
Computing Spectral Measures and Spectral Types: New Algorithms and Classifications
Despite new results on computing the spectrum, there has been no general...
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Improved spectral convergence rates for graph Laplacians on epsilongraphs and kNN graphs
In this paper we improve the spectral convergence rates for graphbased ...
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Stabilization of infinitedimensional linear control systems by POD reducedorder Riccati feedback
There exist many ways to stabilize an infinitedimensional linear autono...
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Convergence rates and structure of solutions of inverse problems with imperfect forward models
The goal of this paper is to further develop an approach to inverse prob...
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L^∞ norm error estimates for HDG methods applied to the Poisson equation with an application to the Dirichlet boundary control problem
We prove quasioptimal L^∞ norm error estimates (up to logarithmic facto...
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On the Convergence of Tsetlin Machines for the IDENTITY and NOT Operators
The Tsetlin Machine (TM) is a recent machine learning algorithm with sev...
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Spectral convergence of probability densities
The computation of probability density functions (PDF) using approximate...
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On the infinitedimensional QR algorithm
Spectral computations of infinitedimensional operators are notoriously difficult, yet ubiquitous in the sciences. Indeed, despite more than half a century of research, it is still unknown which classes of operators allow for computation of spectra and eigenvectors with convergence rates and error control. Recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation, and no convergence rates nor error control is possible. This begs the question: which classes of operators allow for computations with convergence rates and error control? In this paper we address this basic question, and the algorithm used is an infinitedimensional version of the QR algorithm. Indeed, we generalise the QR algorithm to infinitedimensional operators. We prove that not only is the algorithm executable on a finite machine, but one can also recover the extremal parts of the spectrum and corresponding eigenvectors, with convergence rates and error control. This allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture. The algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches (that are notorious for providing false solutions).We also find that in some cases the IQR algorithm performs better than predicted by theory and make conjectures for future study.
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