Lecture 3 Reduced-Order Modeling

What is the main goal of Model Order Reduction (MOR)?

To reduce the size of a large ODE system (e.g., from N~10⁶ to m≪N degrees of freedom) so it can be solved much faster, while keeping acceptable accuracy.

What is the core idea of projection-based ROM?

Approximate the full solution u(t) ≈ Q q(t), where Q is a small orthonormal basis matrix (N×m), then project the equation of motion onto that subspace to get a much smaller system of size m×m.

What is modal analysis used for?

To characterize the free vibration behavior of a dynamic system by computing eigenfrequencies ωᵢ and eigenmodes φᵢ — the natural frequencies and shapes the system "wants" to vibrate in.

What is modal reduction and when does it work well?

Using the first m eigenmodes as the reduction basis Q. Works well for 2nd-order, linear problems. The resulting reduced equations are decoupled and can be integrated very efficiently.

What does it mean for the reduced modal equations to be "decoupled"?

Each reduced equation involves only one modal coordinate qᵢ(t) — the equations are independent of each other, so each can be solved separately, which is computationally very cheap.

Why are only the first k≪N eigenfrequencies computed in practice?

Because modal analysis is expensive for large N, and the smallest eigenfrequencies dominate the dynamic behavior — higher modes contribute negligibly.

What is a snapshot matrix in the context of POD?

A matrix 𝕊 = [u₁, u₂, …] ∈ ℝᴺˣᴹ whose columns are solution vectors u(t) recorded at different time steps (and possibly different loadings) from a full-order simulation.

What is Proper Orthogonal Decomposition (POD) and how is the basis obtained?

POD finds the orthogonal matrix Q that best approximates the snapshot matrix. The POD basis is the first m left singular vectors from the SVD of the snapshot matrix: Q = [𝕦₁, …, 𝕦ₘ].

What do the singular values in SVD tell you about a POD basis?

Larger singular values σᵢ mean that direction captures more of the data's energy. The POD error can be estimated as the sum of discarded singular values divided by the total — this tells you how much information you lose by truncating at m modes.

What is the key difference between modal reduction and POD as reduction strategies?

Modal reduction is physics-based (basis comes from the eigenvalue problem, no data needed) and only works for 2nd-order linear systems. POD is data-driven (basis comes from simulation snapshots via SVD) and works for any type of problem including nonlinear ones.

What is the major remaining challenge when applying POD-based ROM to nonlinear problems?

The projected nonlinear force term f̃(q) ≈ Qᵀ f(Qq) still requires evaluating the full-order nonlinear operator f at every time step — so the computational bottleneck of the high-dimensional system is not actually removed.

What are the offline and online phases of ROM?

Offline: expensive one-time computations (snapshot generation, SVD, reduced basis, pre-computed reduced matrices) — done on HPC. Online: fast simulation using only the small reduced system — enables real-time or faster-than-real-time use for optimization, control, and digital twins.

What is Dynamic Mode Decomposition (DMD) and what problem does it solve?

DMD identifies a best-fit linear map A such that xₖ₊₁ ≈ A xₖ directly from snapshot data, without knowing the governing equations. It compresses the high-dimensional dynamics into r dominant modes and eigenvalues.

What do DMD eigenvalues and DMD modes each represent physically?

DMD eigenvalues λᵢ encode the temporal behavior of each mode (growth, decay, oscillation frequency). DMD modes Φ encode the spatial structure — the pattern in state space associated with each temporal behavior.

What are the main limitations of DMD?

It assumes a linear flow map, so it cannot capture transient (short-lived, startup) phenomena, cannot capture physical invariances (e.g., rotation symmetry), and requires a fixed time step Δt between snapshots.