# Abstracts

**I. High Performance Scientific Computing**

Laura Grigori (INRIA/ EPFL) *Randomization techniques for solving linear systems of equations and eigenvalue problems*In this talk we discuss recent progress in using randomization for solving linear systems of equations and eigenvalue problems. We present a randomized version of the Gram-Schmidt process for orthogonalizing a set of vectors and its usage in the Arnoldi iteration.

This leads to introducing new Krylov subspace methods for solving large scale linear systems of equations and eigenvalue problems. The new methods retain the numerical stability of classic Krylov methods while reducing communication and being more

efficient on modern massively parallel computers.

Gerhard Wellein (Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)) *Application Performance Modeling for Fun and Profit*Scalable and hardware-efficient software is an indispensable prerequisite for large scale simulations. Expensive HPC-systems need to be utilized to the maximum of their capabilities, but deep insight into the bottlenecks of a particular hardware-software combination is often lacking on the users’ side. Analytic, first-principles performance models such as the roofline model can provide such insight. They are built on simplified descriptions of the machine, the software, and how they interact. This goes, to some extent, against the general trend towards automation in computer science; the individual conducting the analysis does require some knowledge of the application and the hardware in order to make performance engineering a scientific process instead of blindly generating data with tools that are poorly understood.

This talk uses introduces the basic idea of analytic, first principles performance modelling and uses examples from parallel high-performance computing to demonstrate how such models may provide insight to hardware-software interaction and may support structured performance engineering.

*Performance Engineering for Sparse Matrix-Vector Multiplication: Some new ideas for old problems*

The sparse matrix-vector multiplication (SpMV) kernel is a key performance component of numerous algorithms in computational science. Despite the kernel’s apparent simplicity, the sparse and potentially irregular data access patterns of SpMV and its intrinsically low computational intensity haven been challenging the development of high-performance implementations over decades. Still these developments are rarely guided by appropriate performance models.

This talk will report on recent advancements to boost the performance of SpMV with symmetric matrices and cache blocking in the computation of matrix power kernels (MPK) for sparse matrices. Reformulating the SpMV as a graph traversal problem as used by RACE [1] allows us to handle dependencies in parallelization and cache blocking in an hardware efficient way. On the compute node-level the RACE implementation of sparse MPK achieves speed-ups of up to 2x-5x compared to state-of-the art implementations [2]. Various numerical schemes like s -step Krylov solvers, polynomial preconditioners and power clustering algorithms may directly benefit from these developments.

[1] Alappat, C. L. et al.: A Recursive Algebraic Coloring Technique for Hardware-efficient Symmetric Sparse Matrix-vector Multiplication. In: ACM TOPC 7 (2020), Article No.: 19. ISSN: 2329-4949. DOI: 10.1145/3399732

[2] Alappat, C.L. et al.: Level-Based Blocking for Sparse Matrices: Sparse Matrix-Power-Vector Multiplication. In: IEEE Transactions on Parallel and Distributed Systems, vol. 34, no. 2, pp. 581-597, 1 Feb. 2023, doi: 10.1109/TPDS.2022.3223512.

**II.** **Data Assimilation**

Sebastian Reich (University of Potsdam)*Data Assimilation: From Optimisation to Bayesian Inference*Abstract: Data assimilation (DA) started out as an optimisation problem with thegoal to fit a computational model optimally to available observations. This originalapproach has now been largely superseded by probabilistic approaches which can be implemented using Monte Carlo techniques. The talk will give an introduction to these techniques using an interacting particle and mean-field approach.

The talk is based on a joint survey paper with Eduardo Calvello and Andrew Stuart (arXiv:2209.11371).

*Data assimilation and control*

Abstract. In the second lecture, I will introduce and discuss the popular ensemble Kalman filter (EnKF) from a frequentist

estimation perspective. This will be followed by an example demonstrating limitations of the EnKF which motivates the

second part of the talk on combining control and data assimilation. The proposed control approach is closely linke to

coupling of measures in the form of Schrödinger bridge problems. A computable approximation using an appropriate adaptation

of the EnKF is introduced and numerically tested. The second part of the talk is based on Data assimilation: A dynamic homotopy-based coupling approach (arXiv:2209.05279).

Susana Gomes (University of Warwick) *Mean field limits for interacting particle systems, their inference, and applications*Part 1:

*Mean field limits and inference.*

In this talk, I will present interacting particle systems which are systems of Stochastic Differential Equations (SDEs) evolving in a confining potential, which can have multiple wells, and in the presence of noise. I will discuss how to obtain the mean-field limit of this system, as well as how to use this to analyse its longtime behaviour. I will then move on to discuss inference for this type of SDEs, in particular parameter estimation for coefficients of interest.

Part 2:

*Inference for more complicated interactions, and some applications*

In the second part, I will apply the concepts introduced before to applications in the life and social sciences, in particular pedestrian dynamics and cell dynamics. I will explain a framework to estimate parameters on SDEs and PDEs for a population using data from individual trajectories (rather than aggregate data such as density), and will discuss issues that arise when using existing data for the above scenarios.

The talks will be mostly based on the following references (and some work in progress). I will give other appropriate references throughout.

[1] S.N. Gomes, G.A. Pavliotis, Mean field limits for interacting diffusions in a two-scale potential, Journal of Nonlinear Science 28(3): 905-941, 2018.

[2] S.N. Gomes, S. Kalliadasis, G.A. Pavliotis, P. Yatsyshin, Dynamics of the Desai-Zwanzig model in multi-well and random energy landscapes, Physical Review E, 99: 032109, 2019.

[3] S.N. Gomes, A.M. Stuart, M.T. Wolfram, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM Journal on Applied Mathematics 79(4):1475-1500, 2019.

[4] S.N. Gomes, G.A. Pavliotis, U. Vaes, Mean-field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods, Multiscale Modeling and Simulation, 18(3), 1343–1370, 2020.

**III.** **Numerical Solution of Stochastic Optimal control**

Jean Francois Chassagneux (Université Paris Cité)

*Probabilistic numerical methods for stochastic control problem: a FBSDE approach.*In this lecture, I will discuss probabilistic numerical methods for stochastic control problems, motivated by problems in mathematical finance or economics. These numerical methods will be obtained mainly through the approximation of Forward Backward Stochastic Differential Equations (FBSDE). A PDE approach to this question would lead to the approximation of semi-linear (as special HJB equation) or quasi-linear parabolic PDEs. Thus, approximations of FBSDEs yield naturally probabilistic numerical methods for these non-linear PDEs appearing in various scientific domains.

The outline of the two talks is as follows.

1. Link between FBSDE and stochastic control problem: FBSDEs allow to characterise the optimal value and/or optimal control of various type of stochastic control. I will focus on two particular problems:

- stochastic target problems, leading to BSDEs and mainly motivated by the super-hedging problem in finance;

- the maximum stochastic principle approach to 'classical' control problem, leading to fully coupled FBSDEs. This last point will be further motivated by the modeling of carbon market. A goal of this lecture is to design an efficient method to approximate the singular FBSDE representing the dynamics of the price of emission permits in carbon markets.

2. Numerical approximation of Forward-Backward SDEs: Focusing mainly on the Markovian framework, I will first review the numerical analysis of the Euler scheme for FBSDEs. I will then discuss a shooting method, which, implemented using deep neural networks (

*_deep BSDE solver_*), appears as an efficient forward method in a high dimensional setting.

3. Probabilistic numerical methods for quasi-linear PDEs: I will first show how the previous methods can be adapted to tackle the approximation of fully coupled FBSDEs. I will then conclude with the approximation of the singular fully coupled FBSDEs arising in carbon market models.

Peter Forsyth (University of Waterloo)

*A Stochastic Control Approach to Defined Contribution PlanDecumulation: “The Nastiest, Hardest Problem in Finance”*

There is an inevitable move away from Defined Benefit (DB) pension plans to Defined Contribution (DC) plans throughout the world. This is simply because governments and corporations no longer want to take on the funding risk associated with a DB pension plan. Under a DC plan, the employer and employee contribute to an account during the working life of the employee. Upon retirement, the retiree now has to determine (i) a withdrawal strategy and (ii) an investment strategy. In order to fund a reasonable lifestyle, it is necessary to invest in risky assets (i.e. stocks).

The retiree is now exposed to two sources of risk (i) market risk and (ii) longevity risk. William Sharpe (Nobel laureate) has termed this problem “The nastiest, hardest problem in finance." There is a standard decumulation strategy, which is commonplace advice given by financial planners. This strategy suggests that a retiree can withdraw 4% of initial capital each year (adjusted for inflation), with an investment portfolio consisting of 50% bonds and 50% stocks, rebalanced annually. Backtests indicate that this 4% rule would have never run out of cash over any historical 30-year period (Bengen (1994)).

Abstract: Part I

Part I will give an overview of the decumulation problem, and the significance of the numerical results. We pose the decumulation strategy for a Defined Contribution (DC) pension plan as a problem in optimal stochastic control. The controls are the withdrawal amounts and the asset allocation strategy. We impose maximum and minimum constraints on the withdrawal amounts, and impose no-shorting no-leverage constraints on the asset allocation strategy. Our objective function measures reward as the expected total withdrawals over the decumulation horizon, and risk is measured by Expected Shortfall (ES) at the end of the decumulation period.

Expected shortfall is the mean of the worst 5% of the outcomes. Based on a parametric model of market stochastic processes, this impulse control problem is solved numerically. We find that, compared to the Bengen 4% rule, the optimal strategy has average withdrawals of about 5% of initial capital (adjusted for inflation), with much less risk. Tests on bootstrapped resampled historical market data indicate that this strategy is robust to parametric model misspecification.

∗David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1, paforsyt@uwaterloo.ca, +1 519 888 4567 ext. 34415.1

Abstract: Part II

Part II will discuss the numerical algorithms used to produce the results discussed in Part I. Our objective here is to determine the Pareto optimal points based on an expected shortfall, expected withdrawals criteria. This is decomposed into an inner and outer maximization problem. The inner maximization is solved classically using dynamic programming, which results in a linear time

advance (backwards) followed by a numerical optimization at each rebalancing time. The linear time advance requires solution of a Partial Integro Differential Equation (PIDE). An -monotone Fourier method is used for the time advancement. In order to resolve the expected shortfall, an outer maximization is carried out, which requires solution of the HJB equation at each inner iteration. The optimal controls are stored, and then statistics of interest determined using Monte Carlo simulation. An alternative approach which does not use dynamic programming is described. We approximate the controls directly using Neural Networks (NNs). This technique is also known as policy function approximation (PFA). Expectations are approximated using stochastic path samples. The optimal parameters of the NNs are determined using stochastic gradient descent. We compare the efficient frontiers determined by the HJB equation solution and the Neural Network solution. Rather surprisingly, the NNs are able to approximate the bang-bang withdrawal controls quite well (bang-bang controls are discontinuous functions). The optimal control heat maps are in good agreement except for the Warren Buffet1 regions of the state space. Going forward, the advantage of an NN approach compared to the HJB equation method is that (i) it is feasible to solve high dimensional problems and (ii) we do not need a parametric model of the underlying processes for stocks and bonds. However, for low dimensional problems, with a known parametric model of the underlying stochastic processes, the NN approach is considerably more computationally expensive compared to the HJB equation technique.

References

Background on the nastiest, hardest problem in finance

1. W. Bengen, “Determining withdrawal rates using historical data”, Journal of Financial Planning,” 7 (1994) 171-180.

2. J. T. Guyton and W. J. Klinger. “Decision rules and maximum initial withdrawal rates”, Journal of Financial Planning 19:3 (2006) 48–58.

3. B.-J. MacDonald, B. Jones, R. J. Morrison, R. L. Brown and M. Hardy, “Research and reality: A literature review on drawing down retirement financial savings”, North American Actuarial Journal, 17 (2013) 181-215.

4.W. D. Pfau “Making sense out of variable spending strategies for retirees”, Journal of Financial Planning 28:10 (2015) 42–51.

5.T. Bernhardt and C. Donnelly, “Pension Decumulation Strategies: A State of the Art Report”, Technical Report, Risk Insight Lab, Heriot Watt University (2018) https://www.actuaries. org.uk/system/files/field/document/STAR%20decumulation.pdf.

1 Warren Buffet is the fifth richest man in the world, with an estimated net worth of USD 104 billion. At age 92, it is inconceivable that Buffet would ever run out of cash. It also does not matter if Buffet invests 100% in stocks or 100 % in bonds. In this case, the decumulation problem is ill-posed. We should all be so lucky.

2 HJB equation approach to the nastiest, hardest problem in finance P.A. Forsyth, G. Labahn, “-Monotone Fourier methods for optimal stochastic control in finance,” Journal of Computational Finance 22:4 (2019) 25-71.

P. A. Forsyth, K. Vetzal, G. Westmacott, “Optimal control of the decumulation of a retirement portfolio with variable spending and dynamic asset allocation,” ASTIN Bulletin 51:3 (2021) 905-938.

P. A. Forsyth, “A stochastic control approach to defined contribution plan decumulation: the nastiest, hardest problem in finance,” North American Actuarial Journal 26:2 (2022) 227-251. Neural Network approximation of an optimal control, without Dynamic Programming

Han, J. and E. Weinan, “Deep learning approximation for stochastic control problems”, NIPS Deep Reinforcement Learning Workshop (2016) .

H. Buehler, L. Gonon, J. Teichmann and B. Wood, “Deep hedging”, Quantitative Finance 19:8 (2019) 1271–1291.

Y. Li and P.A. Forsyth, “A data driven Neural Network approach to optimal asset allocation for target based defined contribution pension plans,” Insurance: Mathematics and Economics 86 (2019) 189-204.

Tsang, K. H. and H. Y. Wong (2020). “Deep-learning solution to portfolio selection with serially dependent returns”, SIAM Journal on Financial Mathematics 11:2 (2020) 593–619.

C. Hure, H. Pham, A. Bachouch and N. Langrene, “Deep neural networks algorithms for stochastic control problems on finite horizon: Convergence analysis”, SIAM Journal on Numerical Analysis 59:1 (2021) 525-557.

C. Ni, Y. Li, P. A. Forsyth, R. Carroll, “Optimal asset allocation for outperforming a stochastic benchmark target,” Quantitative Finance 22:9 (2022) 1595-1626.

P. M. van Stadan, P. A. Forsyth, Y. Li, “Beating a benchmark: dynamic programming may not be the right numerical approach,” SIAM Journal on Financial Mathematics 14:2 (2023) 407-451.

P. M. van Staden, P. A. Forsyth and Y. Li, “A parsimonious neural network approach to solve portfolio optimization problems without using dynamic programming, (2023) arXiv:2303.08968 3