Semidiscrete Inverse Source Problem

• Semidiscrete inverse source problems are inverse problems where one variable is discretized while others remain continuous, enabling the formulation of coupled finite-dimensional subproblems.
• They appear in various settings, including time-discretized PDE evolution, piecewise constant source models, spatial discretization in stochastic frameworks, and frequency-truncated Helmholtz formulations.
• Analytic tools such as Rothe’s method, Carleman estimates, and regularization strategies are employed to ensure uniqueness, stability, and convergence of the inverse reconstructions.

A semidiscrete inverse source problem is an inverse problem in which the source is reconstructed under a mixed continuous–discrete representation: one variable is discretized while the remaining variables remain continuous, or a continuous model is reduced to a finite-dimensional inverse system through a semidiscretization procedure. In the recent literature, this term covers several non-equivalent but structurally related settings: time-discretized reconstruction of a time-dependent source coefficient in an evolution equation, piecewise constant-in-time source identification in parabolic dynamics, space-semidiscrete stochastic parabolic inversion, and frequency-semidiscrete reformulations of multi-frequency Helmholtz source recovery (Bockstal et al., 7 Feb 2025, Lin et al., 2021, Lecaros et al., 3 Sep 2025, Nguyen et al., 2019). A central theme is that semidiscretization converts the inverse problem into a coupled system of algebraic, elliptic, or finite-dimensional subproblems while preserving enough analytic structure to establish existence, uniqueness, stability, or convergence.

1. Scope and defining formulations

The phrase “semidiscrete inverse source problem” is used in at least three precise senses in the cited literature.

First, it can mean a time-discrete inverse source problem for a continuous-in-space PDE. In the semilinear pseudo-parabolic setting, time is partitioned by Rothe’s method, derivatives are replaced by backward differences, and at each time level one reconstructs a discrete source coefficient together with the state (Bockstal et al., 7 Feb 2025).

Second, it can mean that the unknown source itself is semi-discrete, typically piecewise constant in time and continuous in space. For the linear parabolic equation,

$$F(x,t)=\sum_{k=1}^K p_k(x)\,\chi_{t\in[t_{k-1},t_k)},$$

the unknown consists of the switching times and the spatial source states $p_k\in L^2(\Omega)$, while the governing PDE remains continuous (Lin et al., 2021).

Third, it can mean a space semidiscrete inverse problem, where the underlying stochastic parabolic equation is discretized in space but remains continuous in time. In that setting, the unknown is a random source term in the diffusion part of the semidiscrete SPDE, identified from terminal data and partial boundary flux traces (Lecaros et al., 3 Sep 2025).

A related usage appears in inverse potential and Helmholtz problems. There the forward model is continuous, but the inverse reconstruction is carried out over a finite-dimensional source space or a truncated frequency basis. The potential-theoretic formulation approximates a continuous boundary source by piecewise constant coefficients on boundary segments and solves a Nonnegative Least Squares problem (Vabishchevich, 6 Oct 2025). The multi-frequency Helmholtz formulation expands the $k$-dependence in a finite orthonormal basis and reduces the inverse source problem to a coupled elliptic system for finitely many modal coefficients (Nguyen et al., 2019). This suggests that “semidiscrete” is best understood structurally rather than purely by whether time or space is discretized.

2. Canonical PDE models and observation operators

A representative time-discrete model is the semilinear pseudo-parabolic equation on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$,

$$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$

where the source is separated into a known spatial factor $p(t,\mathbf{x})$ and an unknown scalar coefficient $h(t)$ (Bockstal et al., 7 Feb 2025). The inverse data are nonlocal: $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$ Integrating the PDE over $\Omega$ and using the Neumann boundary condition yields an explicit reconstruction formula for $h(t)$, provided

$p_k\in L^2(\Omega)$0

so identifiability is tied directly to a nonvanishing spatial average of the source shape (Bockstal et al., 7 Feb 2025).

In the parabolic semi-discrete source model, the forward equation is

$p_k\in L^2(\Omega)$1

with homogeneous Dirichlet boundary condition and zero initial condition. The measurements are sparse boundary fluxes

$p_k\in L^2(\Omega)$2

where $p_k\in L^2(\Omega)$3 is an arbitrary nonempty open subset (Lin et al., 2021). Here the discrete structure lies in time, but the profiles $p_k\in L^2(\Omega)$4 remain continuous in space.

In the space-semidiscrete stochastic parabolic problem, the unknown is the random source term $p_k\in L^2(\Omega)$5 in

$p_k\in L^2(\Omega)$6

on the spatial mesh $p_k\in L^2(\Omega)$7, with homogeneous discrete Dirichlet boundary condition (Lecaros et al., 3 Sep 2025). The observation operator is

$p_k\in L^2(\Omega)$8

combining terminal data with the trace of the discrete normal derivative on a subset of the discrete boundary (Lecaros et al., 3 Sep 2025).

Other observation models illustrate further semidiscrete patterns. The inverse potential problem uses values of the potential on an observation curve $p_k\in L^2(\Omega)$9 and reconstructs a nonnegative boundary density on a candidate enclosing boundary $k$0 (Vabishchevich, 6 Oct 2025). The multi-frequency inverse source problem for the Helmholtz equation assumes boundary data for all wavenumbers $k$1 and projects the frequency dependence onto a finite orthonormal basis $k$2, producing finitely many spatial coefficient functions $k$3 (Nguyen et al., 2019).

3. Time semidiscretization, Rothe’s method, and stepwise reconstruction

The most explicit semidiscrete construction in the cited corpus is the Rothe scheme for the pseudo-parabolic inverse source problem. Let $k$4, $k$5, $k$6, and

$k$7

The time-discrete inverse problem at level $k$8 is: find $k$9 and $\Omega\subset\mathbb{R}^d$0 such that, for all $\Omega\subset\mathbb{R}^d$1,

$\Omega\subset\mathbb{R}^d$2

with

$\Omega\subset\mathbb{R}^d$3

This produces a two-stage update: first reconstruct $\Omega\subset\mathbb{R}^d$4 explicitly from the previous state $\Omega\subset\mathbb{R}^d$5 and measured data, then solve a linear elliptic variational problem for $\Omega\subset\mathbb{R}^d$6 (Bockstal et al., 7 Feb 2025).

The analysis is stepwise. Once $\Omega\subset\mathbb{R}^d$7 is known, $\Omega\subset\mathbb{R}^d$8 is explicit, and the bilinear form

$\Omega\subset\mathbb{R}^d$9

is coercive: $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$0 Hence the Lax–Milgram lemma yields a unique $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$1, and induction gives a unique sequence $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$2 (Bockstal et al., 7 Feb 2025).

The same paper derives the discrete energy estimate

$$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$3

for $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$4, and then introduces piecewise linear and piecewise constant Rothe interpolants $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$5. Compactness and convergence arguments show that a subsequence converges to the unique weak solution of the continuous inverse problem (Bockstal et al., 7 Feb 2025).

A different notion of time semidiscretization appears in the parabolic source-sequence model. There the source is itself piecewise constant in time, so the unknown is not generated by discretizing a continuous source coefficient but prescribed as

$$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$6

This structure enables interval-by-interval spectral analysis and a sequential Bayesian interpretation in which the spatial states $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$7 are estimated state by state (Lin et al., 2021). The distinction is significant: in the Rothe formulation, the semidiscrete problem is an approximation method for a continuous inverse problem; in the parabolic sequence model, the semidiscrete structure is part of the model itself.

4. Spatial semidiscretization and finite-dimensional source representations

Space-semidiscrete inverse source problems replace differential operators by discrete operators while retaining continuous time. In the stochastic parabolic setting, the discrete difference operators $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$8, averaging operators $$\left\{ \begin{array}{rl} \partial_t u(t,\mathbf{x}) - \nabla \cdot\left(\eta(t,\mathbf{x}) \nabla \partial_t u(t,\mathbf{x})\right) - \nabla \cdot \left( \kappa(t,\mathbf{x}) \nabla u (t,\mathbf{x})\right) & = f(u(t,\mathbf{x})) + p(t,\mathbf{x})\,h(t),\ \kappa(t,\mathbf{x}) \nabla u(t,\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) = g(t,\mathbf{x}),\ u(0,\mathbf{x}) = \tilde{u}_0(\mathbf{x}), \end{array} \right.$$9, and discrete trace operators $p(t,\mathbf{x})$0 are defined on a mesh $p(t,\mathbf{x})$1, and the semidiscrete operator is

$p(t,\mathbf{x})$2

The inverse problem is to recover the adapted random source $p(t,\mathbf{x})$3 from the observation operator $p(t,\mathbf{x})$4 (Lecaros et al., 3 Sep 2025).

The principal analytic tool is a global Carleman estimate for the semidiscrete stochastic parabolic operator. For homogeneous boundary data, the estimate controls weighted norms of $p(t,\mathbf{x})$5, discrete first and second derivatives, and the stochastic term $p(t,\mathbf{x})$6, by the interior source terms, boundary flux terms on $p(t,\mathbf{x})$7, and the terminal term at $p(t,\mathbf{x})$8 (Lecaros et al., 3 Sep 2025). Applied to the difference of two solutions, it yields the Lipschitz estimate

$p(t,\mathbf{x})$9

with $h(t)$0 independent of $h(t)$1, under the structural condition

$h(t)$2

for $h(t)$3 (Lecaros et al., 3 Sep 2025). This is a genuinely semidiscrete stability theorem rather than a continuous theorem subsequently discretized.

A different finite-dimensional source representation appears in inverse potential theory. There, a continuous volume potential is approximated by a single-layer potential on a candidate boundary $h(t)$4, and $h(t)$5 is partitioned into $h(t)$6 boundary segments $h(t)$7. Approximating the single-layer density by piecewise constants $h(t)$8 yields

$h(t)$9

and the inverse problem becomes the discrete Nonnegative Least Squares problem

$$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$0

The paper explicitly describes this as a semidiscrete inverse source problem in which the forward model is continuous but the inverse reconstruction is performed over a finite-dimensional source space (Vabishchevich, 6 Oct 2025).

The multi-frequency Helmholtz formulation is semidiscrete in the auxiliary frequency variable. Writing

$$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$1

and projecting the differentiated Helmholtz equation onto the basis $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$2 leads to the coupled elliptic system

$$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$3

The source term is eliminated before spatial discretization, and the inverse source problem is recast as an overdetermined boundary value problem for $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$4 (Nguyen et al., 2019). This suggests that semidiscretization can act not only on physical coordinates but also on spectral or parameter variables.

5. Identifiability, uniqueness, and regularization mechanisms

Identifiability in semidiscrete inverse source problems is tied to the chosen representation.

For the pseudo-parabolic problem, uniqueness follows from the representation of $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$5 through the measurement $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$6 and the condition $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$7. Theorem 3.1 states that there exists at most one couple $$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$8 such that

$$\int_\Omega u(t,\mathbf{x})\,d\mathbf{x} = m(t),\quad t>0.$$9

and Theorem 4.4 establishes existence and uniqueness of the weak solution pair $\Omega$0 under the stated assumptions on $\Omega$1 (Bockstal et al., 7 Feb 2025).

For the parabolic semi-discrete source model, the main theorem is an identifiability result from sparse boundary measurements: the flux data on any nonempty open subset $\Omega$2 uniquely determine the semi-discrete source

$\Omega$3

More precisely, equality of the measured flux traces implies equality of the time grid, the number of intervals, and all spatial profiles $\Omega$4 (Lin et al., 2021). No stability estimate is proved there, so the result is uniqueness without quantitative continuity.

In the space-semidiscrete stochastic problem, the Carleman estimate yields a stability theorem rather than merely uniqueness. The source is not only identifiable but determined in a Lipschitz-stable way from terminal and boundary-flux observations (Lecaros et al., 3 Sep 2025).

Regularization strategies differ correspondingly. The pseudo-parabolic Rothe method regularizes indirectly through time discretization, coercivity, energy bounds, and compactness (Bockstal et al., 7 Feb 2025). The inverse potential formulation uses a priori nonnegativity, implemented through NNLS, and also discusses Tikhonov regularization

$\Omega$5

for noisy data (Vabishchevich, 6 Oct 2025). The Helmholtz frequency-semidiscrete method applies the quasi-reversibility method to a coupled elliptic system and proves convergence of the minimizers by means of a new Carleman estimate (Nguyen et al., 2019). The hyperbolic inverse source paper transforms the problem to a Volterra integro-differential equation and proves Lipschitz-like convergence of the quasi-reversibility method: $\Omega$6 (Nguyen, 2018).

A recurrent misconception is that semidiscretization is purely numerical. The literature indicates otherwise. In some papers, semidiscreteness is an approximation device for a continuous inverse problem (Bockstal et al., 7 Feb 2025); in others, it is the actual source model (Lin et al., 2021); in others still, it is the natural analytic framework for a spatially discrete operator (Lecaros et al., 3 Sep 2025) or a finite-dimensional boundary source representation (Vabishchevich, 6 Oct 2025). This suggests that the term denotes a class of inverse formulations rather than a single numerical technique.

6. Algorithms, numerical evidence, and broader research directions

The pseudo-parabolic Rothe algorithm is explicitly stepwise: at each time level, compute $\Omega$7 from the discrete reconstruction formula and then solve the linear elliptic variational problem for $\Omega$8 (Bockstal et al., 7 Feb 2025). In one-dimensional experiments on $\Omega$9 with

$h(t)$0

Backward Euler with $h(t)$1, P1 Lagrange finite elements on 200 elements, and implementation in FEniCSx (DOLFINx 0.9.0), the paper reports that the reconstructed $h(t)$2 closely matches the exact source for all tested noise levels and that, in the noise-free case,

$h(t)$3

indicating first-order convergence in time (Bockstal et al., 7 Feb 2025).

The parabolic semi-discrete source paper adopts a Bayesian sequential prediction viewpoint. In the numerical section, the source states are parameterized as moving square patches on $h(t)$4, the time grid is known, and a Sequential Importance Resampling particle filter is used with $h(t)$5 particles. Across two multiscale permeability fields and two source trajectories, the estimated positions closely match the true trajectories (Lin et al., 2021). This does not numerically reconstruct general $h(t)$6, but it demonstrates that the semidiscrete temporal state-sequence model is compatible with sequential Bayesian inversion.

In inverse potential theory, the NNLS formulation is embedded in a moving-window search over candidate source domains. The residual

$h(t)$7

and the total mass

$h(t)$8

act as localization indicators. The paper reports that low residuals form a plateau when the window contains the true sources, whereas residuals increase when part of the source lies outside the candidate domain (Vabishchevich, 6 Oct 2025).

The frequency-semidiscrete Helmholtz method combines a basis truncation in $h(t)$9, a coupled elliptic PDE system in $p_k\in L^2(\Omega)$00, and a discretized quasi-reversibility solver. In 2D tests with 5% noise, the reconstructed shapes and amplitudes of the source are reported to agree well with the truth both for Cauchy data and Dirichlet-only data, although convergence is proved only for the overdetermined formulation (Nguyen et al., 2019).

The hyperbolic quasi-reversibility paper gives a fully discrete implementation of the Volterra-based reformulation. Noisy boundary data are differentiated in time by a Tikhonov-regularized linear system, then a regularized normal equation is solved for the discrete approximation of the auxiliary field $p_k\in L^2(\Omega)$01, and finally the source is reconstructed from the discrete analogue of the identity relating $p_k\in L^2(\Omega)$02 to $p_k\in L^2(\Omega)$03. Numerical tests in two dimensions with noise levels $p_k\in L^2(\Omega)$04, $p_k\in L^2(\Omega)$05, and $p_k\in L^2(\Omega)$06 show visually accurate recovery of shapes and amplitudes (Nguyen, 2018).

Broader directions already present in the literature include stochastic semidiscrete inverse sources in arbitrary dimensions (Lecaros et al., 3 Sep 2025), time-fractional inverse source problems with discrete random noise and spectral cutoff regularization (Huy et al., 2016), discrete Helmholtz multi-frequency inverse sources with phased and phaseless data (Novikov et al., 2024), and temporal-source recovery in coupled subdiffusion systems, where the paper explicitly states that the continuous framework is one “you would want to semidiscretize in space” (BenSalah et al., 12 Mar 2026). A plausible implication is that semidiscrete inverse source problems now function as an interface between PDE identifiability theory, finite-dimensional regularization, and computational state-space inversion rather than as a narrowly numerical subtopic.