# An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem

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## Abstract

**:**

## 1. Introduction

#### 1.1. Nomenclature

## 2. Nonlinear Reference Problem

#### 2.1. Strong Formulation

**p**. The nonlinearity of the problem is introduced by an isotropic Fourier-type heat conductivity $\mu (u;\mathit{p})$ that depends on the current temperature u and the parameter vector. The Dirichlet and Neumann boundary data are denoted by ${u}_{*}$ and ${q}_{*}$, respectively. Note that, in the example, ${q}_{*}=0$ is considered. However, the formulation of the weak form is considered with ${q}_{*}$ being arbitrary for the sake of generality. The corresponding Dirichlet and Neumann boundaries are denoted by ${\Gamma}_{1}$ and ${\Gamma}_{2}$, respectively. The strong formulation of the boundary value problem is

#### 2.2. Weak Formulation

**p**via ${\overline{u}}_{*}$, the space of test functions ${\mathcal{V}}_{0}$ is independent of the parameters

**p**. For the heat conductivity $\mu $, an explicit dependence on the temperature via the nonlinear constitutive model

**p**in the present context is

#### 2.3. Discrete Formulation

**p**are given by

**N**denotes a row vector of finite element ansatz functions for the temperature, and

**G**and ${\mathit{G}}^{T}$ are the discrete gradient and divergence matrices, respectively. The FE approximation of the temperature u and its gradient $\nabla u$ are given by

**J**as an approximation of the differential stiffness matrix ${\mathit{J}}_{*}$. The resulting iteration scheme is commonly referred to as successive substitution (Ref. [22], p. 66). Note also that

**J**is well defined for arbitrary thermal fields, while the exact Jacobian is not defined, or more precisely the Jacobian is semi-smooth, for the critical temperature ${u}_{\mathrm{c}}=({\mu}_{1}-{\mu}_{0})/c,$ which implies a non-differentiability of the conductivity $\mu $.

Algorithm 1: Finite Element Solution |

#### 2.4. Reduced Basis Ansatz

**C**are defined by the discrete snapshot solutions ${w}^{\left(i\right)}$ via the unit mass matrix

**M**as follows:

**M**.

Algorithm 2: Snapshot Proper Orthogonal Decomposition (Snapshot POD) |

## 3. Sampling-Based Reductions

#### 3.1. Galerkin Reduced Basis Approximation

**p**during the subsequent iteration, which provides the new iterate ${\mathbf{\gamma}}^{(\alpha +1)}$ as the solution of

**J**and the residual

**r**leads to the linear system

Algorithm 3: Galerkin Reduced Basis Solution (Online Phase) |

**V**) can replicate the FEM solution to a high accuracy (see Section 4). It also provides a significant reduction of the memory requirements: instead of $\mathit{u}\in {\mathbb{R}}^{n}$, only $\mathbf{\gamma}\in {\mathbb{R}}^{m}$ needs to be stored. Despite the significant reduction of the number of unknowns from n to m, the Galerkin RB cannot attain substantial accelerations of the nonlinear simulation due to a computationally expensive assembly procedure with complexity $\mathcal{O}\left({n}_{\mathrm{gp}}\right)$ for the residual vector

**r**and for the fixed point operator

**J**(compare ${c}_{\mathrm{rhs}}$ and ${c}_{\mathrm{Jac}}$ in Algorithm 1 and Algorithm 3). Here, ${n}_{\mathrm{gp}}$ is the number of quadrature points in the mesh. However, if the linear systems are not solved with optimal complexity, e.g., using sparse $LU$ or Cholesky decompositions with at least $\mathcal{O}\left({n}^{2}\right)$, then a reduction of complexity can still be achieved. It shall be pointed out that, for very large n (i.e., for millions of unknowns), the linear solver usually dominates the overall computational expense. Then, the Galerkin RB may provide good accelerations without further modifications.

#### 3.2. Discrete Empirical Interpolation Method (DEIM)

**r**of the form

**r**: As the residual

**r**is zero for all snapshots, we would try to find a basis for a zero-dimensional space $\mathrm{span}\left(\mathcal{Y}\right)=0$, which is not possible for $M>0$. However, the residual at the intermediate (non-equilibrium) iterates is non-zero and this is also a good target quantity for the (D)EIM, as these terms appear on the right-hand side of the linear system during the fixed point iteration. Hence, a reasonable set $\mathcal{Y}$ is obtained via

**U**can be recognized: If $\mathit{J}(\mathit{V}\mathbf{\gamma};\mathit{p})\mathit{V}\delta \mathbf{\gamma}\in \mathrm{colspan}\left(\mathit{U}\right)$ and $\mathit{r}(\mathit{V}\mathbf{\gamma};\mathit{p})\in \mathrm{colspan}\left(\mathit{U}\right),$ then we are exactly solving the Galerkin–POD reduced linearized system

**J**or $\mathit{J}\mathit{V}$ and

**r**.

Algorithm 4: Offline Phase of the Discrete Empirical Interpolation Method (DEIM) |

Algorithm 5: Online Phase of the Discrete Empirical Interpolation Method (DEIM) | |

Input : parameters $\mathit{p}\in \mathcal{P}$ reduced basis V, POD-DEIM sampling matrix $\mathit{X}:={\mathit{W}}^{T}\mathit{U}{\left({\mathit{P}}^{T}\mathit{U}\right)}^{-1}$ and magic point index set IOutput: reduced vector $\mathbf{\gamma}$ and nodal temperatures $\tilde{\mathit{u}}$ (optional) | |

1 set ${\mathbf{\gamma}}^{\left(0\right)}=0$; ${\tilde{\mathit{u}}}^{\left(0\right)}={\overline{\mathit{u}}}_{*}\left(\mathit{p}\right)$; $\alpha =0$ ; | // initialize |

2 $\overline{\mathit{J}}\leftarrow {(\mathit{J}({\tilde{\mathit{u}}}^{\left(\alpha \right)};\mathit{p})\mathit{V})}_{I}$ ; | // ${c}_{\mathrm{Jac}}$; evaluate M rows of right-projected Jacobian |

3 $\overline{\mathit{r}}\leftarrow {(\mathit{r}({\tilde{\mathit{u}}}^{\left(\alpha \right)};\mathit{p}))}_{I}$ ; | // ${c}_{\mathrm{rhs}}$; evaluate M rows of right hand side |

4 solve $\mathit{X}\overline{\mathit{J}}\delta {\mathbf{\gamma}}^{\left(\alpha \right)}=-\mathit{X}\overline{\mathit{r}}$ ; | // ${c}_{\mathrm{sol}}$; fixpoint iter. for $\delta {\mathbf{\gamma}}^{\left(\alpha \right)}$ |

5 update ${\mathbf{\gamma}}^{(\alpha +1)}\leftarrow {\mathbf{\gamma}}^{\left(\alpha \right)}+\delta {\mathbf{\gamma}}^{\left(\alpha \right)}$; | // update |

6 compute ${\mathit{P}}_{\overline{M}}{\tilde{\mathit{u}}}^{(\alpha +1)}\leftarrow {\mathit{P}}_{\overline{M}}{\tilde{\mathit{u}}}^{\left(\alpha \right)}+{\mathit{P}}_{\overline{M}}\mathit{V}\delta {\mathbf{\gamma}}^{\left(\alpha \right)}$ and set $\alpha \leftarrow \alpha +1$ ; | // ${c}_{\mathrm{reloc}}$ |

7 converged ($\parallel \delta {\tilde{u}}^{\left(\alpha \right)}{\parallel}_{{L}^{2}\left(\Omega \right)}={\parallel \delta {\mathbf{\gamma}}^{\left(\alpha \right)}\parallel}_{{l}^{2}}<{\u03f5}_{\mathrm{max}}$)? → end; else: goto 2 |

#### 3.3. Hyper-Reduction (HR)

**Z**. More precisely, if $\mathcal{F}=\{{i}_{1},{i}_{2},\dots ,{i}_{l}\}$ with $l:=\mathrm{card}\left(\mathcal{F}\right)$, then

**J**is given by (15) and ${\mathbf{\gamma}}^{(\alpha +1)}:={\mathbf{\gamma}}^{\left(\alpha \right)}+\delta {\mathbf{\gamma}}^{\left(\alpha \right)}$.

**Z**, we introduce also the operator $\overline{\mathit{Z}}\in {\mathbb{R}}^{n\times \overline{l}}$ that is a truncated projection operator onto the $\overline{l}$ points contained in the RID. In practice, the discrete unknowns are computed at these $\overline{l}\ge l$ points in order to compute the residual at the inner points l. Note that often $\overline{l}$ is significantly larger than l, especially if the RID consists of disconnected (scattered) regions.

**J**on the left-hand side term scale with $2\zeta lm+2l{m}^{2}$, where $\zeta $ is the maximum number of non-zero entries per row of

**J**. For the right-hand side, the computational complexity is $2lm$. For both products, the complexity reduction factors are $n/l$. To obtain a well-posed hyper-reduced problem, one requires to fulfill the following condition $l\ge m$. If this condition is not fulfilled, the linear system of Equation (45) is rank deficient. In case of rank deficiency, one has to add more surrounding elements to the RID. The closer l to m, and the lower m, the less complex is the solution of the hyper-reduced formulation. The RID construction must generate a sufficiently large RID. If not, the convergence can be hampered, the number of correction steps can be increased and, moreover, the accuracy of the prediction can suffer. When ${\Omega}_{Z}=\Omega $, then

**Z**is the identity matrix and the hyper-reduced formulation coincides with the usual system obtained by the Galerkin projection. An a posteriori error estimator for hyper-reduced approximations has recently been proposed in [31] for generalized standard materials.

#### 3.4. Methodological Comparison

**Z**is a square permutation matrix, hence being invertible and yielding $\mathit{Z}{\mathit{Z}}^{T}=\mathit{I}$, thus (45) reduces to the POD–Galerkin reduced system (27). For the (D)EIM, this implies that the magic points consist of all grid points. We similarly obtain that

**P**and

**U**are invertible and thus $\mathit{U}{\left({\mathit{P}}^{T}\mathit{U}\right)}^{-1}{\mathit{P}}^{T}=\mathit{I}$ and (38) also reproduces the POD–Galerkin reduced system (27).

Algorithm 6: Offline Phase of the Hyper-Reduction (HR) |

Algorithm 7: Online Phase of the Hyper-Reduction (HR) |

**u**) also approximates

**r**and

**J**well. This is a very reasonable assumption in symmetric elliptic problems and—in a certain way—it mimics the idea of having the same ansatz and test space as in any Galerkin formulation. However, from a mathematical point of view, it may not be valid in some more general cases, as in principle

**U**and

**V**are completely independent. For example, we can multiply the vectorial residual Equation (11) by an arbitrary regular matrix, hence arbitrarily change

**r**(and thus

**U**for the DEIM), but not changing

**u**at all (i.e., not changing the POD-basis

**U**). Hence, the collateral basis in the (D)EIM is first an additional technical ingredient and difficulty, which in turn allows for adopting the approximation space to the quantities that needs to be approximated well.

#### 3.5. Computational Complexity

- the computation of the local unknowns and of their gradients ${c}_{\mathrm{reloc}}$ (gradient/temperature computation),
- the evaluations of the (nonlinear) constitutive model ${c}_{\mathrm{const}}$,
- the assembly of the residual ${c}_{\mathrm{rhs}}$ and of the Jacobian ${c}_{\mathrm{Jac}}$,
- the solution of the (dense) reduced linear system ${c}_{\mathrm{sol}}$.

**Finite Element Simulation**(${n}_{\mathrm{gp}}$: number of integration points; ${n}_{\mathrm{el}}$: number of elements; ${n}_{\mathrm{el}}^{\mathrm{DOF}}$: degrees of freedom per element)$$\begin{array}{ccc}\hfill {c}_{\mathrm{reloc}}& =2{n}_{\mathrm{gp}}{n}_{\mathrm{el}}^{\mathrm{DOF}}\hfill & (\mathrm{gradient}/\mathrm{temperature}\phantom{\rule{4.pt}{0ex}}\mathrm{computation})\hfill \\ \hfill {c}_{\mathrm{const}}& \sim {n}_{\mathrm{gp}}\hfill & \left(\mathrm{constitutive}\phantom{\rule{4.pt}{0ex}}\mathrm{model}\right)\hfill \\ \hfill {c}_{\mathrm{rhs}}& =2{n}_{\mathrm{gp}}{n}_{\mathrm{el}}^{\mathrm{DOF}}\hfill & \left(\mathrm{residual}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\right)\hfill \\ \hfill {c}_{\mathrm{Jac}}& ={n}_{\mathrm{gp}}\left[{\left({n}_{\mathrm{el}}^{\mathrm{DOF}}\right)}^{2}+4{n}_{\mathrm{el}}^{\mathrm{DOF}}\right]\hfill & \left(\mathrm{Jacobian}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\right)\hfill \\ \hfill {c}_{\mathrm{sol}}& \sim {n}^{2}.\hfill \end{array}$$**Galerkin–POD**$$\begin{array}{ccc}\hfill {c}_{\mathrm{reloc}}& =3{n}_{\mathrm{gp}}m\hfill & (\mathrm{gradient}/\mathrm{temperature}\phantom{\rule{4.pt}{0ex}}\mathrm{computation})\hfill \\ \hfill {c}_{\mathrm{const}}& \sim {n}_{\mathrm{gp}}\hfill & \left(\mathrm{constitutive}\phantom{\rule{4.pt}{0ex}}\mathrm{model}\right)\hfill \\ \hfill {c}_{\mathrm{rhs}}& =2m{n}_{\mathrm{gp}}\hfill & \left(\mathrm{direct}\phantom{\rule{4.pt}{0ex}}\mathrm{residual}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\right)\hfill \\ \hfill {c}_{\mathrm{Jac}}& =(4m+{m}^{2}){n}_{\mathrm{gp}}\hfill & \left(\mathrm{direct}\phantom{\rule{4.pt}{0ex}}\mathrm{Jacobian}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\right)\hfill \\ \hfill {c}_{\mathrm{sol}}& \sim {m}^{3}.\hfill \end{array}$$**Hyper-Reduction**In the following, ${n}_{\mathrm{gp}}^{\mathrm{RID}}$ is the number of integration points in the RID. Furthermore, ${c}_{\mathrm{FE}}^{N,B}$ denotes the cost for the evaluation of u and $\nabla u$ using the FE matrices**N**and**G**and ${c}_{\mathrm{FE}}^{r}$ is the related to the cost for the residual computation on element level (both at least linear in the number of nodes per element + scattered assembly + overhead) and ${c}_{\mathrm{FE}}^{K}$ is the cost related to the contribution to the element stiffness at one element (∼number of nodes per element squared + scattered assembly + overhead). Lastly, ${c}_{\mathrm{A}}$ is the cost for the Jacobian assembly (i.e., matrix scatter operations).$$\begin{array}{ccc}\hfill {c}_{\mathrm{reloc}}& =\overline{l}m+{n}_{\mathrm{gp}}^{\mathrm{RID}}{c}_{\mathrm{FE}}^{N,B}\hfill & (\mathrm{get}\phantom{\rule{4.pt}{0ex}}u,{\partial}_{x}u,{\partial}_{y}u\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}{\Omega}_{Z})\hfill \\ \hfill {c}_{\mathrm{const}}& \sim {n}_{\mathrm{gp}}^{\mathrm{RID}}\hfill & \left(\mathrm{constitutive}\phantom{\rule{4.pt}{0ex}}\mathrm{model}\right)\hfill \\ \hfill {c}_{\mathrm{rhs}}& ={n}_{\mathrm{gp}}^{\mathrm{RID}}{c}_{\mathrm{FE}}^{r}+ml\hfill & \left(\mathrm{residual}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{projection}\right)\hfill \\ \hfill {c}_{\mathrm{Jac}}& =(m\omega +{m}^{2})l+{n}_{\mathrm{gp}}^{\mathrm{RID}}{c}_{\mathrm{FE}}^{K}+{c}_{\mathrm{A}}\hfill & \left(\mathrm{Jacobian}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{projection}\right)\hfill \\ \hfill {c}_{\mathrm{sol}}& \sim {m}^{3}.\hfill \end{array}$$**Discrete Empirical Interpolation Method**The computational cost for the DEIM is closely related to the one of the HR by substituting M for l and $\overline{M}$ for $\overline{l}$ (denoting the number of nodes which are needed to evaluate the residual at the M magic points). Similar to the other methods, we denote ${n}_{\mathrm{gp}}^{\mathrm{DEIM}}$ as the number of quadrature points to evaluate the entries of the residual and Jacobian. In the cost notation of the Algorithm 5, we obtain$$\begin{array}{cc}\hfill {c}_{\mathrm{reloc}}& \sim m+nm\hfill \\ \hfill {c}_{\mathrm{const}}& ={n}_{\mathrm{gp}}^{\mathrm{DEIM}}\hfill & \left(\mathrm{constitutive}\phantom{\rule{4.pt}{0ex}}\mathrm{model}\right)\hfill \\ \hfill {c}_{\mathrm{rhs}}& \sim M\overline{M}\hfill & \left(\mathrm{residual}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{projection}\right)\hfill \\ \hfill {c}_{\mathrm{Jac}}& \sim M\overline{M}m\hfill & \left(\mathrm{Jacobian}\phantom{\rule{4.pt}{0ex}}\mathrm{assembly}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{projection}\right)\hfill \\ \hfill {c}_{\mathrm{sol}}& \sim {m}^{3}.\hfill \end{array}$$

## 4. Numerical Results

#### 4.1. ONLINE/OFFLINE Decomposition and RB Identification

#### 4.2. Test Cases

**[A]**- A diagonal in the parameter space is considered with$$\begin{array}{cccccc}\hfill {\mathit{p}}^{\left(j\right)}& :={\mathit{p}}_{0}+{\beta}^{\left(j\right)}(\widehat{\mathit{p}}-{\mathit{p}}_{0}),\hfill & \hfill {\mathit{p}}_{0}& :=[0,0,1,1,1/2],\hfill & \hfill \widehat{\mathit{p}}& :=[1,1,2,1,1/2].\hfill \end{array}$$A total of 101 equally spaced values of ${\beta}^{\left(j\right)}$ was chosen, i.e., ${\beta}^{\left(j\right)}=\frac{j}{100}$ for $j=0,1,\cdots ,100$.
**[B]**- A set of 1000 random parameters ${\mathit{p}}^{\left(j\right)}$ was generated using a uniform distribution in parameter space, i.e., a uniform distribution $\mathcal{U}\left(\right[0,1\left]\right)$ was chosen for ${g}_{\mathrm{x}}$, ${g}_{\mathrm{y}}$ and the parameter c was assumed to be distributed via $\mathcal{U}\left(\right[1,2\left]\right)$.

#### 4.3. Certification of the Galerkin RB Method

**[A]**and

**[B]**. The minimum, the mean and the maximum of $\eta \left(\mathit{p}\right)$ were determined for the 101 and 1000 test of case

**[A]**and

**[B]**, respectively. The results shown in Table 2 state the POD approximation error is found close to the projection error. This confirms the quality of the chosen RB.

**[B]**only a finite number of random parameter vectors was chosen which does not necessarily contain the extreme values of $\eta \left(\mathit{p}\right)$. The numerical data in Table 2 for test case

**[A]**shows that indeed,

**[A]**contains parameters leading to larger values of $\eta \left(\mathit{p}\right)$. When increasing the size of the random parameter set for

**[B]**, the maximum values of $\eta \left(\mathit{p}\right)$ in case

**[B]**should be equal or larger than the maximum values of case

**[A]**.

**[A]**the minimum error is truly zero for ${g}_{\mathrm{x}}={g}_{\mathrm{y}}=0$, which implies a homogeneous zero temperature. For an RB of dimension 32, the mean error is well below 10

^{−3}for all tests and the maximum error over all tests is 3.23 × 10

^{−3}. This basis provides a compromise between accuracy and computational cost and is therefore used for the comparison of the methods in the sequel.

#### 4.4. Application to Uncertainty Quantification

**[C]**and assume that the random variables are independent and uniformly distributed as already introduced in test case

**[B]**

**[B]**and ${u}_{\star}$ and ${m}_{\star}^{k}$ are the approximations of the solution and its k-th centered moment ($k>1$) obtained from Finite Elements, Galerkin RB, DEIM and HR, i.e., $\star \in \{\mathrm{h},\mathrm{RB},\mathrm{DEIM},\mathrm{HR}\}$, respectively (e.g., ${m}_{\mathrm{h}}^{3}$ is the third statistical moment obtained from Finite Element computations). For the DEIM, we selected a fixed number of $M=400$ magic points, which is a conservative choice (see Figure 4 for a discussion). In practice, one may choose an adaptive selection strategy.

#### 4.5. Accuracy of the HR and DEIM vs. Number of Interpolation Points

**[A]**and

**[B]**. The resulting relative errors are compared in Figure 4 in terms of the statistical distribution function $P\left(t\right)$ of the relative error, i.e., the probability of finding a relative error $\delta $ that is smaller than or equal to t. Obviously the number of interpolation points has a pronounced impact on the distribution. Generally, the error function for a low number of points states a significant increase of the computational error due to DEIM in comparison with the POD. With an increasing number of points, the distribution function approaches the one of the POD. In our test, the use of more than 300 sampling points can only improve the accuracy in a minor way. We must note that, in general, the accuracy of the DEIM must not be a monotonic function of interpolation points number.

**[A]**and

**[B]**. The resulting relative errors are compared in Figure 6 in terms of the statistical distribution function $P\left(t\right)$ of the relative error. With increasing number of layers, the distribution function approaches the one of the POD. The number of internal points, l, increases rapidly when increasing the number of layers. In the case of the DEIM, the growth of interpolation points is much more progressive. More than two layers of elements do not improve the accuracy significantly.

**[A]**, black line) and Figure 6 (left, test case

**[A]**, red line), where 80% of the samples lead to errors below $\approx 0.002$ for the DEIM and $\approx 0.01$ for the HR. Nevertheless, the accuracy of the hyper-reduced predictions is generally of the same order of magnitude as the accuracy of the DEIM.

#### 4.6. Computing Times

**[B]**is approx. 10 times slower than

**[A]**.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Geometry of the planar benchmark problem (

**left**) and nonlinearity of the temperature dependent conductivity $\mu $ (

**right**).

**Figure 2.**Parameter dependent conductivity $\mu (u;\mathit{p})$ (top row) and solution $u(x;\mathit{p})$ (bottom row) for three different snapshot parameters.

**Figure 3.**Decay of the spectrum of C (normalized to largest eigenvalue ${\xi}_{1}$) and the relative projection error ${E}_{m}$ of the snapshot data defined by (49).

**Figure 4.**Statistical distribution function of the relative error of the DEIM for different numbers of magic points $M\in \{150,200,250,300\}$ (dimension of POD basis: $m=32$).

**Figure 5.**${L}^{2}\left(\Omega \right)$ error of the centered moments w.r.t. the finite element solution. Please note that we have set ${m}_{\star}^{1}:={\overline{u}}_{\star}$ for simplicity of notation.

**Figure 6.**Statistical distribution function of the relative error of the hyper-reduction for different layers of elements added to the RID $\ell \in \{1,2,3,4\}$ (dimension of POD basis: $m=32$).

**Figure 7.**Position of the magic points (blue points) and the additional points required for the evaluation of the residual (green points); Hyper-reduction (left) vs. DEIM (middle, right) for $m=32$.

Dimension of RB | 16 | 24 | 32 | 48 | 60 |
---|---|---|---|---|---|

${E}_{m}$ | 6.629 × 10^{−3} | 4.120 × 10^{−3} | 3.180 × 10^{−3} | 2.017 × 10^{−3} | 1.486 × 10^{−3} |

**Table 2.**Computed values of $\eta \left(\mathit{p}\right)$ for different modes sets and for test cases

**[A]**,

**[B]**; the last row represents the upper bound ${\eta}_{\mathrm{UB}}^{\mathrm{h}}\ge \eta \left(\mathit{p}\right)$.

Test Case | [A] | [B] | ||||
---|---|---|---|---|---|---|

min. | mean | max. | min. | mean | max. | |

$m=16$ | 1.000 | 1.4708 | 1.8248 | 1.000 | 1.3185 | 2.0239 |

$m=24$ | 1.000 | 1.9088 | 2.8926 | 1.000 | 1.3187 | 2.6559 |

$m=32$ | 1.000 | 1.7273 | 2.7441 | 1.000 | 1.3679 | 2.4393 |

$m=48$ | 1.000 | 1.5386 | 2.0232 | 1.000 | 1.3051 | 1.9371 |

$m=60$ | 1.000 | 1.5096 | 1.9333 | 1.000 | 1.3447 | 1.8922 |

${\eta}_{\mathrm{UB}}^{\mathrm{h}}$ cf. (52) | 62.12 | 64.693 | 70.581 | 62.51 | 74.728 | 82.504 |

Test Case | [A] | [B] | ||||
---|---|---|---|---|---|---|

min. | mean | max. | min. | mean | max. | |

$m=16$ | 0.00 10^{−3} | 2.54 10^{−3} | 7.28 10^{−3} | 0.67 10^{−3} | 1.75 10^{−3} | 7.49 10^{−3} |

$m=24$ | 0.00 10^{−3} | 1.07 10^{−3} | 3.80 10^{−3} | 0.16 10^{−3} | 0.88 10^{−3} | 5.29 10^{−3} |

$m=32$ | 0.00 10^{−3} | 0.82 10^{−3} | 3.23 10^{−3} | 0.18 10^{−3} | 0.63 10^{−3} | 3.13 10^{−3} |

$m=48$ | 0.00 10^{−3} | 0.31 10^{−3} | 1.14 10^{−3} | 0.06 10^{−3} | 0.33 10^{−3} | 2.07 10^{−3} |

$m=60$ | 0.00 10^{−3} | 0.20 10^{−3} | 0.70 10^{−3} | 0.05 10^{−3} | 0.26 10^{−3} | 1.55 10^{−3} |

**Table 4.**Elapsed time and number of failures of the nonlinear solver for the solution of all computations of test case

**[A]**(101 solves) and

**[B]**(1000 solves) for Finite Element (FE), Hyper-Reduction (HR) with varying number of additional element layers $\ell \in \{1,2,3,4\}$, DEIM for $M\in \{200,250,300,400\}$ magic points; all computations are carried out using $N=32$ POD modes.

Test Case [A] | Test Case [B] | |||
---|---|---|---|---|

Time | Fail | Time | Fail | |

FE | 59.9 s | - | 660.5 | - |

HR, $\ell =1$ | 10.6 s | 0 | 118.9 s | 5 |

HR, $\ell =2$ | 17.4 s | 0 | 180.6 s | 2 |

HR, $\ell =3$ | 24.5 s | 0 | 247.2 s | 1 |

HR, $\ell =4$ | 30.9 s | 0 | 303.5 s | 0 |

DEIM, $M=200$ | 20.6 s | 0 | 292.3 s | 108 |

DEIM, $M=250$ | 21.4 s | 0 | 240.6 s | 7 |

DEIM, $M=300$ | 24.4 s | 0 | 249.0 s | 3 |

DEIM, $M=400$ | 27.6 s | 0 | 272.7 s | 2 |

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## Share and Cite

**MDPI and ACS Style**

Fritzen, F.; Haasdonk, B.; Ryckelynck, D.; Schöps, S.
An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem. *Math. Comput. Appl.* **2018**, *23*, 8.
https://doi.org/10.3390/mca23010008

**AMA Style**

Fritzen F, Haasdonk B, Ryckelynck D, Schöps S.
An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem. *Mathematical and Computational Applications*. 2018; 23(1):8.
https://doi.org/10.3390/mca23010008

**Chicago/Turabian Style**

Fritzen, Felix, Bernard Haasdonk, David Ryckelynck, and Sebastian Schöps.
2018. "An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem" *Mathematical and Computational Applications* 23, no. 1: 8.
https://doi.org/10.3390/mca23010008