The paper asks how to make the modified Born series (an iterative sum of repeated scattering terms with a complex relaxation parameter) converge for high-frequency, strongly scattering Helmholtz problems in one, two and three dimensions, where the classical Born series diverges because the scattering operator's spectral radius exceeds one. The authors propose selecting a single complex relaxation parameter from an estimated spectrum: eigenvalues precomputed for a constant reference medium are propagated along a homotopy to the radial medium profile using a leading-order WKB approximation, whic

Karamehmedović & Triki · arXiv 2026

To predict a wave inside an object, you can add up its echoes. When the object scatters strongly, the echoes grow louder and the sum explodes. This paper shows how to choose, cheaply and in advance, the one complex dial that makes the sum settle anyway.

Settling…

How to read this. The disk is an object that slows waves down; the dashed ring on its left is a source sending waves into it. Colours show the wave at one instant: amber crests, cyan troughs. What you are watching is a running total: the source's own wave, plus its first echo off the object, plus the echo of that echo, and so on. The meter shows how far the total still is from obeying the wave equation. Switch to Plain sum to watch the very same echoes blow up; as it grows, its colours are toned down so the runaway pattern stays visible, and the meter shows its true size.

01 Every echo, added up

A wave inside an object is a sum of echoes

Send a wave of a single pitch into an object that slows it down. Inside, the wave scatters off the material again and again before it gets out.

One classic way to predict the result is to write it as an endless sum: the wave the source would make in empty space, plus the part of it that scatters once off the object, plus the part that scatters twice, and so on. Each term is one more round of scattering, an echo of the one before. Physicists call this the Born series; on this page it is the plain sum.

How strongly the object slows the wave is its contrast. When the contrast is weak, every echo is fainter than the last and the sum settles on the right answer. Turn the contrast up and watch the echoes.

Turn up the contrast

Predict first. The paper's own object sits at the far right of this slider (1.00). Before you drag: how far right do you think you can go before the sum stops settling?

The running totalecho 0

Echo ladder: how loud is echo n?log scale

Each bar compares one echo with the source's own wave. Bars above the line are louder than the wave itself.

Loudest pattern comes back ×0.57 each round → the sum settles

The tipping point sits at about 0.21 of the paper's contrast. There the loudest echo pattern comes back exactly as strong as it went in: ×1.00 per round. Beyond it, that pattern grows every round and the sum explodes, however many echoes you add.

The paper's actual 2D object sits at 1.00 , where the loudest pattern comes back ×4.7 louder each round. Higher frequencies push the same way: the paper's one-dimensional examples reach ×42.7 . So the question is not whether plain adding fails, but what to do instead.

02 One pattern, one multiplier Illustration

Why echoes either fade or run away

Some wave patterns inside the object have a special property: after one round of scattering they come back as an exact copy of themselves, only multiplied by a number λ.

Call λ the pattern's echo multiplier (mathematicians say eigenvalue). It is a complex number, which sounds worse than it is: think of it as a stretch and a turn , drawn as a point on a plane. Its distance from the centre, |λ|, says how much louder or fainter the pattern comes back; its angle says how far the wave's timing shifts. Two rounds multiply by λ twice, λ². The pattern's share of the total is 1 + λ + λ² + λ³ + …

Drag the multiplier λ

The answer exists; plain adding just can't reach it.

The star at 1/(1 − λ) is the value the endless sum is supposed to reach, and it exists for any λ other than 1. But when |λ| is bigger than 1, each echo is louder than the last and the running total never arrives. The real object has thousands of such patterns at once.

03 A cloud of multipliers

Thousands of patterns, one cloud

The disk has one echo multiplier per pattern. Here are all 3,228 of them for this page's disk model, found the expensive way: by computing every eigenvalue of a 3,228 × 3,228 matrix. The plain sum settles only if every single point sits inside the white circle.

Scale the cloud with the contrast

Linked to the slider in chapter 01: double the contrast and every multiplier doubles.

At the paper's contrast, dozens of patterns sit outside the circle and the plain sum has no chance. But that upward lean of the whole cloud is exactly the opening the next idea uses.

04 The mixing dial

One dial that tucks every multiplier inside

Starting in the late 1980s, Kleinman and colleagues found a way around the explosion: the modified Born series.

Instead of adding each new echo outright, blend it. At every step, keep a share (1 − γ) of the current term and mix in a share γ of its echo. The mixing dial γ (the paper's overrelaxation parameter) is one complex number for the whole object: again, a stretch and a turn.

Blending changes every multiplier λ into a new one, μ = 1 − γ + γλ. In pictures: look at λ from the point 1, then shrink that arrow by |γ| and turn it by γ's angle. The blended sum γ(1 + μ + μ² + …) adds up to exactly the same answer 1/(1 − λ), provided every |μ| is below 1.

Blend one runaway pattern

|γ| = 1 at 0° is the plain sum. Try a turn of about 60° and a size around 0.6.

Now the real thing. Because the whole cloud leans above the axis, a dial pointing almost straight up (γ close to the positive imaginary direction) turns it about a quarter-turn around the point 1 and tucks it inside the circle. The authors proved in earlier work that a dial that works always exists. Your turn to find one.

Find the convergence island

The dial γdrag anywhere

Blended multipliers μpivot: 1

Mismatch vs echoes addedlog scale

Ready: pick a dial, then run.

Why even the best dial leaves the sum slow

Shrinking harder pulls the big multipliers in. But the tiniest multipliers (λ close to 0) all land at μ ≈ 1 − γ, and a small γ parks them right next to the rim. The best dial balances the two, so the slowest pattern ends up just inside the circle. Here it shrinks by ×0.9952 per step and the series needs about three thousand terms. In the paper's hardest one-dimensional example, the dial the method picks gives 0.99984: close to 96,000 terms. Slow, but it gets there, and each term is cheap.

05 The catch

To place the dial you need the cloud, and the cloud is expensive

To choose γ you have to know where the multipliers are. Computing all of them for a realistic object is a giant eigenvalue problem, far more work than simply running the series. The existing rule from Kleinman's group depends on the geometry of that cloud, which, in the paper's words, makes it "less readily applicable in practice."

This paper's contribution is a cheap prediction of the cloud. It takes four moves:

1 · Start uniform An object with the same peak contrast everywhere. Its multipliers are computed once, and reused for every object of that size, frequency and contrast.

2 · Morph and track Morph the uniform object into the real round profile and follow each multiplier. At high frequency this takes one equation per multiplier.

3 · Nudge for the bump If the object is not quite round, shift each multiplier by a small first-order correction.

4 · Choose γ Pick the dial that makes the worst predicted multiplier shrink fastest, with a small disk of radius δ standing in for the tiny untracked ones. This is a convex problem: quick and reliable.

Then run the series. None of the first four moves looks at the source of the wave, which will matter at the end. First, let's watch moves 1, 2 and 4 in the simplest setting: a slab.

06 Morph lab Recomputed from the paper's equations

Morph a uniform slab into the real one and watch the multipliers slide

In one dimension the object is a slab: a layer of material between x = 0 and x = 1. The paper's three slabs are probed at a very high frequency, ten wavelengths across the slab in empty space, and that is what makes them hard: their loudest patterns come back ×42.7, ×11.5 and ×9.5 per round.

Start from a uniform slab whose contrast equals the real slab's peak (open circles show its multipliers). Then morph it, qt = (1 − t)·max q + t·q, as t runs from 0 to 1. At high frequency each pattern behaves like a ray whose local wavelength is set by √(1 + qt/λ). The paper's key observation: while the medium morphs, the pattern's total phase budget across the slab (how many wiggles it fits in, and how much it grows) stays fixed. That turns tracking each multiplier into a single equation, solved by Newton's method.

The slab and one pattern

Echo multipliers zoom to |λ| < 3

Near some turning points Newton's method fails to converge. The paper then "completes" those branches numerically by following the discretised eigenproblem instead: twelve branches in the hardest slab (triangles here). Everything else comes from the one-line equation.

Choose the dial from the prediction, and run

The predicted multipliers, plus a tail disk of radius δ = 0.015 for the untracked tiny ones, go into the convex minimax problem. Out comes γ. Then the series runs on the selected slab.

Mismatch vs echoes addedboth axes log

The wave across the slabRe u(x)

Press run: the series will use the dial chosen from the predicted multipliers.

07 Not quite round

A small bump, a small correction

In two and three dimensions the morph trick needs a round object. Then each pattern is labelled by how many times it winds around the centre, and tracking its multiplier is again one equation per pattern.

Real objects are rarely perfectly round, so the paper allows a round profile plus a small bump: q = qround + ηQ, with the bump Q no bigger than 1 and η small. Its 2D example, the disk on this page, has a four-lobed bump with η = 0.12 .

For the bump, each predicted multiplier gets a first-order nudge. It shifts by λ times the bump averaged over where that pattern lives, divided by the round part averaged the same way:

λη ≈ λ + η · λ ∫Qφ² / ∫qroundφ²

Add the bump

Reported in the paper For the 2D disk, the predicted multipliers (dots) land on the exact ones (crosses): median difference 1.32 % , worst 2.94 % . Figure 7 from Karamehmedović & Triki (2026), arXiv:2610.23428, CC BY 4.0; resized.

08 What the paper shows Reported in the paper

Every test settled, and agreed with a direct solver

The authors ran the full recipe on three slabs and the disk, choosing γ from the predicted multipliers only, and compared the result with a direct solve of the same equations (LU factorisation) and with an independent finite-difference solver. The plain sum exploded every time; the modified sum with the predicted dial settled every time.

The hardest case. The dial chosen from the prediction (white circle, right panel) lands right next to the dial chosen from the exact multipliers (red cross), and the slowest blended pattern shrinks by ×0.999837 per step.

09 Any source, same dial

Move the source, keep the dial

Drag the source

Source at (−0.36, 0.08) · same dial γ = 0.048 + 0.199i

The four setup moves never look at the source.

The dial depends only on the object and the frequency. So once γ is chosen, it works for every source you shine at the object: drag the source anywhere in the disk, and the series reruns with the same γ and no new setup. Faint rings mark where you have tried it before, with the number of terms each run needed.

That is the practical pay-off. With many sources, the cost is a small fixed setup followed by nothing but repeated matrix–vector products, the kind of work graphics processors do well. The paper calls the method "well-suited for GPU implementation."

∞ Back to the opening question

A sum of echoes that grow louder every round can still be trusted, if each echo is blended in with the right dial.

This paper makes choosing that dial cheap and independent of the source, for high-frequency, strongly scattering objects that are round or nearly round. The method was tested in one and two dimensions; the three-dimensional formulas are derived but not yet tested.

One step in the derivation (a division in the paper's equation 7) has no formal justification; the authors assume it is well defined and check it numerically.

A dial that keeps every predicted multiplier inside the circle does not by itself guarantee that every true one is inside, since the predictions carry errors. In all of the paper's tests it was enough.

The ray-style tracking can fail near turning points; those branches are completed numerically, and that extra work is counted in the setup time. The 3D version of the tracking equation is derived but has no numerical example yet.

Source

Mirza Karamehmedović and Faouzi Triki. Accelerated Born series for high-frequency strong scattering by weakly non-radial media. arXiv:2610.23428 [math.NA], 2026. arxiv.org/abs/2610.23428 · CC BY 4.0

Interactive scenes are simplified recomputations: the 2D disk uses a 64 × 64 grid (the paper's runs use finer polar grids), so its numbers differ slightly from the paper's (for example ×4.73 rather than ×4.71 for the loudest pattern, 3,277 rather than 5,543 terms). The 1D slabs use the paper's 768-cell setup. Figures reproduced under CC BY 4.0, resized. The research was funded by Villum Foundation grant 58857.