[SIAM NEWS] Probing the Deep Mathematics of Nonlinear Inverse Problems

By Samuli Siltanen

June 01, 2012

[Samuli Siltanen is a professor at the University of Helsinki and the SIAM News liaison for the SIAM Activity Group on Imaging Science.]

As the SIAM News liaison for the SIAM Activity Group on Imaging Science, Samuli Siltanen suggested that readers might be interested in the work of the most recent recipient of the Calderón Prize, Guillaume Bal. The prize, given biannually by the Inverse Problems International Association, recognizes researchers under the age of 40 who have made significant advances in the mathematics of inverse problems; it was awarded to Bal at the 2011 Applied Inverse Problems Conference at Texas A&M University.

作为SIAM图像科学活动组的SIAM News联络人，Samuli Siltanen认为读者可能会对Calderón Prize最新得主Guillaume Bal的工作感兴趣。这一奖项由Inverse Problems International Association每两年颁发一次，用于嘉奖在反问题数学中取得重大进展的40岁以下研究者；Bal在2011年德州农工大学举办的应用反问题会议上被授予该奖。

Inverse problems are about deciphering indirectly measured data of an unknown quantity, such as the inner structure of a patient. The aim is to go from an effect to the cause, as opposed to the simpler direct problem of going from cause to effect. In medical X-ray tomography, for example, the direct problem is to determine what kind of X-ray images (effect) we would get from different directions when imaging a patient whose inner structure (cause) we know. The inverse problem is the more difficult task of producing a three-dimensional reconstruction of the patient based on several two-dimensional X-ray images.

反问题研究的是间接破译一个未知量的测量数据，例如病人的内部结构。其目的是从结果到原因，而不是更简单的从原因到结果的直接问题。例如，在医学X射线断层摄影术中，直接问题是在成像病人的内部结构（原因）时，我们将从不同方向得到的X射线图像（效果）中确定哪一个。而反问题是基于多个二维X射线图像进行病人三维重建这样一个更难的任务。

X-ray tomography is a linear inverse problem. For many medical measurements, by contrast, the body is probed with energy whose propagation depends on the medium, leading to nonlinear inverse problems. An example is electrical impedance tomography, which is based on feeding harmless electric currents into the body and reconstructing the inner conductivity distribution from the resulting voltages at the skin. The boundary measurements depend on the conductivity in a nonlinear way. Another example is the classic diagnostic method of manual palpation, which can be viewed as boundary measurements based on elastic deformations. Quite deep and interesting mathematics has been needed, and created, for the analysis of nonlinear inverse problems.

X射线断层成像是一个线性反问题。相比之下，在许多医学测量中，我们用能量来探测人体，其能量传播依赖于介质，这导致了非线性反问题。电阻抗断层成像便是其中一个例子，它基于向体内引入无害电流，并从皮肤产生的电压中重建内部电传导分布。边界测量以一种非线性方式依赖于电导率。另一个例子是经典的手工触诊诊断方法，它可以被视作是基于弹性变形的边界测量。对于非线性反问题的分析，已经有了非常深入和有趣的数学。

For Guillaume Bal, hybrid inverse problems, which often arise from nonlinear PDE models, are one of the most exciting areas of his recent work.

In recent years, Guillaume Bal has made a variety of mathematical contributions to the study of nonlinear inverse problems, concentrating for the most part on cases in which the propagation of the probing energy is modeled by partial differential equations. His studies cover a broad scientific domain, including integral geometry, direct and inverse transport models, Monte Carlo methods, asymptotic models for equations with random coefficients, and time-reversal imaging in random media. He has a special interest in hybrid inverse problems arising from coupled-physics measurements.

近年来，Guillaume Bal对非线性反问题的研究作出了许多数学贡献，主要集中在利用偏微分方程来模拟探测能量的传播情况。他的研究涵盖了广泛的科学领域，包括积分几何、直接和反传输模型、蒙特卡洛方法、随机系数方程的渐近模型以及随机介质中的时间反演成像等。他对耦合物理测量中产生的混成反问题特别感兴趣。

The solution of hybrid inverse problems is typically done in two steps. In the first, one solves an inverse boundary-value problem that provides high-resolution information about an internal structure. The internal functional revealed in this way can be viewed as another indirect measurement. The second step consists of solving the inverse problem of interpreting the latter data. The advantage of this procedure is that the second step would be extremely sensitive to measurement errors if the data were collected at the boundary; the virtually collected inner data allows a more stable solution and leads to better resolution.

混成反问题的求解通常分两步进行。第一步，求解一个反边值问题，它给出了内部结构的高分辨信息。以这种方式揭示的内部功能可以被视作是另一种间接测量。第二步，求解解释后面数据的反问题。此过程的优点是，如果在边界处收集数据，那么第二步将对测量误差极为敏感；而实际收集的内部数据允许更稳定的解并导致更好的分辨率。

Photo-acoustic tomography, or PAT, is a prime example of a hybrid inverse problem. In the first step, infrared light pulses are sent into tissue, leading to local thermal expansion at the sites of energy absorption. This rapid expansion creates ultrasound waves that can be detected when they arrive at the skin, allowing a high-resolution reconstruction of the distribution of optical energy absorption. In the second step, optical properties of the tissue are recovered from the internal information provided by the first step. Combining the high resolution of acoustic waves with the large contrast of optical waves, PAT is sometimes called the lightning-and-thunder method. Bal’s recent work has greatly advanced the understanding of the fundamental properties and possibilities of PAT.

光声层析成像（Photo-acoustic tomography，PAT）是混成反问题的一个主要例子。第一步，将红外光脉冲传送到组织中，这导致能量吸收点处的局部热膨胀。这种快速膨胀产生了超声波，该超声波能在到达皮肤时被检测，从而允许光能吸收分布的高分辨率重建。第二步，从第一步给出的内部信息中恢复组织的光学性质。结合声波的高分辨率与光波的大反差，PAT有时被称为雷电法。Bal最近的工作大大促进了对PAT基本性质和可能性的理解。

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In the inverse problems research community, Bal is known as an energetic mathematician whose enthusiasm is catching. In September 2011, during a conference in Edinburgh, I had the chance to interview him for SIAM News.

在反问题研究界，Bal是一位精力充沛的数学家，他的热情极富感染力。在2011年9月爱丁堡的一次会议上，我有机会为SIAM News采访了他。

Among mathematical disciplines, what is it about inverse problems in particular that drew you into the field?

在数学学科中，是什么特别吸引你进入反问题这个领域?

It all started when Oliver Dorn and I met as postdocs at Stanford and discussed the mathematics of inverse problems. Also, an important role was played by the MSRI theme year on inverse problems in 2001. I was working with a time-reversal problem related to transport equations, and the talks at a workshop in November 2001 revealed a variety of new and fascinating mathematical questions.

这一切都要从我在Stanford读博后期间和Oliver Dorn相遇并讨论反问题数学开始。另外，2001年MSRI反问题主题年也发挥了重要作用。当时我正在研究一个与传输方程有关的时间反演问题，而2001年11月的一个研讨会上的报告揭示了许多新的迷人的数学问题。

The main reason for my continuing interest is the fruitful interaction between applications and theory, which is a characteristic feature of inverse problems.

我持续不断的兴趣的主要原因是应用和理论之间富有成效的交互，这是反问题的一个典型特征。

What is the role of hybrid, or combined, imaging modalities among traditional methods?

混成或联合成像方式在传统方法中的作用是什么?

From the practical point of view, it is advantageous to combine two measurements, one having high contrast, the other high resolution. There are a bunch of candidates for practically useful hybrid imaging methods. Judging their potential is based on analyzing the strength of the physical interactions involved: Is there enough signal to yield information?

从实际角度来看，把一种具有高对比度，另一种具有高分辨率的两种测量方法结合起来是很有好处的。有一串非常有用的混成成像方法可供选择。人们基于所涉及的物理相互作用的强度分析来判断它们的潜力：有足够的信号来产生信息吗？

From the theoretical point of view, hybrid inverse problems lead to the study of so-called internal functionals. There are lots of open problems and new fundamental questions about them, so there is no lack of exciting research work to be done. On one hand, new tools are needed for analyzing these internal functionals, and on the other hand, some old tools are readily applicable. So both challenges and ways to proceed are available.

从理论上来看，混成反问题导致了所谓的内部泛函研究。有许多开放问题和新的基本问题，所以并不缺令人兴奋的研究工作要做。一方面，需要新的工具来分析这些内部泛函，另一方面，也可以使用一些旧工具。因此，挑战和前途并存。

(Warming to the topic, Bal continues his comments on hybrid inverse problems.)

（Bal继续谈论混成反问题，且越说越起劲。）

The connection to the theory of nonlinear PDEs is interesting as well—traditionally, nonlinear inverse problems arise from linear PDE models, but in hybrid cases the models are often nonlinear.

与非线性PDEs理论的联系也是有趣的，传统的非线性反问题来自线性PDE模型，但在混成情况下，模型本身往往就是非线性的。

A long-term research objective is to provide a comprehensive classification of inverse problems with internal functionals.

一个长期研究目标是给出一个内部泛函反问题的全面分类。

Which one of your published results are you the most proud of?

你发表的成果中哪一个是你最引以为傲的？

It’s really hard to pick out just one paper. I mean, could you pick one of yours? If I have to, it would be the article “Ray Transforms in Hyperbolic Geometry,” concerning nonlinear tomography in hyperbolic geometry, with applications to single-photon emission computed tomography (SPECT). There the tomographic reconstruction task is reduced to a Riemann–Hilbert problem.

真的很难挑出一篇来。我的意思是，你能挑一个吗？如果我必须这样做，那就是那篇“Ray Transforms in Hyperbolic Geometry”，那篇文章是关于双曲几何中的非线性层析成像，以及在单光子发射计算断层扫描（single-photon emission computed tomography，SPECT）中的应用。那里的层析重建任务被简化为Riemann–Hilbert问题。

Rather than promote a single paper, though, I would say that the recent activity on hybrid inverse problems is the most exciting cohesive work I’ve done in the field of inverse problems theory.

但我想说的是，最近在混成反问题上的活动是我在反问题理论领域所做的最激动人心的内聚工作，而不是为了宣传一篇论文。

How do you see the future of inverse problems?

你如何看待反问题的未来？

Well, new measurement techniques do create new questions all the time, and the theory of inverse problems provides tools for answering them. There will surely be a continuous feed of interesting problems offered by future technologies.

新的测量技术总是产生新的问题，而反问题理论为这些问题提供了工具。而未来的技术肯定会不断带来一些有趣的问题。

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Guillaume Bal received a PhD at the University of Paris VI in 1997 under the supervision of Yvon Maday. For his PhD thesis, “Coupling of Equations and Homogenization in Neutron Transport,” he received the Jean-Pierre Lepetit Prize for the best PhD thesis defended in 1997–1998 at the Direction des Études et Recherches d’Electricité de France (EDF). He is currently a professor of applied mathematics at Columbia University.

Guillaume Bal在Yvon Maday的指导下，于1997年巴黎六大获得博士学位。他的博士论文，“Coupling of Equations and Homogenization in Neutron Transport”，获得了Direction des Études et Recherches d’Electricité de France (EDF)1997-1998年度Jean-Pierre Lepetit最佳博士论文奖。他目前是哥伦比亚大学应用数学教授。

Previous winners of the Calderón Prize were Matti Lassas (2007) and Martin Burger (2009). Lassas is known as a versatile mathematician who has provided innovative solutions to analytic and geometric inverse problems, to questions of invisibility, and to practical imaging problems, such as three-dimensional dental X-ray tomography. Martin Burger, a similarly multi-faceted mathematician, has created and analyzed reconstruction algorithms for medical imaging and image processing. He is an expert on the theory and application of sparsity-promoting inversion, such as total variation regularization and level set methods.

Matti Lassas（2007）和 Martin Burger（2009）是Calderón Prize前任获得者。Lassas是一个多才多艺的数学家，他给出了全新的解析和几何反问题、不可见性问题（译者注：如水印）以及实际成像问题（如三维牙科X射线层析成像）的解。Martin Burger同样是一位多方面的数学家，他创建并分析了医学成像和图像处理中的重建算法。他是稀疏提升反演（如全变分正则化和水平集方法）理论和应用的专家。

Written on August 1st , 2014 by 李军