Proper Orthogonal Decomposition Basis Research Articles

Automated optimal experimental design strategy for reduced order modeling of aerodynamic flow fields

Aerodynamic flow fields reveal essential physical insights (such as shocks) that substantially affect aerodynamic performance. Conventional flow field computations, however, adopt high-fidelity but time-consuming simulation models which prohibit real-time decision makings. Alternatively, reduced-order model (ROM) arises for fast flow field predictions but random sampling could result in an excessive training cost. In this paper, we propose a fully automated optimal experimental design (Auto-OED) strategy on proper orthogonal decomposition (POD) for ROM-based rapid flow field predictions. In particular, Auto-OED leverages two individual optimal experimental design (OED) strategies, automatically selects the number of POD bases for the first sampling strategy, and intelligently switches to the second strategy on the fly. We showcased the Auto-OED strategy on airfoil flow field predictions within a transonic regime where the Mach number and angle of attack varied within the ranges of [0.55, 0.75] and [-1, 3] deg, respectively. OED-based ROM's were constructed over 20 trials for robustness tests with a total of 40 training samples on each trial including 16 random initial samples by Latin hypercube sampling (LHS) and 24 OED samples. Results exhibited that the ROM predictions completely based on LHS had an error of 1.6×10−3 while the existing OED strategies-based ROM's over 20 trials obtained a mean error (μerror) of 7.53×10−4 with a standard deviation (σerror) of 9.00×10−5. In contrast, the best Auto-OED ROM achieved the lowest μerror of 7.49×10−4 and lowest σerror of 6.20×10−5. The self-evident error reductions confirm the viability of the proposed Auto-OED based ROM in fluid field predictions and potentially other engineering applications of the similar type.

A domain decomposing reduced-order method for solving the multi-domain transient heat conduction problems

In this article, a domain decomposition algorithm combined with a reduced-order model (ROM) is proposed to deal with the transient heat conduction problems in multi-domain structures. Using the domain decomposition algorithm and the finite element method (FEM), the original problem is decomposed into multiple non-overlapping subdomains, which can be solved independently under appropriate boundary conditions. However, additional inter-regional connected calculation will be brought through the simple geometry decomposition. Thus, the proper orthogonal decomposition (POD), a projection-based ROM technique, is utilized to further reduce the number of unknowns and the calculation time for each subdomain. To be more specific, the local POD bases of each subdomain are obtained by decomposing a snapshot matrix that gathered according to time by the application of the virtual boundary conditions on interfaces of the adjacent subdomains and other boundaries. Furthermore, the problems under different boundary conditions can be linked by the variable condensation technique on the interfaces, which enhances the efficiency of the numerical solver dramatically. Finally, the accuracy and efficiency of the proposed method are demonstrated by comparing the results calculated using both the proposed algorithm and the direct finite element simulation.

Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

Fast reconstruction of boiler numerical physical field based on proper orthogonal decomposition and conditional deep convolutional generative adversarial networks

ABSTRACT Obtaining real-time physical field data inside the boiler furnace is crucial for combustion diagnostics and optimization. In this work, we propose a fast reconstruction technique for the physical fields in a tangentially-fired coal boiler. The method combines the Proper Orthogonal Decomposition (POD) and Conditional Deep Convolutional Generative Adversarial Networks (cDCGAN) based on the boiler physical field obtained by computational fluid dynamics (CFD). Firstly, CFD simulations are performed using data from the boiler Distributed Control System (DCS). After verifying the CFD results, we expanded the simulation to 105 operational condition cases sample sets. Then, the physical fields of the sample set undergo POD to determine the appropriate number of POD bases, thereby ensuring the accuracy of the reconstruction. Finally, the cDCGAN method is employed to establish the correlation between the POD bases and the operational conditions of the boiler. Results show that the POD-cDCGAN method achieves rapid and accurate reconstruction of temperature and velocity fields, with mean relative error (MRE) below 1.5% and root relative squared error (RRSE) below 5.7%. The reconstruction time is reduced to 10.57s, significantly faster than direct CFD methods. The POD-cDCGAN method offers an efficient approach for online operational diagnosis and digital twin construction of boiler equipment.

Dissipation-based proper orthogonal decomposition of turbulent Rayleigh–Bénard convection flow

We present a formulation of proper orthogonal decomposition (POD) producing a velocity–temperature basis optimized with respect to an H1 dissipation norm. This decomposition is applied, along with a conventional POD optimized with respect to an L2 energy norm, to a dataset generated from a direct numerical simulation of Rayleigh–Bénard convection in a cubic cell (Ra=107, Pr=0.707). The dataset is enriched using symmetries of the cell, and we formally link symmetrization to degeneracies and to the separation of the POD bases into subspaces with distinct symmetries. We compare the two decompositions, demonstrating that each of the 20 lowest dissipation modes is analogous to one of the 20 lowest energy modes. Reordering of modes between the decompositions is limited, although a corner mode known to be crucial for reorientations of the large-scale circulation is promoted in the dissipation decomposition, indicating suitability of the dissipation decomposition for capturing dynamically important structures. Dissipation modes are shown to exhibit enhanced activity in boundary layers. Reconstructing kinetic and thermal energy, viscous and thermal dissipation, and convective heat flux, we show that the dissipation decomposition improves overall convergence of each quantity in the boundary layer. Asymptotic convergence rates are nearly constant among the quantities reconstructed globally using the dissipation decomposition, indicating that a range of dynamically relevant scales is efficiently captured. We discuss the implications of the findings for using the dissipation decomposition in modeling and argue that the H1 norm allows for a better modal representation of the flow dynamics.

Multi-fidelity reduced-order surrogate modelling

High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.

Flow fields prediction for data-driven model of parallel twin cylinders based on POD-RBFNN and POD-BPNN surrogate models

Flow around cylinders is an important phenomenon in many different engineering fields. In this paper, the fast prediction of the pressure fields of parallel twin cylinders is implemented based on a data-driven algorithm. Firstly, the pressure fields of parallel twin cylinders with a low Reynolds number are obtained through the Computational Fluid Dynamics (CFD) method. The pressure fields at different time steps are collected to form a snapshot matrix. The Proper Orthogonal Decomposition (POD) algorithm is then applied to obtain the POD basis vectors of the snapshot matrix, enabling the reconstruing the pressure fields. Subsequently, two reduced-order models (ROM) called the POD-RBFNN and POD-BPNN surrogate models are proposed in this paper. The POD-RBFNN surrogate model uses the Radial Basis Function Neural Network (RBFNN) to train the POD mode coefficients obtained from the POD algorithm, while the POD-BPNN surrogate model uses the Backpropagation Neural Network (BPNN) for the same purpose. Linearly combining the POD mode coefficients predicted by the POD-RBFNN or POD-BPNN surrogate models with the POD basis vectors obtained from the POD algorithm enables fast and efficient prediction of pressure fields for non-sample points. Finally, comparisons are made between the predicted pressure fields obtained from these two surrogate models and the actual values obtained through CFD simulations. It is found that both the POD-RBFNN and POD-BPNN surrogate models proposed in this paper not only significantly improve efficiency but also maintain a high level of accuracy. However, the training time of the POD-RBFNN surrogate model is significantly shorter than that of the POD-BPNN surrogate model. Additionally, the POD-RBFNN surrogate model exhibits smaller Root Mean Square Errors (RMSE) and Mean Absolute Error (MAE). For the data-driven model of parallel twin cylinders described in this paper, the POD-RBFNN surrogate model is more suitable to predict the pressure fields. The research results in this paper are believed to hold significant value for CFD calculations of parallel twin cylinder models, offering essential guidance for a deeper understanding of cylinder flow problems and optimizing engineering design.

Surrogate Modelling of Field Performance Analysis and Optimization via Hierarchical Proper Orthogonal Decomposition and Adaptive Sampling Radial Basis Function

Surrogate Modelling of Field Performance Analysis and Optimization via Hierarchical Proper Orthogonal Decomposition and Adaptive Sampling Radial Basis Function

Non-Intrusive Reduced-Order Modeling Based on Parametrized Proper Orthogonal Decomposition

A new non-intrusive reduced-order modeling method based on space-time parameter decoupling for parametrized time-dependent problems is proposed. This method requires the preparation of a database comprising high-fidelity solutions. The spatial bases are extracted from the database through first-level proper orthogonal decomposition (POD). The algebraic relationship between the time trajectory/parameter positions and the projection coefficient is described by the linear superposition of the second-level POD bases (temporal bases) and the second-level projection coefficients (parameter-dependent coefficients). This decomposition strategy decouples the space-time parameter effects, providing a stable foundation for fast predictions of parametrized time-dependent problems. The mappings between the parameter locations and the parameter-dependent coefficients are approximated as Gaussian process regression (GPR) models. The accuracy and efficiency of the PPOD-ROM are demonstrated through two numerical examples: flows past a cylinder and turbine flows with a clocking effect.

A data-driven analysis of short and long laminar separation bubbles

This work investigates the statistical response of short and long laminar separation bubbles to external flow parameters, such as Reynolds number, free-stream turbulence intensity and streamwise pressure gradient, known to govern bubble formation and characteristics. A parametric experimental campaign has been performed using particle image velocimetry on a flat plate to provide a comprehensive database for the characterization of separation-induced transition in both short and long separation bubbles. The proper orthogonal decomposition (POD) was applied to the data set of all dividing streamlines commonly used to identify a laminar separation bubble. This provides an optimal state-space basis for the data-driven classification of the state of a laminar separation bubble, with the leading modes capturing the change in length and height of the laminar separation bubble in response to changes in the flow parameters. When projected onto the POD subspace constituted by the first three leading modes, the normalized data from the present study and the results from prior investigations not used in the modal analysis collapse on the same trajectory in the low-dimensional space. The present POD basis can be therefore adopted for the description of the general response of the time-mean shape of a laminar separation bubble to changes in the main influencing parameters. A well-defined pattern was observed in the case of short laminar separation bubbles in the reduced-order space defined by the first three POD coefficients, whereas a higher dispersion in the long-bubble regime indicates an increased sensitivity of long bubbles to the external flow characteristics.

Reduced-order model-based variational inference with normalizing flows for Bayesian elliptic inverse problems

We propose a reduced-order model-based variational inference method with normalizing flows for Bayesian elliptic inverse problems. The coefficient of the elliptic PDE is represented by a finite number of parameters. We aim to estimate the posterior distributions of these parameters in the framework of variational inference, and the approximation of the posterior distribution is constructed through a normalizing flow. Moreover, as data of inputs and outputs of the forward problem are accumulated during the training of the normalizing flow, we can naturally exploit the low-dimensional intrinsic structure of the forward elliptic PDE using reduced-order models based on these data, in which information of the true posterior is incorporated. We construct a low-dimensional set of data-driven basis functions in the solution space using proper orthogonal decomposition (POD) and train a neural network that maps the parameters to the coefficients of these data-driven basis functions. The surrogate forward map, which is the combination of the reduced-order model and the parameter-to-coefficient neural network, is then applied to the Bayesian elliptic inverse problem, where in each iteration the forward problem is evaluated in the space spanned by the POD basis functions using the surrogate forward map. Thus the inversion will be significantly accelerated as the surrogate forward map is essentially a low-dimensional model. We present numerical examples to show the accuracy and efficiency of the proposed method.

A predictive surrogate model for hemodynamics and structural prediction in abdominal aorta for different physiological conditions

Background and objectiveThis study investigates the application of a Predictive Surrogate Model (PSM) for the prediction of the fluid and solid variables in the abdominal aorta by integrating Proper Orthogonal Decomposition (POD) and Long Short-Term Memory (LSTM) techniques. MethodsThe Fluid-Structure Interaction (FSI) solver, which serves as the Full-Order Model (FOM), can capture the blood hemodynamics and structural mechanics precisely for a variety of physiological states, namely the rest and exercise conditions. ResultsDetailed analyses have been conducted on velocity components, pressure, Wall Shear Stress (WSS), and Oscillatory Shear Index (OSI) variables. Firstly, the reconstruction error has been derived based on a specific number of POD bases to assess the Reduced Order Model (ROM). Notably, the reconstruction error for velocity components in the rest condition is one order of magnitude higher than that in the exercise condition, yet both remained below 10%. This error for pressure is even more minimal, being less than 1%. ConclusionsThe PSM is evaluated against rest and exercise conditions, exhibiting promising results despite the inherent complexities of the physiological conditions. Despite the inherent complexities of phenomena in the aorta, the predictive model demonstrates consistent error magnitudes for velocity components and wall-related indices, while solid variables show slightly higher errors.

Accelerating inverse inference of ensemble Kalman filter via reduced-order model trained using adaptive sparse observations

The implementation of data assimilation often struggles with low computational efficiency due to the complexity of high-fidelity numerical modeling and the large size of the observation vectors. To address this challenge, it is necessary to employ reduced-order strategies for both the full-order dynamical model and observation operator. Proper orthogonal decomposition (POD) based reduced-order model (ROM) and discrete empirical interpolation method (DEIM) based sparse observation identification are often considered as primary solutions. However, their effective combination requires the determination of high-quality POD reduced-order basis vectors. The current research focuses on incorporating the ROM and DEIM into a Kalman filter framework, resulting in a reduced-order Kalman filter with sparse observations. The key aspect is identifying the optimal POD basis, which can be achieved iteratively by introducing a relative root-mean-square-error (RMSE) for a near-optimal exploration of both the desired low-order POD basis and observation locations. The singular value decomposition (SVD) of the snapshots is performed in a low-dimensional subspace, leading to significant computational savings. This improved POD basis not only enhances the reconstruction of the ROM, but also helps build the DEIM observation matrix, resulting in simultaneous dimensionality reduction of the state and observation space. The effectiveness of this algorithm is demonstrated through the retrieval of initial conditions for one-dimensional tsunami wave equations with real data scenarios. Experiments are conducted for both smooth and nonsmoothed initial conditions. The results show that when the observation data is available at 3 and 6 spatial observation locations, our method only requires fewer vectors with the first 45 and 90 dominant modes derived using the RMSE, compared to as many as 210 and 252 derived from the relative energy criterion. Our method enables a low-modal ROM to effectively reconstruct the wave height field, leading to considerable prediction capability for potential long-wave dynamics with comparable accuracy. The CPU time savings achieved through our method is at least one to two orders of magnitude compared to the CPU time required by the full-order model.

A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations

High-fidelity computer simulations of childbirth remain prohibitively expensive and time consuming, making them impractical for guiding decision-making during obstetric emergencies. Cheap computer simulations that preserve the accuracy of high-fidelity models can be developed using surrogate modeling. Two common approaches to surrogate modeling are physics-based reduced order modeling (ROM) and machine learning (ML), with the latter gaining popularity as the scientific computing community seeks to leverage advances from other, mostly non-physics-based, computational strategies. Although ROM and ML have been compared for various problems, to our knowledge, such a comparison for simulations of vaginal deformations is currently missing. This study provides a baseline numerical comparison between methods from these two fundamentally different approaches. Since there are many methods falling into each modeling approach, to provide a fair and natural comparison, we select a basic model from each category, with each allowing (i) a straightforward implementation in commercial software packages, and (ii) use by practitioners with limited experience in the field. As a benchmark for the numerical comparison of the ROM and ML approaches, we use the finite element (FE) modeling of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. From the ROM strategies, we consider a simplified Galerkin ROM (G-ROM) that is based on the linearization of the underlying nonlinear equations. From the ML strategies, we select a feed-forward neural network to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the proper orthogonal decomposition (POD) coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical investigation of G-ROM, ML, and POD-ML takes place in the reconstructive regime. The numerical results show that the G-ROM outperforms the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The G-ROM achieves superior error performance to the best ML model with 11 POD basis functions. With higher-dimensional POD bases, the G-ROM achieves a relative error 3 orders of magnitude lower than that of the best ML model with an online CPU time still on the same order of magnitude as the best ML model. The POD-ML model improves on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model does not improve on the error performance of the ML and is still outperformed by the G-ROM for POD bases of size greater than 11. This baseline numerical investigation serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, and material properties of the human vagina, as well as their changes during labor.

Deep learning methods for blood flow reconstruction in a vessel with contrast enhanced x-ray computed tomography.

The reconstruction of blood velocity in a vessel from contrast enhanced x-ray computed tomography projections is a complex inverse problem. It can be formulated as reconstruction problem with a partial differential equation constraint. A solution can be estimated with the a variational adjoint method and proper orthogonal decomposition (POD) basis. In this work, we investigate new inversion approaches based on PODs coupled with deep learning methods. The effectiveness of the reconstruction methods is shown with simulated realistic stationary blood flows in a vessel. The methods outperform the reduced adjoint method and show large speed-up at the online stage.

An adaptive model order reduction technique for parameter-dependent modular structures

AbstractThis work is concerned with an adaptive reduced order model of modular structures assembled from parameter-dependent substructures. The substructures are reduced by proper orthogonal decomposition (POD) and connected by means of a tied contact formulation. We present a method to adapt the matrices of the substructures to parameter changes. We employ interpolation on Grassmann manifolds for the parametric adaption of the projection matrices. For the adaptation of the stiffness matrices, we use the direct empirical interpolation method (DEIM). Manifold interpolation of the reduced stiffness matrices, cannot be applied here since it would require semi-positive definiteness, which is here not fulfilled because of necessary rigid body motion modes. The novelty of this work is the application of these interpolation methods to the special problem class of POD-based tied contact model order reduction. Furthermore, we show a methodology to compute significant snapshots on the substructure level to compute a POD basis that can be used in different global structures.

A rapid method to predict biaxial fatigue life of automotive wheels using proper orthogonal decomposition and radial basis function algorithm

This paper presents a rapid method for predicting the biaxial fatigue life of automotive wheels using a combination of proper orthogonal decomposition and radial basis function algorithm. Currently, numerical simulations of biaxial fatigue tests are being developed to evaluate wheel performance. However, these simulations are computationally expensive due to the need to simulate multiple discrete loading cases within a given biaxial spectrum. To address this issue, we propose a novel approach that utilizes proper orthogonal decomposition and radial basis function algorithm to improve computational efficiency. By leveraging high-fidelity simulation results from a small number of loading cases, a reduced order model is constructed to accurately predict the tire-rim interface force fields required for wheel strength calculations. The reduced order model significantly reduces the computational time by 65.4% for simulating all loading cases, while maintaining a maximum predicted error of less than 2% compared to the high-fidelity model. Subsequently, the predicted interface forces are mapped onto the rim surface for strength calculation, and the wheel fatigue life is determined using the Brown-Miller multiaxial damage criterion. Comparative analysis with experimental results demonstrates the desirable accuracy of our method in simulating the stress-strain history, crack initiation position, and minimum fatigue life of the wheel. Overall, the proposed method offers a powerful tool for the rapid fatigue analysis of spectrum-loaded wheels, providing an efficient and accurate means of predicting biaxial fatigue life.

A simulation-based software to support the real-time operational parameters selection of tunnel boring machines

With the fact that the main operational parameters of the construction process in mechanized tunneling are currently selected based on monitoring data and engineering experience without exploiting the advantages of computer methods, the focus of this work is to develop a simulation-based real-time assistant system to support the selection of operational parameters. The choice of an appropriate set of these parameters (i.e., the face support pressure, the grouting pressure, and the advance speed) during the operation of tunnel boring machines (TBM) is determined by evaluating different tunneling-induced soil-structure interactions such as the surface settlement, the associated risks on existing structures and the tunnel lining behavior. To evaluate soil-structure behavior, an advanced process-oriented numerical simulation model based on the finite cell method is utilized. To enable the real-time prediction capability of the simulation model for a practical application during the advancement of TBMs, surrogate models based on the Proper Orthogonal Decomposition and Radial Basis Functions (POD-RBF) are adopted. The proposed approach is demonstrated through several synthetic numerical examples inspired by the data of real tunnel projects. The developed methods are integrated into a user-friendly application called SMART to serve as a support platform for tunnel engineers at construction sites. Corresponding to each user adjustment of the input parameters, i.e., each TBM driving scenario, approximately two million outputs of soil-structure interactions are quickly predicted and visualized in seconds, which can provide the site engineers with a rough estimation of the impacts of the chosen scenario on structural responses of the tunnel and above ground structures.

A new method to interpolate POD reduced bases–Application to the parametric model order reduction of a gas bearings supported rotor

SummaryThe proper orthogonal decomposition (POD) is successfully employed in a variety of projection‐based methods for parametric model order reduction (pMOR) of large dynamical systems. It extracts the most energetic modes describing the dynamics of a system from time snapshots of the solution. In practice, these snapshots are computed at user‐defined parameters of the system with a high fidelity model. Then either all the snapshots are concatenated to extract a global POD basis, or a different POD basis is extracted for each parameter value. In the latter case, for unseen target parameters, the POD bases need to be adapted with a dedicated technique. An established method addressing this task is the interpolation on the tangent space to the Grassmann manifold (ITSGM). It interpolates the linear subspaces spanned by the known POD bases, discarding by construction the knowledge on the modes ordering by energy levels. In this paper, we formalize an ordered reduced basis interpolation (ORBI) technique which preserves such ordering. This approach improves the adaptation accuracy of the resulting POD basis as more information is taken into account in the construction of the interpolation operator. Numerical investigations are conducted on the parametric reduced order model of a dynamical system representing a small‐scale gas‐bearings supported rotor. Results show that the proposed method is more accurate than the ITSGM method at a similar computation cost. Trained at only three points of the parameters space, the developed hyper reduced order model (h‐ROM) performs to faster simulations with satisfactory accuracy, even far from the training points.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.