PIE-Bench: Benchmarking PIE for PDEs

• PIE-Bench is a comprehensive test suite designed for benchmarking the conversion of linear PDEs with spatial-integral terms into Partial Integral Equations.
• It employs a systematic change-of-variable and operator mapping workflow to transform PDE parameters into solvable operator-valued LMIs for stability analysis.
• Validated on prototypical examples like the McKendrick population PDE and reaction-diffusion observers, it enables precise determination of stability margins.

PIE-Bench refers to a rigorous suite of computational tasks and test cases designed for benchmarking the Partial Integral Equation (PIE) framework as applied to linear Partial Differential Equations (PDEs) with spatial integral terms. The PIE framework enables the conversion of infinite-dimensional linear PDEs—including those with polynomial-kernel integral operators—into PIEs, permitting stability analysis through operator-valued Linear Matrix Inequality (LMI) optimization. PIE-Bench systematically exercises all critical process stages: PDE parametrization with integral and boundary conditions, transformation via change of variable and construction of operator maps, encoding and solution of stability LMIs, and validation against analytically known or previously established benchmark results (Shivakumar et al., 2022).

1. PDEs with Spatial-Integral Terms – Parametrization

The PIE-Bench suite encompasses linear PDEs defined on domains $s \in [a, b]$ (typically $[0, 1]$) where the state is partitioned by regularity: $x = \operatorname{col}(x_0, x_1, x_2)$ with $x_0 \in W_0^{n_0}$, $x_1 \in W_1^{n_1}$, $x_2 \in W_2^{n_2}$. The full derivative vector is

$$x_D(t, s) = \operatorname{col}(x_0, x_1, x_2, \partial_s x_1, \partial_s x_2, \partial_s^2 x_2) \in \mathbb{R}^{n_x \times 1}$$

Boundary conditions combine mixed point and integral forms:

$0 = \int_a^b B_I(s) x_D(t,s) \,ds - B x_b(t)$

where $B_I(s) \in \mathbb{R}^{n_{BC} \times n_S}$ and $B \in \mathbb{R}^{n_{BC} \times 2n_S}$, subject to invertibility of the “boundary-matrix” $$B_T = B [T(0)\ | \ T(b-a)] - \int_a^b B_I(s) U_2 T(s-a)\,ds$$. The in-domain dynamics typically adopt the form

$$\dot{x}(t,s) = A_0(s) x_D(t,s) + \int_a^s A_1(s, \theta) x_D(t,\theta)\,d\theta + \int_s^b A_2(s, \theta) x_D(t, \theta)\,d\theta$$

with $A_0(s) \in \mathbb{R}^{n_x \times n_S}$, and $A_1, A_2$ as separable matrix-valued polynomial kernels.

2. Change-of-Variable and PIE Construction

The PIE-Bench methodology employs a systematic change of variables for conversion to the PIE setting. Define $v(t, s) := D x(t, s)$ with

$$D = \operatorname{diag}(I_{n_0}, \partial_s I_{n_1}, \partial_s^2 I_{n_2})$$

ensuring $v \in L_2^{n_x}$ has no continuity requirements. The state $x$ can be reconstructed via an explicit 3-PI integral transform:

$$x(t,s) = (T v)(s) = G_0(s) v(s) + \int_a^s G_1(s, \theta) v(\theta)\,d\theta + \int_s^b G_2(s, \theta) v(\theta)\,d\theta$$

where $G_0, G_1, G_2$ are explicitly defined (see Table 1 for summary).

Kernel	Construction
$G_0(s)$	$[I_{n_0}, 0]$
$G_1(s,\theta)$	$Q_1(s-\theta) + G_2(s, \theta)$
$G_2(s,\theta)$	$[0 \;\; T_1(s-a)\cdot B_Q(\theta)]$

The change of variable is invertible under the specified boundary conditions. The underlying PDE is equivalently expressed as

$T \dot{v}(t) = A v(t)$

where $T$ and $A$ are 3-PI operators with kernels constructed directly from the PDE data.

3. Explicit Operator Mapping Workflow

Transformation from PDE parameters $(n, B, B_I, A_0, A_1, A_2)$ to PIE parameters $(T, A)$ is explicit and algorithmic. The practitioner:

• Assembles block-matrices $T(s)$, $U_1$, $U_2$, $Q(s)$.
• Computes the invertible boundary-matrix $B_T$ and $B_Q(s)$ using the prescribed formulae.
• Constructs 3-PI kernels $G_0, G_1, G_2$ and subsequently $\hat{A}_0, \hat{A}_1, \hat{A}_2$.
• Forms operator-valued mappings $T = P_{\{G_i\}}$, $A = P_{\{\hat{A}_i\}}$ with the integral expressions detailed in the appendix.

A plausible implication is the suitability of PIE-Bench for automated and reproducible benchmarking, as these steps are implementable in environments such as PIETOOLS.

4. Stability Analysis as Operator-Valued LMI

The central experiment of PIE-Bench involves testing exponential stability for $T \dot{v} = A v$ by seeking a 3-PI Lyapunov operator $R = P_{\{R_0, R_1, R_2\}} \succ 0$ satisfying:

• $R \succeq \alpha I$ on $L_2$
• $A^* R T + T^* R A \preceq -\delta T^* T$ for some $\alpha,\delta > 0$

Equivalently, defining $P=R$ and $H := -(A^* P T + T^* P A)$, the LMI constraints are:

• $P \in [\Pi_3]^+$
• $P - \alpha I \succeq 0$
• $H - \delta T^* T \succeq 0$
• $H + A^* P T + T^* P A = 0$

All operator-valued positivity constraints are encoded as sum-of-squares (SOS) LMIs and solved with standard SDP solvers (MOSEK) or PIETOOLS, exploiting the parameterization of operator kernels as polynomials of given degree.

5. Prototypical Benchmark Examples

PIE-Bench incorporates canonical extensions exemplified by:

• McKendrick population PDE with spatial integral boundary: $\dot{x} = -\partial_s x + c x$, $x(t,0) = \int_0^1 (1-s) s x(t,s) ds$. PIE conversion and subsequent operator-LMI analysis yield the critical mortality $c_0 \approx -0.740625$, with stability for $c < c_0$ (extinction threshold).
• Reaction-diffusion observer with polynomially-approximated integral feedback: $\dot{x} = \lambda x + \partial_s^2 x$ and observer with integral feedback in $\partial_s^2$ error, boundary conditions $x(0) = x(1) = \hat{x}(0) = \hat{x}(1) = 0$. For each $\lambda$, polynomial approximation degree $n$ controls successful stability certification; $n=1$ suffices for $\lambda \leq 5$, $n=4$ for $\lambda=6$.

These examples demonstrate the method's ability to identify sharp stability margins and facilitate direct comparison with established analytical results.

6. Implementation Strategies and Validation

PIE-Bench requires:

• Representation of all 3-PI kernels as a finite polynomial basis in $(s, \theta)$.
• Assembly of $T$ and $A$ using block formulas.
• Declaration of decision-variables $R_i(s, \theta)$, enforcement of $R \succ 0$, and the dissipation condition $H \succeq \delta T^* T$.
• Translation of constraints to SOS-LMIs solved via tools such as SOSTOOLS+MOSEK or PIETOOLS.
• Validation by extracting smallest $\alpha, \delta$ ensuring positivity and comparing to published critical parameters ($c_0$, $\lambda$-bounds).

A plausible implication is that PIE-Bench serves both as a functional compliance suite for software and as a comparative metric platform for new PDE-to-PIE conversion methods, LMI solvers, and Lyapunov parameterizations.

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For exhaustive procedural and mathematical detail, see “Computational stability analysis of PDEs with integral terms using the PIE framework” (Shivakumar et al., 2022).