Probabilistic interpretation of conjugate gradient iterations in spectral stochastic finite element method

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[1] [1,Ghosh, Debraj
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Ghosh, D. (dghosh@civil.iisc.ernet.in) | 1600年 / AIAA International卷 / 52期
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The spectral stochastic finite element method (SSFEM) is one of the popular choices for analyzing uncertain systems; mainly due to its computational speed. It is shown that; although the successive iterates are conjugate to each other in the product space; the orthogonality of the successive residuals holds only in an inner product defined deterministically; that is; without involving an expectation operator. These observations imply that the existing result on orthogonality of the residuals in the deterministic CG method does not directly extend to a stochastic case. Some of the properties of the CG in a deterministic setting are found to have their stochastic counterparts. For instance; conjugacy of the search directions and monotonic nonincreasing property of the error also hold in the product space; with the usual inner product and norm; whereas the orthogonality of the residual does not hold in the usual inner product in the product space. However; here the orthogonality holds with a different norm that does not involve the expectation operator;
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