# Research Area C4

# Harmonic analysis and numerical multiscale approximation

## Research Area Leaders: Martin Rumpf, Christoph Thiele

## PIs: Martin Griebel, Martin Rumpf, Christoph Thiele

## Contributions by Alex Amenta, Markus Bachmayr, Carsten Burstedde, Felipe Goncalves, Herbert Koch, Lillian Pierce (Duke University), Olli Saari, Leonardo Tolomeo, Pavel Zorin-Kranich

## Topic and goals

This research area focuses on multiscale phenomena, which are the core of many open problems in pure and applied mathematics. Harmonic analysis, in particular the analysis of function spaces and frames as well as optimal approximation therein, plays a fundamental role in the study of these phenomena. The theory of Riesz bases and multiscale expansions is used in time-frequency analysis and in modern approaches to numerical discretization. We develop and use sharp estimates in suitable choices of function spaces to establish well-posedness of PDEs and to derive efficient a posteriori estimates for nonconvex variational problems. We study dispersive equations from the rough-path perspective, and we investigate the interplay between multiscale frame representations of random fields and the approximation of PDEs with stochastic coefficients.

## State of the art, our expertise

**Riesz frames and multiscale expansions**. The time-frequency analysis of multilinear singular integrals was initiated by Lacey–Thiele in the context of Carleson-type theorems and the bilinear Hilbert transform. This theory uses suitable frames of wave packets to decompose the multilinear objects in question. It has been reformulated in the framework of an Lptheory for outer measures in [DT15]. Being a quite general integration theory, it works in the presence of overdeterminacy in the underlying space and has been applied by DiPlinio–Guo–Thiele–Zorin-Kranich to the analysis of directional operators. Moreover, Demeter–Thiele [DT10] studied entangled multilinear singular integrals, which led to applications in ergodic theory [DKST17] and additive combinatorics. Riesz frames and multilevel expansions in reproducing kernel Hilbert spaces are central tools in numerical multiscale approximation. Wendland, Schaback, and Fasshauer studied kernel translates and obtained quasi-optimal convergence rates in isotropic Sobolev spaces. To address the challenge of efficient evaluation while maintaining quasi-optimal rates for certain Besov spaces, Griebel et al. [GRZ17] showed how to truncate multiscale kernel expansions. A stable Hilbert space splitting based on overlapping frames has recently been proposed in [GO17].

**Sharp estimates and applications in partial differential equations**. Multilinear and multiscale techniques are a recent trend in Kakeya and restriction estimates, combined with decoupling estimates and polynomial partitioning methods. Stationary phase and Strichartz estimates can be used to quantify dispersion of waves. Sharper estimates involve Bourgain’s Fourier restriction spaces Xs;b. The spaces of bounded p variation and their predual were used by Hadac–Herr– Koch to further sharpen these tools. Thereby, scattering became accessible for many dispersive equations. Sharp estimates in suitable tailored function spaces are exploited in the context of thin obstacle problems. Building on the almost sharp regularity results in the seminal work of Uralceva, in a series of papers including [KRS17], Koch et al. have proven higher regularity for the regular part of the free boundary in a series of papers. To this end they combined Carleman estimates, a new partial hodograph transform, and a regularity theory for subelliptic equations. Similar questions for the porous medium equation, thin films, and fast diffusion [DKM15] are understood by now. Multiscale phenomena also arise in elastic shape optimization, caused by a lack of lower semicontinuity of the cost functional. As already analyzed for instance by Tartar and Murat, sequential lamination is a powerful microstructure model with pointwise optimal effective material properties. We used the dual weighted residual approach for optimal control problems to derive a posteriori estimates for a two-scale model in elastic shape optimization. Rigorous a posteriori error estimates usually exploit either a linearization or strict convexity. In PDE-constrained shape optimization, such estimates for the domain error are still out of reach. As a first step, Rumpf et al. [BER17] developed a rigorous a posteriori control for a simple shape optimization problem, the binary Mumford–Shah problem, using Repin’s functional approach together with relaxation and duality techniques. Scalable simulation tools with adaptive mesh refinement for high-order discretizations of time-dependent multiscale features of PDEs were developed by Burstedde et al. [BWG11].

**Multilevel representations and random fields**. There are many new results on Gibbs measures for Hamiltonian dispersive equations. First results for nonlinear wave equations with additive white noise were obtained by Gubinelli–Koch–Oh using renormalization. This connects spaces of bounded p variation with the theory of rough paths also studied in RA B1. Diffusion problems with random diffusivity can be cast as deterministic problems whose coefficients depend on countably many parameters. Improved rates of convergence were established recently by Bachmayr et al. [BCDS17], taking into account the multilevel structure of random coefficient expansions with low regularity. Bachmayr et al. [BCM17] constructed wavelet frame expansions with such structure for a relevant class of Gaussian random fields; see Figure (b).

## Research program

**Riesz frames and multiscale expansions**. We seek to generalize time-frequency analysis to a multiscale analysis of wave packet frames arising through singular value decompositions. The major open problem in this context is to obtain bounds for simplex Hilbert transforms. This is linked to the recent breakthrough by Marcus–Spielman–Srivastava on the Kadison–Singer con- 60 jecture. Multiscale analysis via singular value decomposition is also a key ingredient of numerical low-rank tensor approximation. To describe directional phenomena in harmonic analysis, such as restriction and decoupling inequalities, we plan to use Lptheory of outer measures. Particular problems concern singular integrals with determinantal kernels and hyperbolic cross multipliers, which also underly the investigated numerical error estimates in hyperbolic cross approximation. The construction of dimension-adaptive multilevel frames with quasi-optimal approximation rates will be based on [GRZ17] and [GO17]. Special care is required to enrich and coarsen multiscale expansions in case of highly anisotropic problems. Here, variational and directional Carleson-type theorems as derived by Thiele and coworkers will play an important role in designing appropriate directional error estimators and adaption criteria. Furthermore, we will study the concatenation of multiscale frame expansions, used in the context of non-smooth machine learning problems in RA B2. We aim to generalize the complexity bounds for low-rank solvers applied to vectors reinterpreted as high-order tensors.

**Sharp estimates and applications in partial differential equations**. We plan to combine analytical and numerical tools to compute sharp constants for basic inequalities in harmonic analysis as suggested in [CFOT17]. For example, constant functions are conjectured to extremize the circular restriction inequality. The presence of other local maxima in distant regions of the underlying manifold suggests the use of refined numerical calculations. Using sharp inequalities, we aim for a more global understanding of solutions to nonlinear dispersive equations with strong nonlinear interaction. Specifically, we will address the challenge of global existence of finite-depth water waves for small and decaying initial data in dimension two. These theoretical investigations are accompanied by high-resolution adaptive numerical simulations of the shallow-water equations performed by Burstedde. Moreover, we will study problems with multiple phases and apply methods of singular integrals in non-standard settings. Here, the challenge consists in finding a proper linearization. Of particular interest are thin films with different mobility exponents and problems with triple points of phases. Starting from our work on the binary Mumford–Shah model and on phase field approximation [PRW12], we strive for rigorous a posteriori error estimates in more general problems in elastic shape optimization. To deal with the inherent nonconvexity, we will use the calibration method of Alberti–Bouchitté–Dal Maso and apply primal-dual techniques to derive a strictly convex relaxed functional and its pre-dual. Then Repin’s functional approach will be combined with a suitable error splitting to obtain an estimate for the location of the diffusive interface. Furthermore, we will investigate multiscale branching phenomena arising at material interfaces; see Figure (a). In cooperation with Conti (RA C1) we aim for efficient discretizations of truly two-scale models to better understand the underlying branching-type patterns in elastic structures, such as bones and graphene sheets.

**Multilevel representations and random fields**. We take up the momentum from Hairer’s work on regularity structures and the paracontrolled calculus developed by Gubinelli for the study of dispersive equations with noise. Of particular interest will be the dynamics of dispersive waves and an improved understanding of the structured random patterns on the surface of water. We will investigate multilevel strategies for the numerical approximation of PDEs with stochastic coefficients. Hybrid strategies combining polynomial approximation and random sampling will lead to efficient a posteriori error theory with spatial adaptivity independently tailored for each expansion coefficient. Higher-order convergence rates require new techniques for estimating Besov norms of coefficient functions. Here, multiscale approaches based on overcomplete Fourier and wavelet frames allow appropriate approximations of Gaussian random fields. A particular aim is to generalize these tools to address PDE problems with stochastic nonlinearity. For stochastic or parametric PDEs, the infinite-dimensional hyperbolic cross approach proposed by D˜ung–Griebel will be used to develop tensor-product multiscale spaces, leading to quasi-optimal discretizations. The error analysis will be developed for the L2as well as the L1norm, resulting in efficient stochastic collocation algorithms.

## Summary

This research area will explore interlinked modern tools of harmonic analysis, multiscale approximation, and simulation to push the frontier of sharp regularity results in analysis and optimal numerical discretization. In this endeavor, frames and multiscale expansions, sharp a priori as well as robust a posteriori estimates, and a stochastic PDE perspective each play a decisive role. We share the interest in rough-path tools with RA B1, and the interest in optimal material patterns with RA C1. Multiscale approximation plays a foundational role in data analysis in RA B2.

## Bibliography

[BCDS17] M. Bachmayr, A. Cohen, D. Dinh, and C. Schwab. Fully discrete approximation of parametric and stochastic elliptic PDEs. *SIAM J. Numer. Anal., 55:2151–2186, 2017*.

[BCM17] M. Bachmayr, A. Cohen, and G. Migliorati. Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients. *J. Fourier Anal. Appl., 2017. DOI:10.1007/s00041-017-9539-5*.

[BER17] B. Berkels, A. Effland, and M. Rumpf. A posteriori error control for the binary Mumford-Shah model. Math. Comp., 86(306):1769–1791, 2017. [BWG11] C. Burstedde, L. C. Wilcox, and O. Ghattas. p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. *SIAM J. Sci. Comput., 33(3):1103–1133, 2011*.

[CFOT17] E. Carneiro, D. Foschi, D. Oliveira e Silva, and C. Thiele. A sharp trilinear inequality related to Fourier restriction on the circle. *Rev. Mat. Iberoam., 33(4):1463–1486, 2017*.

[DKM15] J. Denzler, H. Koch, and R. J. McCann. Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach.* Mem. Amer. Math. Soc., 234(1101):vi+81, 2015*.

[DKST17] P. Durcik, V. Kovac, K. Skreb, and C. Thiele. Norm variation of ergodic averages with respect to two commuting transformations.* Ergodic Theory Dynam. Systems, 2017. DOI:10.1017/etds.2017.48*.

[DT10] C. Demeter and C. Thiele. On the two-dimensional bilinear Hilbert transform.* Amer. J. Math., 132(1):201–256, 2010*.

[DT15] Y. Do and C. Thiele. Lp theory for outer measures and two themes of Lennart Carleson united. *Bull. Amer. Math. Soc. (N.S.), 52(2):249–296, 2015*.

[GO17] M. Griebel and P. Oswald. Stable splittings of Hilbert spaces of functions of infinitely many variables. *J. Complexity, 41:126–151, 2017*.

[GRZ17] M. Griebel, C. Rieger, and B. Zwicknagl. Regularized kernel based reconstruction in generalized Besov spaces. *Found. Comput. Math., 2017. DOI:10.1007/s10208-017-9346-z*.

[KRS17] H. Koch, A. Rüland, and W. Shi. The variable coefficient thin obstacle problem: optimal regularity and regularity of the regular free boundary.* Ann. Inst. H. Poincaré Anal. Non Linéaire, 34(4):845–897, 2017*.

[PRW12] P. Penzler, M. Rumpf, and B. Wirth. A phase-field model for compliance shape optimization in nonlinear elasticity.* ESAIM Control Optim. Calc. Var., 18(1):229–258, 2012*.