 Research Article
 Open Access
An efficient quasioptimal spacetime PGD application to frictional contact mechanics
 Anthony Giacoma^{1}Email author,
 David Dureisseix^{2} and
 Anthony Gravouil^{2, 3}
https://doi.org/10.1186/s4032301600677
© Giacoma et al. 2016
 Received: 12 December 2015
 Accepted: 1 April 2016
 Published: 25 April 2016
Abstract
The proper generalized decomposition (PGD) aims at finding the solution of a generic problems into a low rank approximation. On the contrary to the singular value decomposition (SVD), such a low rank approximation is generally not the optimal one leading to memory issues and loss of computational efficiency. Nonetheless, the computational cost of the SVD is generally prohibitive to be performed. In this paper, authors suggest an algorithm to address this issue. First, the algorithm is described and studied in details. It consists in a cheap iterative method compressing a low rank expansion. It will be shown that given a low rank approximation, the SVD of a provided low rank approximation can be reached at convergence. Behavior of the method is exhibited on a numerical application. Second, the algorithm is embedded into a general spacetime PGD solver to compress the iterated separated form for the solution. An application to a quasistatic frictional contact problem is illustrated. Then, efficiency of such a compressing method will be demonstrated.
Keywords
 Low rank approximation
 Proper generalized decomposition
 Singular value decomposition
 Principal component analysis
 Quasistatic contact
Background
Computational mechanics tackles nowadays large models involving huge amount of data to provide fine description of physics or accurate forecasts. For that purpose, several and various numerical methods have to be taken into account in order to perform efficiently these large scale simulations (both accurate and computationally cheap). To address this issue, both computational hardware and algorithms have to progress. During the last decades, a specific class of algorithms based on model reduction methods has been developed. They consist basically in focusing on dominant trends of the problem. Then, a large amount of computational time can be spared and accurate and well representative solution can be captured. These methods rely strongly on basis design for approximated solution which has to span the dominant trends and perhaps weaker ones up to a desired level of accuracy.
Given a collection of data (also called snapshots), the wellknown canonical method to design the optimal basis is the Singular Value Decomposition (SVD). Such a decomposition may lead to prohibitive computational times in an industrial context due to its complexity. In addition reduced order modeling methods often require strategies to adapt online the reduced basis in order to include uncaptured trends of the problem. In other words, if one has a SVD basis and wants to add some vectors, one has to recompute the SVD with new data. For such situations updating strategies are proposed in [1, 2].
Usual SVD algorithms [3] compute SVD modes onebyone “incrementally” until having a basis satisfying a certain level of accuracy. These algorithms iterate until finding a mode. Once the precision criterion is reached, the basis is ensured to be optimal because each found mode is the most representative one.
In this paper, we propose a different approach. Given a set of vectors, a basis is defined. The hereinafter suggested approach iterates over the whole basis in order to make all of its vectors closer to optimal ones until having SVD basis. Doing so, after each iteration a “quasioptimal” basis is computed and few iterations are expected to provide a quite optimal basis. Such an approach ensures to have at each iteration a basis which spans the whole considered space to detriment of its optimality. Such an iterated basis could be sufficient to perform reliable computation or data analysis. One expects that the computational effort to get a quasioptimal basis is low whereas classic SVD algorithms prescribing the optimality property are expensive.
In the following sections, the proposed strategy is first described on a rank2 expansion. Convergence proof, analysis and results are exposed. Second, this strategy is generalized for rankp expansion with a global convergence proof. Afterwards, this strategy is tested by computing the SVD of a matrix. Finally, an application case is performed. It deals with a combination of the suggested method and the proper generalized decomposition (PGD) method. On this basis, the efficiency of quasioptimal approaches will be exemplified.
An iterative process to compute the SVD
In the following, we will denote with \(\mathbf {A} \in \mathbb {R}^{n\times m}\) a real rectangular matrix. Without loss of generality, we will assume that \(n \geqslant m\) (if not the case, we simply consider the transpose of \(\mathbf {A}\)). Given two column vectors of same size \(\mathbf {u}\) and \(\mathbf {v}\), the associated inner product is denoted with \((\mathbf {u} \mid \mathbf {v})\); since in this article we consider the euclidean canonical inner product associated to the euclidean norm \(\Vert \cdot \Vert \), \((\mathbf {u} \mid \mathbf {v}) = \mathbf {u}^T \mathbf {v}\).
In order to obtain a decomposition (1), several methods could be used. They are expected to be able to prescribe specific properties such as orthonormality condition for the involved vectors. Three of them are listed below.
Decomposition according to the canonical basis. Given the matrix \(\mathbf {A}\), each snapshot can be written in the canonical basis leading to \(\mathbf {U} = \mathbf {1}_n\) (square \(n\times n\) identity matrix, \(p=n\)) and \(\mathbf {V} = \mathbf {A}^T\). Hence, vectors \(\mathbf {u}_i\) are orthonormal and columns of \(\mathbf {A}\) correspond to vectors \(\mathbf {v}_i\).
Other standard methods aim also at providing a first guess of the low rank expansion (QR factorization) and may have suitable advantages like numerical complexity or numerical stability. Nevertheless, one has to keep in mind that such preorthogonalization processes have a numerical cost.
Iterative singular value decomposition for a rank2 matrix
Definition of the compression function F
 1.
Right vector \(\mathbf {v}_2\) is written as \(\mathbf {v}_2 = \alpha \mathbf {v}_1 + \varvec{\bar{\mathrm{v}}}_2\) so that \(\mathbf {v}_1^T \varvec{\bar{\mathrm{v}}}_2 = 0\) and \(\alpha = ( \mathbf {v}_1 \mid \mathbf {v}_2) / ( \mathbf {v}_1 \mid \mathbf {v}_1 )\). \(\mathbf {A}_2 = \varvec{\bar{\mathrm{u}}}_1 \mathbf {v}_1^T + \mathbf {u}_2 \varvec{\bar{\mathrm{v}}}_2^T\) with \(\varvec{\bar{\mathrm{u}}}_1 = \mathbf {u}_1 + \alpha \mathbf {u}_2\). One can remark that \(\varvec{\bar{\mathrm{u}}}_1^T \mathbf {u}_2 \ne 0\) a priori.
 2.
Left vectors are reorthogonalized using \(\mathbf {u}_2 = \beta \varvec{\bar{\mathrm{u}}}_1 + \varvec{\bar{\mathrm{u}}}_2\) with \(\varvec{\bar{\mathrm{u}}}_1^T \varvec{\bar{\mathrm{u}}}_2 = 0\) so that \(\beta = (\varvec{\bar{\mathrm{u}}}_1^T \varvec{\bar{\mathrm{u}}}_2) / (\varvec{\bar{\mathrm{u}}}_1^T \varvec{\bar{\mathrm{u}}}_1) = \alpha / ( 1 + \alpha ^2 )\). \(\mathbf {A}_2 = \varvec{\bar{\mathrm{u}}}_1 \varvec{\bar{\mathrm{v}}}_1^T + \varvec{\bar{\mathrm{u}}}_2 \varvec{\bar{\mathrm{v}}}_2^T\) with \(\varvec{\bar{\mathrm{v}}}_1 = \mathbf {v}_1 + \beta \varvec{\bar{\mathrm{v}}}_2\).
 3.
Denoting \(\gamma = \sqrt{1+\alpha ^2}\), since \(\Vert \varvec{\bar{\mathrm{u}}}_1 \Vert = \gamma \) and \(\Vert \varvec{\bar{\mathrm{u}}}_2 \Vert = 1/\gamma \), the left vectors are normalized with \(\varvec{\tilde{\mathrm{u}}}_1 = \varvec{\bar{\mathrm{u}}}_1 / \gamma \), \(\varvec{\tilde{\mathrm{u}}}_2 = \gamma \varvec{\bar{\mathrm{u}}}_2\) and \({\varvec{\tilde{\mathrm{v}}}}_1 = \gamma \varvec{\bar{\mathrm{v}}}_1\), \({\varvec{\tilde{\mathrm{v}}}}_2 = \varvec{\bar{\mathrm{v}}}_2 / \gamma \).
Algorithm study
Since one also has \( 1 + \alpha _\xi ^2(2+\eta _\xi )> 1 + \alpha _\xi ^2 > 1  \alpha _\xi ^2 / \eta _\xi \), the following property holds: \(0< \eta _{\xi +1} < \eta _\xi \). As a decreasing and lowerbounded \(\eta _\xi \) serie, it converges to a value \(\eta \) (the convergence rate increases).
Algorithm properties
Numerical example

The higher \(\alpha _0\), the lower becomes \(\eta \) and the higher the convergence rate is.

The lower \(\eta _0\), the faster the convergence is.

The lower \(\alpha _0\) (right vectors are poorly correlated) and the higher \(\eta _0\) (amplitudes of right vectors are similar), the lower the convergence rate is.
Generalization to higher rank expansions
Orthogonality (\(\clubsuit \)) and compression (\(\spadesuit \)) properties have to be checked for all vectors. Ordering property (\(\lozenge \)) can be always ensured by sorting dyads at the end of the application of F.
Various combinations of \(p2\) dimensional subspace rotations can be chosen. It is a compromise between efficiency and computational sustainability (parallel computation).
Algorithm
Algorithm 2 consists in applying compression function F to two rank2 approximations composing the whole approximation of \(\mathbf {A}\). This is achieved in a such way that conditions (\(\clubsuit \)), (\(\spadesuit \)) and (\(\lozenge \)) are fulfilled. Doing so, previously given proofs can be reused. This algorithm may run until producing the SVD of \(\mathbf {A}\).
With the proposed algorithm, loops dependencies do not enable their execution in parallel. Several forloop strategies can be implemented, but are not studied herein.
Rank adaptation and downsizing
Let n be the size of \(\mathbf {u}\) and m the size of \(\mathbf {v}\). The complexity of one instance of the compression function F is \(c_F = 6 n + 10 m + 6\). One loop (indexed by \(\xi \)) involves \(\frac{1}{2}p(p1)\) occurrences of F. All in all, complexity of Algorithm 3 can be estimated to \(c = \xi _\text {max} [3 n p(p  1) + 5 m p(p1) + 3 p (p1)]\). This complexity is evaluated assuming that expansion size \(q=p\) remains constant. Nonetheless, during iterative process, one is able to eliminate pairs of vectors of poor contribution by prescribing a threshold \(\epsilon \) for the norms of the right vectors. Thus, a computational expense could be spared and the analysis focused on dimensions of interest.
Numerical application
To illustrate the previously described algorithm, the singular value decomposition of a given matrix is performed. This matrix is picked up from Matrix Market^{1} and is called rbs480a.mtx. First, its SVD is computed using the standard Matlab solver. Singular values (\(\sigma _i^\mathrm{ref}\)), reference left (\(\mathbf {u}_i^\mathrm{ref}\)) and right (\(\mathbf {v}_i^\mathrm{ref}\)) singular modes are therefore provided.
Previously, convergence properties have been enlightened according to mode amplitude properties. To exemplify those convergence behaviors, several configurations are built to affect amplitude ratio between modes (\(\sigma _i^\mathrm{ref}\) is transformed into \(\sigma _i^\mathrm{mod}\)); left and right singular modes remain the same and only mode contribution is affected. For each configuration, a whole modified matrix is rebuilt.

No modification. Original singular value amplitudes of the matrix decrease slowly along the 400 first modes.

Medium slope for singular value amplitudes. In a semilog diagram, a linear slope for mode amplitudes is prescribed.

Strong initial slope for singular value amplitudes. A small amplitude ratio is prescribed for first successive modes.

No modification. As it may have been expected, the convergence is quite slow during the first iterations, because the first successive modes do have a high (close to 1) amplitude ratio. Nevertheless, the last 80 modes are more rapidly found.

Medium slope. The amplitude ratios are all the same, and all iterations provide a similar convergence rate.

Strong initial slope. The small amplitude ratio for the first successive modes lead to a large convergence rate during first iterations. The converse if obtained for the latest iterations.
Application to SVDfree quasioptimal spacetime PGD

a posteriori approaches (POD/SVD, surrogate modeling) using prior knowledge about the solution (sampling, snapshots, etc.) to compute desired new solutions.

a priori approaches which do not require previous knowledge about the solution and aim at computing a desired solution in a convenient form (memory and cost efficient).

Reducibility: \(\mathcal {S}\) can be represented on a lowdimensional basis, i.e. \(\mathcal {S}\) can be written accurately (up to a certain level) with a linear combination of a few vectors

Dominant trends (scale separability): some vectors of the basis (which are supposed to be the first ones) are highly contributory to generate \(\mathcal {S}\) whereas other ones are less important. These vectors depict the different scales of the problem [4].
The obtained basis can be used within the a posteriori approach to generate Reduced Order Models (ROMs). Indeed, a first approach consists in projecting \(\mathcal {P}\) (Galerkin projection) into the spanned subspace [6–8]. Secondly, this basis can be considered as a filter for data due to the basis truncation. Indeed, noise is expected to generated by the high order SVD modes. Therefore the basis can be used to generate surrogate models relying on regression methods (ARMA, ARIMA processes [9, 10]), time series analysis [11], etc. These resulting models are expected to be easy to use and computationally efficient. The quality of the snapshots depends highly on the initial chosen vectors and the process to generate the model; error criterion could be difficult to exhibit.
A widespread a priori approach is the Proper Generalized Decomposition (PGD) [12–14]. This approach aims at finding \(\mathcal {S}\) directly into a separated form or low rank expansion as in equation (1) without prior knowledge. PGD solvers are incremental processes which consists in enriching progressively a low rank expansion to make an iterated solution \(\mathcal {S}_i\) more accurate. The ideal PGD solver should be able to find each vectors of the SVD decomposition of \(\mathcal {S}\), i.e. the first iterated vector is the most contributory one of \(\mathcal {S}\), then the second, etc. Basically, PGD does not prescribe orthogonality for left or right vectors of the low rank expansion. In practice, such a condition is often applied for a sake of numerical efficiency. In practice, the low rank expansion generated by the PGD is generally not the optimal none. Nevertheless, as the computational effort is concentrated on rank1 tensors, a great amount of computations and memory can be spared.
We propose to illustrate the previously described algorithm into a spacetime PGD solver aiming at solving a frictional contact solid mechanic quasistatic problem. We suggest to embed into a PGD iteration, one iteration of Algorithm 3 in order to compress progressively the iterated low rank expansion. Doing so, one can expect to make it close to the optimal one and stem inflation of iterated expansion [15].
Quasistatic frictional contact problems
Reference problem
The large time increment method
 (P1)
Separation of the linear and nonlinear behaviors. We denote by \(\mathbf {u}\) the displacement field over \(\Omega \times [0,\ T]\) and \(\varvec{{\lambda }}\) the contact force field over \(\partial _3 \Omega \times [0,\ T]\). \(\mathcal {A}\) denotes the set of solutions \(\mathbf {s} = (\mathbf {u}, \varvec{{\lambda }})\) satisfying linear constitutive law, kinematic admissibility and static admissibility. These are defined on the whole spacetime domain \(\Omega \times [0,\ T]\). \({{\varvec{\Gamma }}}\) denotes the set of solutions \(\hat{\mathbf{s }} = ( \hat{\mathbf{ v }}, \hat{{\varvec{\lambda }}} )\) verifying frictional contact conditions and are defined locally at the contacting interface and on the whole time interval \(\partial _3 \Omega \times [0,T]\). The solution of the problem is \(\mathbf {s} \in \mathcal {A} \cap {{\varvec{\Gamma }}}\).
 (P2)
A twostaged iterative algorithm. The solution of the problem is searched with the construction of two sequences of approximations belonging alternatively to \(\mathcal {A}\) and \({{\varvec{\Gamma }}}\). At the \(i\mathrm{th}\) iteration, the local stage consists in finding \(\hat{\mathbf{{s} }}_{i} = ( \hat{\mathbf{v }}_{i}, \hat{\varvec{{\lambda }}}_{i} ) \in {{\varvec{\Gamma }}}\) with a search direction \(( \hat{\mathbf{{s} }}_{i}  \mathbf {s}_{i1}) = (\hat{\mathbf{{v} }}_{i}  \mathbf {v}_{i1}, \hat{\varvec{{\lambda }}}_{i}  {\varvec{\lambda }}_{i1} ) \in \mathbf {E}^+\). Note that \( \mathbf {s}_{i1} = ( \hat{\mathbf{{v} }}_{i1}, \hat{\varvec{{\lambda }}}_{i1} )\) is known from the previous iteration. Then, the global stage consists in finding \(\mathbf {s}_{i} = ( \mathbf {v}_{i}, \varvec{{\lambda }}_{i} ) \in \mathcal {A}\) with another search direction \((\mathbf {s}_{i}  \hat{\mathbf{{s} }}_{i} ) = ( \mathbf {v}_{i}  \hat{\mathbf{{v} }}_{i}, \varvec{{\lambda }}_{i}  \hat{\varvec{{\lambda }}}_{i} ) \in \mathbf {E}^\). Note that \( \hat{\mathbf{{s} }}_i = ( \hat{\mathbf{{v} }}_i,\ \hat{\varvec{{\lambda }}}_i )\) is known from the previous local stage.
 (P3)
Radial approximation or spacetime separation. Unknown fields are represented as a sum of products between a space function and a time function to limit memory usage. An orthonormality condition is prescribed for space modes (i.e. left vectors).
Numerical results
We consider the LATIN method (including only the first and second principle) as the reference nonlinear solver.
On Figs. 8, 9, MAC matrices are plotted to assess the quality of iterated basis for both methods. LATINP3 catches roughly the trends of the solution. But optimal vectors are obviously not computed. On the other hand, the LATINPGD computes painlessly dominant trends and iterated vector are very close to SVD optimal vectors of the solution. Even if a given iterated vector is not the most suited one (in regard to the converged solution), it is quickly corrected through next iterations. The LATINPGD computes nearly the solution of the numerical problem into its optimal SVD expansion. Additional numerical experiments confirm that the combination of the proposed algorithm with a LATINP3 method achieves a strong solver to design quasioptimal basis for the solution with a reduced computational effort. The basis enriching strategy allowed by the PDG is completed with an onthefly compression strategy provided by the proposed algorithm.
Conclusion
In this paper, an iterative SVD algorithm is proposed. It relies on rotations around subspace which compress a given low rank approximation to its SVD form. Different strategies can be proposed as far as rotations are concerned (selection, order, simultaneity ...) provided that appropriate conditions are fulfilled. Nonetheless, its interest does not rely on SVD expansions but on quasioptimal bases which are expected to be close to. Indeed, the proposed algorithm feature is to provide such quasioptimal bases after a few iterations. This efficiency depends on low rank expansion characteristics (ratios of right vector norms).
It provides an interesting tool for basis enrichment strategies. Usually, reduced order modeling techniques do not require a computationally expensive optimal basis (i.e. quasioptimal is enough). These enrichment strategies can be embedded into PGD methods as shown herein. But a posteriori or SVD approaches within big data framework could also be concerned. Indeed, to design a relevant basis upon which ROMs or surrogate models are built, snapshots are stored and an associated generated basis has to be updated. The basis update can be expensive if one considers the optimal basis. Using the proposed algorithm makes a compromise by refreshing cheaply the basis but weaks the optimality property. Moreover, the suggested algorithm enables to consider specific inner product and to control the quasioptimality.
An interesting extension of such an algorithm could be designed for higher order rank one tensors. This extension could be useful for PGD multiparametric studies and may converge to recent works concerning the High Order SVD (HOSVD) or similar tensor decomposition [28–32].
Declarations
Authors' contributions
The three authors contributed to the implementation of the suggested quasioptimal LATINPGD method. DD and AGi participated to the development of mathematical proves and numerical studies of the suggested compression algorithm. AGi has drafted the manuscript. DD and AGr have supervised the different studies and the corrections of the draft. All authors read and approved the final manuscript.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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