13,071 research outputs found
Randomized Dimension Reduction on Massive Data
Scalability of statistical estimators is of increasing importance in modern
applications and dimension reduction is often used to extract relevant
information from data. A variety of popular dimension reduction approaches can
be framed as symmetric generalized eigendecomposition problems. In this paper
we outline how taking into account the low rank structure assumption implicit
in these dimension reduction approaches provides both computational and
statistical advantages. We adapt recent randomized low-rank approximation
algorithms to provide efficient solutions to three dimension reduction methods:
Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and
Localized Sliced Inverse Regression (LSIR). A key observation in this paper is
that randomization serves a dual role, improving both computational and
statistical performance. This point is highlighted in our experiments on real
and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized
eigendecompositon, low-rank, supervised, inverse regression, random
projections, randomized algorithms, Krylov subspace method
Matrix-interpolation-based parametric model order reduction for multiconductor transmission lines with delays
A novel parametric model order reduction technique based on matrix interpolation for multiconductor transmission lines (MTLs) with delays having design parameter variations is proposed in this brief. Matrix interpolation overcomes the oversize problem caused by input-output system-level interpolation-based parametric macromodels. The reduced state-space matrices are obtained using a higher-order Krylov subspace-based model order reduction technique, which is more efficient in comparison to the Gramian-based parametric modeling in which the projection matrix is computed using a Cholesky factorization. The design space is divided into cells, and then the Krylov subspaces computed for each cell are merged and then truncated using an adaptive truncation algorithm with respect to their singular values to obtain a compact common projection matrix. The resulting reduced-order state-space matrices and the delays are interpolated using positive interpolation schemes, making it computationally cheap and accurate for repeated system evaluations under different design parameter settings. The proposed technique is successfully applied to RLC (R-resistor, L-inductor, C-capacitance) and MTL circuits with delays
Structure Preserving Model Reduction of Parametric Hamiltonian Systems
While reduced-order models (ROMs) have been popular for efficiently solving
large systems of differential equations, the stability of reduced models over
long-time integration is of present challenges. We present a greedy approach
for ROM generation of parametric Hamiltonian systems that captures the
symplectic structure of Hamiltonian systems to ensure stability of the reduced
model. Through the greedy selection of basis vectors, two new vectors are added
at each iteration to the linear vector space to increase the accuracy of the
reduced basis. We use the error in the Hamiltonian due to model reduction as an
error indicator to search the parameter space and identify the next best basis
vectors. Under natural assumptions on the set of all solutions of the
Hamiltonian system under variation of the parameters, we show that the greedy
algorithm converges with exponential rate. Moreover, we demonstrate that
combining the greedy basis with the discrete empirical interpolation method
also preserves the symplectic structure. This enables the reduction of the
computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy,
and stability of this model reduction technique is illustrated through
simulations of the parametric wave equation and the parametric Schrodinger
equation
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