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The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena
Julius Reiss, Philipp Schulze, Jörn Sesterhenn, Volker Mehrmann
TL;DR
Transport-dominated phenomena, especially those involving multiple velocities, challenge common mode-based reduction methods. The paper introduces shifted proper orthogonal decomposition, which combines time-dependent shifts with transport separation. Tests in one and two dimensions report better performance than POD, including fewer modes for a given accuracy, while the method has non-uniqueness and does not guarantee a minimum total mode count.
Problem
An efficient general model-reduction methodology for phenomena with multiple transport velocities is still missing because a single shift is insufficient.
Method
sPOD extends POD with time-dependent shifts, separates different transports, and determines dominant velocities using tracking or shifted-snapshot singular-value dependence.
Results
sPOD outperforms POD in one- and two-dimensional tests by requiring fewer modes for a specified accuracy and can recover analytic solutions up to machine precision in suitable examples.
Takeaways & Limitations
In co-moving frames, sPOD modes provide a clearer and more intuitive description of sharp moving structures than POD modes in the laboratory frame.
Takeaways & Limitations
The sPOD separation is not unique, and the iteration does not guarantee a decomposition with the minimum total number of modes.
Abstract
from arXiv · showhide
Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kind of systems. It extends the proper orthogonal decomposition (POD) by introducing time-dependent shifts of the snapshot matrix. The approach, called shifted proper orthogonal decomposition (sPOD), features a determination of the {\it multiple} transport velocities and a separation of these. One- and two-dimensional test examples reveal the good performance of the sPOD for transport-dominated phenomena and its superiority in comparison to the POD.
1 Introduction
Model reduction derives low-dimensional descriptions from data for faster simulation and improved understanding, but transport-dominated phenomena challenge common mode-based approaches. The sPOD addresses multiple transport velocities by combining time-dependent shifts with transport separation.
- Model reduction derives low-dimensional models from experimental or numerical data for fast simulations, control, optimization, parameter studies, and dynamical-system understanding.Its goal is to approximate mappings from inputs such as design parameters, system conditions, or controls to outputs including physical quantities or full flow solutions.
- Input-output interpolation methods focus on transfer-function behavior rather than reducing internal dynamics and are mainly successful for linear systems.Extensions to nonlinear systems are described as rare and subject to drawbacks such as low computational efficiency.
- Mode-based nonlinear reduction commonly uses POD or related decompositions, with the SVD providing the best low-rank matrix approximation in the 2-norm.POD is also known as principal component analysis or the Karhunen-Loeve decomposition.
- Transport-dominated dynamics challenge POD because a few spatial modes cannot accurately capture them in a dyadic structure.Existing approaches often introduce time-dependent shifts to compensate for transport, including symmetry-reduction methods.
- Multiple transport velocities require more than the single shift used in many prior approaches, leaving a need for an efficient general methodology.The paper introduces shifted proper orthogonal decomposition to address this gap.
- sPOD combines time-dependent shifts with procedures that separate transports, determines dominant velocities through tracking or shifted-singular-value analysis, and remains purely data based.The shifts extend POD’s dyadic structure to transported systems and are presented as offering broad applicability compared with symmetry reduction.
2 One-dimensional model problems
The one-dimensional problems show why standard POD struggles with transported structures and how sPOD uses time-dependent shifts to separate transports and obtain compact approximations. Across single- and multiple-transport examples, sPOD improves representation quality and can substantially outperform POD.
- Single transported quantity: POD may require many modes for a transported high-gradient quantity because its dyadic structure poorly represents diagonal space-time structures.A sharp transported Gaussian pulse has a slowly decaying singular spectrum despite having an analytic one-pulse description.
- Multiple transported quantities: sPOD combines shifted snapshot data with iterative least-squares fitting to construct and separate low-rank structures associated with different transport velocities.Residual decomposition supplies additional frame-specific modes to reduce cross-influence between transports.
- Multiple transported quantities: Interference where oppositely traveling pulses overlap creates a large localized error and makes separation ambiguous; adding modes can counteract the separation.Broad residual structures can slow convergence, while more residual modes may improve convergence for less-sharp initial conditions.
- Multiple transported quantities: Two sPOD modes approximate the two-transport acoustic solution excellently after 40 iterations, agreeing with the analytic two-mode representation.The construction used only density data, suggesting it did not implicitly rely on the equation’s hyperbolic structure.
- Comparison with POD: For the acoustic pulse, two sPOD modes achieve error less than 3 × 10^-14, whereas more than 80 POD modes are required for comparable accuracy.With only two POD modes, the relative mean error is almost 1.
- Determination of shift velocities: Transport velocities can be identified from maxima of the leading singular value versus shift velocity, although spectra are only first indicators when multiple transports interact.For the pressure pulse, the velocities c± = ±1 appear as maxima; for the standing wave, zero velocity is largest while ±c remain local maxima.
3 A two-dimensional model problem
The two-dimensional vortex-pair problem contains two non-trivial transport velocities, which sPOD separates using shifted snapshot matrices. The resulting modes provide a physically meaningful approximation with a relative mean error below 1%.
- Problem setup: The incompressible Navier–Stokes test contains a primary vortex pair moving upward and a weaker secondary pair moving downward at a smaller velocity.The simulation uses a periodic 512^2-point domain and includes viscosity ν = 1/1000.
- Velocity determination: The dominant velocities are identified from maxima in the singular values of y-shifted snapshot matrices, at −1.1193 for the secondary pair and 2.7663 for the primary pair.These two velocities are then used in the decomposition, although the actual velocities are not strictly constant.
- Convergence: A decent approximation uses 14 modes for the primary and 10 for the secondary vortex pair, while the iterative procedure starts with one mode per frame and adds modes progressively.With constant shifts and complex transport, the relative mean error decreases to slightly less than 1%.
- Convergence: The sPOD convergence criterion uses residual and approximation singular values, orthogonality error, and mean error, with mean error providing the clearest convergence indicator.The singular-value criterion is satisfied from iterations 9 and 12 for the primary and secondary vortex pairs, respectively.
- Mode interpretation: The first sPOD mode describes each vortex pair, while higher modes capture time-dependent shape changes and transport-velocity variation.For the secondary pair, the first two modes have nearly similar weights; after separation at about t > 0.15s, the modal amplitudes become simpler to interpret.
- Quantitative comparison and limitations: The sPOD approximation is compared with the full-order solution and POD using 9 + 6 sPOD modes and 15 POD modes, while the decomposition can remain non-unique.Stripe structures are attributed to non-uniqueness rather than incomplete separation, and the method may become ineffective for a large number of modes.
4 Summary and outlook
The sPOD generalizes POD with time-dependent shifts and iterative separation of multiple transports, enabling low-dimensional representations of transport-dominated phenomena. Tests show improved mode efficiency and intuitive co-moving modes, while practical use depends on estimating transport velocities and currently assumes periodic boundaries.
- Method: sPOD generalizes POD by applying time-dependent shifts to modes and decomposing snapshots using low-rank shifted components.Its iterative algorithm uses truncated SVDs of shifted snapshot and residual matrices.
- Method: Multiple transports can be separated with the iterative sPOD algorithm, which shifts components according to their transport velocities.The approach is designed to handle more than one shift within the same system.
- Results: For a linear wave equation with sharp moving structures, sPOD uses the minimal number of modes and recovers the analytic solution up to machine precision.The common POD performs poorly on these examples because of slow singular value decay.
- Results: In one-dimensional crossing shock waves and a two-dimensional vortex pair, sPOD requires fewer modes than POD for a specified accuracy.Its modes also have a clearer, more intuitive interpretation in co-moving frames than POD modes in the laboratory frame.
- Practical scope: Transport velocities can be estimated from snapshot data through peak or front tracking or singular value maximization when physical values are unavailable.Velocity perturbations require a few additional modes, but singular value decay remains steeper than for POD in the tested wave-equation example.
- Outlook: An extension to non-periodic boundaries is necessary for settings with reflecting or non-reflecting boundaries.A direct optimization reformulation could guarantee local optimality but would likely be more expensive than the proposed algorithm.