with being the i-th eigenvalue of matrix One of the main features of the POD basis is its D. λi orthonormality, which means that . Given a computed displacement field , thanks to the orthonormality of the basis, the reduced representation in terms of amplitudes can be evaluated inverting (A.1) by the expression: (A.5) As regards the choice of the number of significant it is also possible to prove that the K overall error may be expressed by the relation: E (A.6) and so, dividing by and taking the square root, we obtain the root-mean-square error of the snapshots in a non-dimensional form: (A.7) where the equality comes from (A.6) considering that, for the samples considered, taking M all modes means expressing the displacement fields in just another different basis, leading M to E=0. For a different set of samples, the expression for (A.7) is not applicable, and so: P s (A.8) Acknowledgements The first author is grateful to Ms Deborah Agbedjro from King’s College London for the fruitful discussions about the relationship between the variances of the displacement field and experimental data. References Antonov, I. A. & Saleev, V. M., 1979. An economic method of computing LP tau-sequence. 19(1), pp. 252-256. USSR Computational Mathematics and Mathematical Physics, Artyukhin, E. A., 1985. Experimental Design of Measurement of the Solution of Inverse Heat Conduction Problem. 48(3), pp. 372-376. Journal of Engineering Physics, Balk, P., 2013. A priori information and admissible complexity of the model of anomalous objects in the solution of inverse problems of gravity. Izvestiya, Physics of the Solid Earth, 49(2), pp. 165-176. Barbone, P. & Bamber, J., 2002. Quantitative elasticity imaging: What can and cannot be inferred from strain images. 47(12), pp. 2147-2164. Physics in Medicine and Biology,
with λi being the i-th eigenvalue of matrix D. One of the main features of the POD basis is its orthonormality, which means that . Given a computed displacement field , thanks to the orthonormality of the basis, the reduced representation in terms of amplitudes can be evaluated inverting (A.1) by the expression: (A.5) As regards the choice of the number K of significant it is also possible to prove that the overall error E may be expressed by the relation: (A.6) and so, dividing by and taking the square root, we obtain the root-mean-square error of the snapshots in a non-dimensional form: (A.7) where the equality comes from (A.6) considering that, for the M samples considered, taking all modes means expressing the M displacement fields in just another different basis, leading to E=0. For a different set of P samples, the expression for s (A.7) is not applicable, and so: (A.8) Acknowledgements The first author is grateful to Ms Deborah Agbedjro from King’s College London for the fruitful discussions about the relationship between the variances of the displacement field and experimental data. References Antonov, I. A. & Saleev, V. M., 1979. An economic method of computing LP tau-sequence. USSR Computational Mathematics and Mathematical Physics, 19(1), pp. 252-256. Artyukhin, E. A., 1985. Experimental Design of Measurement of the Solution of Inverse Heat Conduction Problem. Journal of Engineering Physics, 48(3), pp. 372-376. Balk, P., 2013. A priori information and admissible complexity of the model of anomalous objects in the solution of inverse problems of gravity. Izvestiya, Physics of the Solid Earth, 49(2), pp. 165-176. Barbone, P. & Bamber, J., 2002. Quantitative elasticity imaging: What can and cannot be inferred from strain images. Physics in Medicine and Biology, 47(12), pp. 2147-2164.
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