Finite element approximation for quantitative photoacoustic tomography in a diffusive regime

Giovanni S. Alberti; Siyu Cen; Zhi Zhou

REVIEW 2 major objections 6 minor 48 references

Finite element approximation for quantitative photoacoustic tomography in a diffusive regime

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A two-stage finite element scheme recovers the diffusion and absorption coefficients in quantitative photoacoustic tomography with L2 error of order h+η+δ, on a high-probability non-zero-gradient event.

desk verdict Solid first FEM error analysis for two-parameter QPAT with vanishing source; the main theorem holds, but the advertised L2(Ω) and L-dependence overstate what is proved. read the letter →

arxiv 2505.05361 v2 pith:W5YPYJS7 submitted 2025-05-08 math.NA cs.NA

Giovanni S. Alberti , Siyu Cen , Zhi Zhou This is my paper · ORCID

classification math.NAcs.NA MSC 65N2165N3035R30

keywords quantitativephotoacoustictomographyinversediffusivityproblemrandomboundaryilluminationsfiniteelementapproximationweightedenergyestimateleast-squaresregularizationconvergencerate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that quantitative photoacoustic tomography in the diffusive regime can be turned into a provably convergent numerical scheme: recover the diffusivity coefficient $q = D u_1^2$ from ratios of internal energy measurements via a regularized least-squares finite element method, then recover $u_1$ from a direct elliptic solve and read off $D$ and $\sigma$ algebraically. The central result is a rigorous $L^2(\Omega)$ error bound for both coefficients of order $h + \eta + \delta$, where $h$ is the mesh size, $\delta$ the data noise, and $\eta$ the error of the first-stage diffusivity reconstruction. The bound holds with high probability, because random boundary illuminations make the gradient of the quotient solutions non-zero on the reconstruction region. A reader should care because the analysis gives an explicit rule for choosing mesh size and regularization parameter from the measured noise level, turning a heuristic two-step pipeline into a scheme with a proven rate.

What carries the argument

The load-bearing object is the quotient $w^{(\ell)}=H^{(\ell+1)}/H^{(1)}=u^{(\ell+1)}/u^{(1)}$, which satisfies the one-parameter elliptic equation $-\nabla\cdot(q\nabla w)=0$ with $q=D u_1^2$. The main estimate is a weighted energy identity obtained by testing the weak form with $\varphi=(q-\tilde q)w/q$; it puts $\frac12\int_{\Omega'}\frac{(q-\tilde q)^2}{q}|\nabla w|^2\,dx$ on the left-hand side, so the non-zero gradient condition (2.4) turns a small misfit in the quotient data into a small $L^2$ error in $q$. The non-zero condition itself is supplied probabilistically: boundary illuminations are drawn as Gaussian series in an $H^{1/2}(\partial\Omega)$ orthonormal basis, and Proposition 2.1 shows that with probability at least $1-L^d e^{-C_1 L}-L e^{-C_2 M}$ some directional derivative of the quotient solutions is bounded away from zero on $\Omega'$. The numerical side is a standard piecewise-linear Galerkin method with an $H^1$-seminorm penalty, and the discrete error analysis combines interpolation, inverse, and duality estimates to transfer the continuous stability bound to the finite element solution.

What would settle it

On a fixed smooth coefficient pair, compute the quantity $\max_{\ell}|\nabla w^{(\ell)}\cdot\nu|$ on a fine grid over $\Omega'$ for many random illumination draws; on the draws where it stays below the $C_0/2$ threshold on a positive-measure set, run the two-stage scheme with noise-free data and check whether the $L^2$ error still decays at the predicted rate in $h$: if it does, the non-zero gradient condition is not necessary, and if it does not, the theorem's key premise is confirmed.

Watch

Extended reading notes

Core claim

The central discovery, stated as Theorem 3.1, is that the two-stage procedure reconstructs both optical coefficients with $L^2(\Omega)$ error of order $h+\eta+\delta$. In the first stage the quotient $w^{(\ell)} = H^{(\ell+1)}/H^{(1)}$ is an observed function satisfying $-\nabla\cdot(q^\dagger \nabla w^{(\ell)})=0$, so the paper recovers $q^\dagger = D^\dagger|u^{(1)}|^2$ by minimizing a regularized least-squares misfit over piecewise-linear finite elements; Theorem 2.2 controls this first-stage error, and balancing $h^2 L^{1/2}\sim\delta$ with $\alpha\sim\delta^2$ yields the rate $L^{7/8}\delta^{1/4-\epsilon}$ in two dimensions. In the second stage, $v=1/u^{(1)}-1$ solves the direct problem $-\nabla\cdot(q^\dagger\nabla v)=H^{(1)}$ with zero boundary data, and replacing $q^\dagger$ and $H^{(1)}$ by their numerical counterparts gives $v_h$, from which $D^* = q^*|v_h+1|^2$ and $\sigma^* = Z_\delta^{(1)}(v_h+1)$ are formed. The proof uses a weighted energy identity with the special test function $\varphi=(q-\tilde q)w/q$, which converts the non-zero gradient condition into a lower bound on the data misfit and hence into an upper bound on the coefficient error. All statements hold with the probability in (2.3), and the error constant is independent of $h$, $\delta$, and $\alpha$.

Load-bearing premise

All the error bounds collapse if the random boundary illuminations do not make the quotient solutions satisfy $\max_{\ell}|\nabla w^{(\ell)}\cdot\nu|\ge C_0$ on $\Omega'$, and the paper proves this only with overwhelming probability, not deterministically.

Editorial extensions

If this is right

Choosing $h^2 L^{1/2}\approx\delta$ and $\alpha\approx\delta^2$ gives a predicted rate of order $\delta^{1/4-\epsilon}$ for both coefficients in two dimensions, and the numerical experiments report exponents between $0.22$ and $0.42$ for the relative $L^2$ errors.
The theorem supplies a parameter-selection rule: mesh size and regularization strength should be scaled as $h\sim\delta^{1/2}$ and $\alpha\sim\delta^2$ once the noise level is known.
The scheme covers the practical case of vanishing source and boundary-only illumination, which earlier two-observation reconstructions could not handle because their positivity conditions failed.
Because the final error is linear in the first-stage diffusivity error $\eta$, any improvement in numerical inverse diffusivity solvers transfers directly to quantitative photoacoustic tomography.
The piecewise-constant and non-smooth experiments still converge at the predicted rates, indicating that the stated regularity assumptions are sufficient but likely not necessary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Since $w^{(\ell)}$ is formed directly from the measured energies, the non-zero gradient condition could be monitored on a computational grid before inverting, enabling an adaptive acquisition rule that adds random illuminations until condition (2.4) is observed rather than relying only on the probabilistic guarantee.
The $\delta^{1/4}$ rate is probably an artifact of the $L^2$ misfit and $H^1$ penalty; using the weighted energy structure itself as the data fidelity term could restore the $\delta^{1/2}$ rate suggested by the conditional stability estimate.
The same quotient reduction to a source-free inverse diffusivity problem should transfer to other hybrid imaging modalities with boundary-only illumination and internal data, such as conductivity or fluorescence imaging.
The experiments with piecewise-constant coefficients suggest the smoothness assumptions are sufficient but not necessary; testing on L-shaped domains or discontinuous coefficients would map the true boundary of validity.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a solid numerical analysis paper and the main theorem is probably right. It is the first FEM error analysis I know for the two-parameter QPAT problem in a diffusive regime with a vanishing source and product measurement H = σu. The trick is to combine the random boundary illuminations of [1] with weighted energy estimates from [30, 18] in a two-stage least-squares scheme: first solve an inverse diffusivity problem for q = D u_1^2, then solve a direct problem for 1/u_1 and recover D and σ. Theorem 3.1 gives ∥D† - D*∥_{L2(Ω)} + ∥σ† - σ*∥_{L2(Ω)} ≤ C(h + η + δ) with η the diffusivity error, and Proposition 3.1 gives the diffusivity error on Ω′. I checked the chain from Proposition 2.1 through Theorem 2.2 to Theorem 3.1; I do not find a load-bearing flaw. The non-zero condition is genuinely probabilistic, but it is explicit with probability (2.3), so the conditional nature is not hidden.

The numerical section is more than decorative: four smooth/nonsmooth examples plus a piecewise-constant case outside the theory, and observed rates align with the predicted δ^{1/4} (some slightly better). No code or data are shipped, so reproducibility is limited but not a fatal issue for this type of paper.

Soft spots, in decreasing order. (1) The abstract and Section 2 claim an L2(Ω) error estimate for the diffusivity reconstruction, but Theorem 2.2 proves the bound only on Ω′. The final QPAT theorem recovers an L2(Ω) statement for D and σ by defining q* outside Ω′ using the datum Zδ^(1)/σ†, which gives an O(δ) contribution; so the abstract's claim is supported at the end, but the intermediate L2(Ω) diffusivity claim is not. (2) The advertised L^{7/8}δ^{1/4-ε} rate in Remarks 2.4 and 3.1 is not actually derived from the stated bounds; what follows is O(L δ^{1/4}) with the same δ exponent, so the L-dependence is weaker than advertised. This matters for users who want to choose L, but it does not break the convergence result. (3) The proof of Theorem 2.2 relies on elliptic regularity and L2-projection stability in Lq for q < 2d/(d-2); the estimates are sketched but I did not find an inconsistency.

Who is this for? Researchers in quantitative photoacoustic tomography and inverse coefficient identification, especially those using FEM or random-illumination strategies. It deserves a serious referee. I would recommend acceptance after the authors correct the domain mismatch and either prove or demote the advertised L-dependence.

Referee Report

2 major / 6 minor

Summary. The paper develops and analyzes a two-stage finite element method for quantitative photoacoustic tomography (QPAT) in the diffusive regime, reconstructing the diffusion coefficient D and absorption coefficient σ from internal deposited-energy data generated by random boundary illuminations. In the first stage, the problem is reduced to an inverse diffusivity problem (IDP) for q = D u_1^2, which is solved by a regularized output least-squares formulation with P1 finite elements. In the second stage, a direct elliptic problem for v = 1/u_1 - 1 is solved and D, σ are recovered algebraically. The main theoretical results are: a high-probability non-zero gradient condition under random boundary data (Proposition 2.1), a conditional stability estimate (Theorem 2.1), an L2(Ω′) error estimate for the discrete diffusivity (Theorem 2.2), and an L2(Ω) error estimate of order h+η+δ for the final coefficients (Theorem 3.1), with the parameter choice h^2 L^{1/2} ∼ δ and α ∼ δ^2 leading to a δ^{1/4} convergence rate. Numerical experiments on smooth and nonsmooth coefficients illustrate the predicted behavior.

Significance. If the estimates are correct, this is a substantial contribution: it provides a rigorous finite element error analysis for QPAT with randomly chosen illuminations, a regime in which most existing analyses are at the continuous level. The proof chain is coherent and uses appropriate tools: the published probabilistic non-zero condition of [1], the weighted energy estimate of [13,30], and the decoupling of QPAT into an inverse diffusivity problem followed by a direct solve. The authors are explicit about the probabilistic nature of the non-zero condition and about the fact that the IDP estimate is on Ω′. The paper also gives concrete guidance for selecting the mesh size and regularization parameter from the noise level, and the numerical rates are consistent with the predicted δ^{1/4} behavior. These strengths make the paper valuable to the numerical analysis and inverse problems communities.

major comments (2)

[§2.2 (Theorem 2.2, Remark 2.4) and §3 (Remark 3.1)] The L2(Ω) convergence rate advertised in Remarks 2.4 and 3.1 is not a direct consequence of the stated theorems. Theorem 2.2 proves an L2(Ω′) estimate for the diffusivity, and substituting h^2 L^{1/2} ∼ δ and α ∼ δ^2 into that estimate gives ∥q†−q*_h∥_{L2(Ω′)} ≤ C L^{7/8}δ^{1/4} for d=2 and C L^{(7+ε)/8}δ^{(1−ε)/4} for d=3, not the stated C L^{7/8}δ^{1/4−ε}. Moreover, the interpolating argument in Remark 2.4 uses H2 regularity for w(q) with q replaced by the discrete reconstruction q*_h, with constants independent of h; this regularity is not established for piecewise-linear q*_h. Since Remark 3.1 repeats the same rate for the final D and σ, the claim should be either proved from Theorem 2.2 and Theorem 3.1 with correct exponents, or explicitly labeled as heuristic.
[Abstract and §2.2] The abstract states that the paper provides 'a rigorous error estimate in L2(Ω) norm for the numerical reconstruction' without specifying that the intermediate diffusivity estimate is proved only on Ω′. This distinction matters because the IDP solver itself is not shown to be accurate in L2(Ω); the full-domain result is obtained in Theorem 3.1 only after setting q* outside Ω′ to the known coefficient value. Please state the Ω′/Ω distinction explicitly in the abstract and in Remark 2.4 to avoid overclaiming the scope of the IDP result.

minor comments (6)

[§2.2, first paragraph] The manuscript states 'Ω ⊂ R^d (d = 2, 2)'; this should read '(d = 2, 3)'.
[§4.1] The text refers to 'Assumption 3.1(iii)', but Assumption 3.1 has only items (i) and (ii); the condition g^(1) ≡ 1 appears in item (ii). Please correct the cross-reference.
[Lemma 2.2 proof] In several displayed estimates the proof writes 'u(ℓ)' where the intended quantity is 'w(ℓ)(q†)'; please make the notation consistent.
[Remark 2.5] For nonhomogeneous Dirichlet problems on polygonal domains, H2 regularity requires compatibility conditions on the boundary data. Please state the required condition or cite the precise theorem that covers the present case.
[§4] The numerical experiments do not describe the optimization solver used for the least-squares problem, the initialization, or the random seeds for the boundary illuminations. Reporting these details would improve reproducibility; the reported convergence rates also appear to come from single realizations.
[Example 4.5] The piecewise-constant coefficients in Example 4.5 do not satisfy Assumption 3.1; the favorable reconstructions there should be framed as additional numerical evidence rather than as validation of the theory.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central error bounds are derived from external probabilistic and FEM inputs, not from the conclusions they target.

full rationale

The derivation chain is self-contained relative to published external theorems. Proposition 2.1 invokes Alberti [1] to obtain the non-zero gradient condition under random boundary data; this is a self-citation by the first author, but it is an independently published theorem whose assumptions (random Gaussian boundary data, elliptic regularity) do not include the numerical reconstruction that the paper later proves. Theorem 2.1 then derives conditional stability by the weighted energy identity, Theorem 2.2 converts the stability estimate and standard FEM approximation results into an L2(Omega') error bound, Proposition 3.1 transfers that bound verbatim to the IDP reformulation of QPAT, and Theorem 3.1 is a direct error propagation for the second stage using the reconstructed diffusivity and the noisy energy datum Z_delta^(1). No equation in the proof is equivalent to its conclusion by construction: the recovered D* and sigma* are algebraic functions of the forward solution of a regularized least-squares problem and a direct elliptic solve, not re-statements of the data or of the fitted parameters. The numerical experiments are presented as illustrations of the predicted rates rather than as inputs to the proof. The only notable gap is a scope/rigor issue, not circularity: Remark 2.4 and Remark 3.1 advertise a full-domain L2(Omega) rate L^{7/8} delta^{1/4-eps}, while Theorem 2.2 and Proposition 3.1 state their bounds on the subdomain Omega'; the extension to Omega\Omega' is not derived from the stated estimates. This affects the advertised domain of convergence, not the independence of the proof from its conclusions.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The mathematical arguments rest on standard elliptic regularity, the probabilistic non-zero gradient result from [1], and domain regularity/known-coefficient assumptions. No fitted constants are introduced into the theory; L, M, h, and α are algorithmic parameters. No new physical entities are postulated.

free parameters (4)

L: number of random boundary illuminations = L=5 in experiments; theory requires L large enough
The error bounds and the probability (2.3) depend on L, and experiments fix L=5, which is small relative to the high-probability regime used in the theorems.
M: truncation order in boundary data expansion = M=5 in experiments; theory requires M large enough
Proposition 2.1 requires M large for the tail probability L exp(-C2 M), but the experiments use only five basis terms.
C0: non-zero condition threshold = 0.1 in experiments
C0 enters all stability constants, and the experiments verify the non-zero condition using the numerical threshold C0=0.1.
mesh size h and regularization parameter α = h ~ δ^(1/2), α ~ δ^2 in experiments
These are algorithmic parameters chosen a priori from the noise level rather than fitted to the exact solution; the error bounds explicitly depend on them.

assumptions (4)

domain assumption Assumption 3.1: C^{1,1} domain, D†∈W^{2,p}, σ†∈Aσ, coefficients known outside Ω′, and boundary data generated by (2.2) with eigenbasis and θ_k decay
Freely imposed regularity and prior knowledge; all error bounds in Section 3, including elliptic regularity for u(1), rely on it.
domain assumption Non-zero gradient theorem of [1]: random boundary data give max_ℓ |∇w_ℓ·ν| ≥ C0 with high probability
Proposition 2.1 states a variant of this theorem and uses it as a black box; the entire stability and error analysis is conditioned on this event.
standard math Standard elliptic regularity and FEM interpolation/projection estimates used throughout Sections 2 and 3
Used to obtain w∈H2∩W1,∞ and C^{1,κ} regularity, interpolation errors O(h^s), and L2 projection stability, as cited from [14,24,25,41].
domain assumption Noise model (3.2) with bounded L2 perturbations and the lower bound 0 < c0 ≤ Zδ^(1) ≤ 1
The quotient data wδ = Zδ^(ℓ+1)/Zδ^(1) is controlled by c0; without the lower bound the noise propagation estimate fails.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Finite element approximation for quantitative photoacoustic tomography in a diffusive regime." pith.science (2026). https://pith.science/paper/W5YPYJS7

@misc{pith2026250505361,
  author       = {Pith},
  title        = {Pith review of: Finite element approximation for quantitative photoacoustic tomography in a diffusive regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5YPYJS7}},
  note         = {Machine review of arXiv:2505.05361}
}

abstract

In this paper, we focus on the numerical analysis of quantitative photoacoustic tomography. Our goal is to reconstruct the optical coefficients, i.e., the diffusion and absorption coefficients, using multiple internal observational data. The foundation of our numerical algorithm lies in solving an inverse diffusivity problem and a direct problem associated with elliptic equations. The stability of the inverse problem depends critically on a non-zero condition in the internal observations, a condition that can be met using randomly chosen boundary excitation data. Utilizing these randomly generated boundary data, we implement an output least squares formulation combined with finite element discretization to solve the inverse problem. In this scenario, we provide a rigorous error estimate in $L^2(\Omega)$ norm for the numerical reconstruction using a weighted energy estimate, inspired by the analysis of a newly proposed conditional stability result. The resulting error estimate serves as a valuable guide for selecting appropriate regularization parameters and discretization mesh sizes according to the noise levels present in the data. Several numerical experiments are presented to support our theoretical results and illustrate the effectiveness of our numerical scheme.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 41 canonical work pages

[1]

G. S. Alberti. Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems. Inverse Problems, 38(12):Paper No. 124005, 26, 2022

work page 2022
[2]

G. S. Alberti, G. Bal, and M. Di Cristo. Critical points for elliptic equations with prescribed boundary conditions. Arch. Ration. Mech. Anal., 226(1):117–141, 2017

work page 2017
[3]

G. S. Alberti, P. Campodonico, and M. Santacesaria. Compressed sensing photoacoustic tomography reduces to compressed sensing for undersampled Fourier measurements. SIAM J. Imaging Sci. , 14(3):1039–1077, 2021

work page 2021
[4]

G. S. Alberti and Y. Capdeboscq. Lectures on elliptic methods for hybrid inverse problems , volume 25. Soci´ et´ e Math´ ematique de France, 2018

2018
[5]

G. S. Alberti and Y. Capdeboscq. Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging. Int. Math. Res. Not. IMRN , (6):4387–4406, 2022

work page 2022
[6]

Alessandrini

G. Alessandrini. An identification problem for an elliptic equation in two variables. Ann. Mat. Pura Appl. (4), 145:265–295, 1986

work page 1986
[7]

Alessandrini, M

G. Alessandrini, M. Di Cristo, E. Francini, and S. Vessella. Stability for quantitative photoacoustic tomography with well-chosen illuminations. Ann. Mat. Pura Appl. (4) , 196(2):395–406, 2017

work page 2017
[8]

Alessandrini and R

G. Alessandrini and R. Magnanini. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal. , 25(5):1259–1268, 1994

work page 1994

Show all 48 references

[9]

S. R. Arridge. Optical tomography in medical imaging. Inverse Problems, 15(2):R41–R93, 1999

1999
[10]

N. Y. Bakaev. Maximum norm resolvent estimates for elliptic finite element operators. BIT, 41(2):215–239, 2001

2001
[11]

Bal and K

G. Bal and K. Ren. Multi-source quantitative photoacoustic tomography in a diffusive regime. Inverse Problems, 27(7):075003, 20, 2011

2011
[12]

Bal and G

G. Bal and G. Uhlmann. Inverse diffusion theory of photoacoustics. Inverse Problems, 26(8):085010, 20, 2010

2010
[13]

Bonito, A

A. Bonito, A. Cohen, R. DeVore, G. Petrova, and G. Welper. Diffusion coefficients estimation for elliptic partial differential equations. SIAM J. Math. Anal. , 49(2):1570–1592, 2017

2017
[14]

S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods , volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008

2008
[15]

Brezis and P

H. Brezis and P. Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35(5):1355–1376, 2018

2018
[16]

K. M. Case and P. F. Zweifel. Linear transport theory . Addison-Wesley Publishing Co., Reading, Mass.- London-Don Mills, Ont., 1967

1967
[17]

S. Cen, B. Jin, Q. Quan, and Z. Zhou. Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic problems. IMA J. Numer. Anal. , 44(5):3059–3093, 2024

2024
[18]

Cen and Z

S. Cen and Z. Zhou. Numerical reconstruction of diffusion and potential coefficients from two observations: decoupled recovery and error estimates. SIAM J. Numer. Anal. , 62(5):2276–2307, 2024. 23

2024
[19]

P. G. Ciarlet and P.-A. Raviart. Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg. , 1:217–249, 1972

1972
[20]

Crouzeix and V

M. Crouzeix and V. Thom´ ee. The stability in Lp and W 1 p of the L2-projection onto finite element function spaces. Math. Comp., 48(178):521–532, 1987

1987
[21]

Ern and J.-L

A. Ern and J.-L. Guermond. Theory and practice of finite elements , volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004

2004
[22]

R. S. Falk. Error estimates for the numerical identification of a variable coefficient. Math. Comp., 40(162):537– 546, 1983

1983
[23]

Finch, S

D. Finch, S. K. Patch, and Rakesh. Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal. , 35(5):1213–1240, 2004

2004
[24]

Giaquinta and L

M. Giaquinta and L. Martinazzi. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs , volume 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) . Edizioni della Normale, Pisa, second edition, 2012

2012
[25]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, volume 224 of Grundleh- ren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer- Verlag, Berlin, second edition, 1983

1983
[26]

Grisvard

P. Grisvard. Elliptic problems in nonsmooth domains , volume 24 of Monographs and Studies in Mathematics . Pitman (Advanced Publishing Program), Boston, MA, 1985

1985
[27]

B. Jin, X. Li, Q. Quan, and Z. Zhou. Conductivity imaging from internal measurements with mixed least- squares deep neural networks. SIAM J. Imaging Sci. , 17(1):147–187, 2024

2024
[28]

B. Jin, X. Lu, Q. Quan, and Z. Zhou. Convergence rate analysis of Galerkin approximation of inverse potential problem. Inverse Problems, 39(1):Paper No. 015008, 26, 2023

2023
[29]

B. Jin, Q. Quan, and W. Zhang. Stochastic convergence analysis of inverse potential problem. arXiv preprint arXiv:2410.14106, 2024

2024 arXiv
[30]

Jin and Z

B. Jin and Z. Zhou. Error analysis of finite element approximations of diffusion coefficient identification for elliptic and parabolic problems. SIAM J. Numer. Anal. , 59(1):119–142, 2021

2021
[31]

Kuchment and L

P. Kuchment and L. Kunyansky. Mathematics of thermoacoustic tomography. European J. Appl. Math. , 19(2):191–224, 2008

2008
[32]

Kunyansky, B

L. Kunyansky, B. Holman, and B. T. Cox. Photoacoustic tomography in a rectangular reflecting cavity.Inverse Problems, 29(12):125010, 20, 2013

2013
[33]

L. A. Kunyansky. Explicit inversion formulae for the spherical mean Radon transform. Inverse Problems , 23(1):373–383, 2007

2007
[34]

M. Lenoir. Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. , 23(3):562–580, 1986

1986
[35]

Li and L

C. Li and L. V. Wang. Photoacoustic tomography and sensing in biomedicine. Physics in Medicine & Biology, 54(19):R59, 2009

2009
[36]

Lions and E

J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. I, volume Band 181 of Die Grundlehren der mathematischen Wissenschaften . Springer-Verlag, New York-Heidelberg,
[37]

S. K. Patch and O. Scherzer. Guest editors’ introduction: Photo- and thermo-acoustic imaging. Inverse Problems, 23(6):S1–S10, 2007

2007
[38]

G. R. Richter. An inverse problem for the steady state diffusion equation. SIAM J. Appl. Math., 41(2):210–221, 1981

1981
[39]

G. R. Richter. Numerical identification of a spatially varying diffusion coefficient. Math. Comp., 36(154):375– 386, 1981

1981
[40]

Safarov and D

Y. Safarov and D. Vassiliev. The asymptotic distribution of eigenvalues of partial differential operators, volume 155 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1997. Translated from the Russian manuscript by the authors

1997
[41]

Thom´ ee.Galerkin finite element methods for parabolic problems , volume 25 of Springer Series in Compu- tational Mathematics

V. Thom´ ee.Galerkin finite element methods for parabolic problems , volume 25 of Springer Series in Compu- tational Mathematics. Springer-Verlag, Berlin, second edition, 2006

2006
[42]

Vershynin

R. Vershynin. High-dimensional probability, volume 47 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018. An introduction with applications in data science, With a foreword by Sara van de Geer

2018
[43]

Wang and J

L. Wang and J. Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete Contin. Dyn. Syst. Ser. B , 14(4):1641–1670, 2010

2010
[44]

L. V. Wang. Multiscale photoacoustic microscopy and computed tomography. Nature photonics, 3(9):503–509, 2009

2009
[45]

Zhang, Z

Z. Zhang, Z. Zhang, and Z. Zhou. Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation. SIAM J. Numer. Anal. , 60(5):2834–2865, 2022

2022
[46]

Zl´ amal

M. Zl´ amal. Curved elements in the finite element method. I. SIAM J. Numer. Anal. , 10:229–240, 1973

1973
[47]

Zl´ amal

M. Zl´ amal. Curved elements in the finite element method. II. SIAM J. Numer. Anal. , 11:347–362, 1974. 24

1974
[1972]

Translated from the French by P. Kenneth

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

G. S. Alberti. Non-zero constraints in elliptic PDE with random boundary values and applications to hybrid inverse problems. Inverse Problems, 38(12):Paper No. 124005, 26, 2022

work page 2022

[2] [2]

G. S. Alberti, G. Bal, and M. Di Cristo. Critical points for elliptic equations with prescribed boundary conditions. Arch. Ration. Mech. Anal., 226(1):117–141, 2017

work page 2017

[3] [3]

G. S. Alberti, P. Campodonico, and M. Santacesaria. Compressed sensing photoacoustic tomography reduces to compressed sensing for undersampled Fourier measurements. SIAM J. Imaging Sci. , 14(3):1039–1077, 2021

work page 2021

[4] [4]

G. S. Alberti and Y. Capdeboscq. Lectures on elliptic methods for hybrid inverse problems , volume 25. Soci´ et´ e Math´ ematique de France, 2018

2018

[5] [5]

G. S. Alberti and Y. Capdeboscq. Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging. Int. Math. Res. Not. IMRN , (6):4387–4406, 2022

work page 2022

[6] [6]

Alessandrini

G. Alessandrini. An identification problem for an elliptic equation in two variables. Ann. Mat. Pura Appl. (4), 145:265–295, 1986

work page 1986

[7] [7]

Alessandrini, M

G. Alessandrini, M. Di Cristo, E. Francini, and S. Vessella. Stability for quantitative photoacoustic tomography with well-chosen illuminations. Ann. Mat. Pura Appl. (4) , 196(2):395–406, 2017

work page 2017

[8] [8]

Alessandrini and R

G. Alessandrini and R. Magnanini. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal. , 25(5):1259–1268, 1994

work page 1994