
This paper introduces a new computational methodology for determining aposteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finiteelement spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multiobjective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multiobjective error in adaptive refinement procedures.
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